#  serenade: A Model for Human-in-the-loop Automatic Chord Estimation

Hendrik Vincent Koops⋆1, Gianluca Micchi⋆1, Ilaria Manco1,2, Elio Quinton1
1Music and Audio Machine Learning Lab, Universal Music Group
2School of EECS, Queen Mary University of London
London, UK
<EMAIL_ADDRESS>
###### Abstract

Computational harmony analysis is important for mir tasks such as automatic
segmentation, corpus analysis and automatic chord label estimation. However,
recent research into the ambiguous nature of musical harmony, causing limited
inter-rater agreement, has made apparent that there is a glass ceiling for
common metrics such as accuracy. Commonly, these issues are addressed either
in the training data itself by creating majority-rule annotations or during
the training phase by learning soft targets. We propose a novel alternative
approach in which a human and an autoregressive model together co-create a
harmonic annotation for an audio track. After automatically generating harmony
predictions, a human sparsely annotates parts with low model confidence and
the model then adjusts its predictions following human guidance. We evaluate
our model on a dataset of popular music and we show that, with this human-in-
the-loop approach, harmonic analysis performance improves over a model-only
approach. The human contribution is amplified by the second, constrained
prediction of the model.

$\star$$\star$footnotetext: These authors contributed equally. For the purpose
of open access, the author has applied a Creative Commons Attribution (CC BY)
license to any Author Accepted Manuscript version arising.

##  I Introduction

Harmony is a fundamental part of Western music and its analysis has been
studied extensively. Computational harmony analysis is important for a number
of Music Information Retrieval (mir) tasks such as automatic segmentation [1],
corpus analysis [2, 3] and automatic chord estimation (ace) [4, 5, 6, 7, 8].
ace is a well-researched topic in mir that studies how to extract a time-
aligned sequence of chords from a given music signal. Commonly, recent ace
systems consist of some type of audio feature extraction and a deep learning
method to classify segmented audio signal into chord classes.

A well-known problem in creating reference annotations for computational
harmony analysis is disagreement between annotators, which makes it hard to
define a general “ground truth”. Differences in annotations can be the result
of personal biases (e.g., an annotator’s preference for annotating chords in
their root position) or of external constraints (e.g., transcribing a piece
for different instrumentation than originally intended), or they can stem from
ambiguity in the music itself [9, 5, 10, 11]. These difficulties in creating a
“ground truth” complicate the evaluation of model output and have slowed down
the progress in computational harmony analysis in the past few years.

In previous research, two main approaches have been proposed to deal with
these challenges. The first approach is to resolve conflict by merging
different perspectives. The effect of training ace algorithms on such
annotations is that their evaluation is bound by a glass-ceiling effect, above
which a model is effectively over-fitted to one particular harmonic point of
view [10, 5, 12, 9].

A second approach to deal with ambiguity and subjectivity is therefore to take
it into account at the model level. For example, Koops _et al._ [12] proposed
a method to personalize chord labels for individual annotators by learning a
shared (soft) representation of harmony. This results in a model that performs
beyond a one-size-fits all solution, by providing chord labels that are
specific for a user or task.

When combining the observations that  a) current ace models are performing at
the level of inter-annotator agreement [13, 5] and  b) further ace performance
improvement depends on the particular user or use case [12],  we argue that
one possible third avenue of ace improvement is a human-in-the-loop approach.

It has already been shown that a human-in-the-loop approach can make mir
systems more useful in real-world scenarios, because the perception of a large
number of musical dimensions are to some extent subjective or personal[13].
For example, Yamomoto [14] proposes a human-in-the-loop user interface for
beat tracking that interactively adapts to a specific user and target music.
It is argued in [14] that the interpretation of a beat differs for each
individual, which means that an ideal beat tracking system needs to produce
different outputs from a single input depending on the context. By iterating
interaction between the user and model, the system adapts the internal neural
network model to the user, allowing it to produce more desirable results,
while only requiring the user to change a small portion of the errors that are
noticed. A human-in-the-loop approach can also help with speeding up labeling
long audio files, as shown by Kim and Pardo in [15]. For more human-in-the-
loop approaches in mir we refer to the discussion presented by Yamamoto in
[14].

Contribution. The contribution of this paper is threefold. First, we introduce
a novel human-in-the-loop approach to automatic chord estimation. Our approach
is similar to in-painting, in which the model is responsible for filling in
the missing chord labels of sparse human annotations. More specifically, we
simulate a human in the loop by experimenting with oracles that provide
different kinds of harmony information. Second, we evaluate our model on a
dataset of popular music and we show that with our human-in-the-loop approach
we can improve over a model-only approach, and can produce an output with a
Return On Investment larger than 1 (cf. the end of Sec. II-C for a definition
of roi). Third, we extend the model introduced in Micchi _et al._ [16] and
apply it to the audio domain.

##  II Method

###  II-A An Introduction to nade

The _Neural Autoregressive Distribution Estimator_ (nade) has been proposed to
model the distribution of random variables that are not independent of each
other[17, 18]. Its core idea is to create a hidden layer that gets updated
every time a new prediction is made, so that subsequent predictions are
informed of the previous outcomes. In simple terms, the nade begins in a state
determined uniquely by the biases. At every step, the logit for the single
variable $x[d]$ is predicted based on the value of the hidden state, then the
variable is sampled, and finally the hidden state is updated based on the
sampled output.

The main equations governing the original formulation of nade are the
following:

$\displaystyle p(x[d])$
$\displaystyle=\sigma\big{(}M_{HV}\sigma(h[d])+b_{v}[d])\big{)}$
$\displaystyle h[d]$ $\displaystyle=M_{VH}[:,d-1]x[d-1]+h[d-1]$ $\displaystyle
h[0]$ $\displaystyle=b_{h}$

where we use _NumPy_ ’s notations for arrays[19], $x$ is the array of sampled
binary outputs from the visible layer (of size $D$), $p(x[d])$ is the
probability of sampling 1 at position $d$ for array $x$, $h$ is the hidden
layer (of size $H$), $b_{v}$ is the bias of the visible layer, $b_{h}$ is the
bias of the hidden layer, and $M_{HV}$ and $M_{VH}$ are the matrices of
weights linking, respectively, hidden layer to visible layer and vice versa
(of size $D\times H$ and $H\times D$ respectively). Teacher forcing can be
used to speed up the training: real target outputs, instead of the model
output, are used as inputs to the next step of the prediction[20].

nade has been sparsely used in mir so far, for example for the task of music
generation[21, 22] and for the task of harmonic analysis[16]. In the latter
case, the classification of harmony is done through the analysis of fixed-
length frames of music through a convolutional recurrent neural network
(crnn), followed by a nade that replaces the more standard shallow
classifiers. The connection between the crnn and the nade is done, following
Boulanger _et al._ [21], by using the output of the recurrent network to
determine the biases of the nade. At each timestep, the label is divided into
6 multiclass sub-labels: key, tonicisation, _droot_ (distance between key root
and chord root), quality, inversion, and root. The basic architecture of nade
is therefore extended to predict a multiclass variable instead of a single
binary output. For example, and as a consequence of the architecture, the
choice of the quality of the chord at timestep $t$ was informed by the sampled
key, tonicisation, and _droot_ , therefore avoiding incoherent outputs.

Figure 1:  A schematic look at the model architecture. The targets/oracle
connection can be absent when the model is performing a standard inference,
hence the dashed lines.

###  II-B serenade: A Separable nade for Harmony Analysis

We modify and extend the nade architecture presented in Micchi _et al._ [16]
and introduce serenade. The main novelties are:

Audio. We apply the nade to audio content instead of symbolic content. Our
audio files are segmented into 60-second excerpts. Chromagrams for both the
bass and global frequency spectrum of the music are extracted using _NNLS
Chroma_ , following Burgoyne _et al._ [2]. The chromagrams are then used as
inputs to the crnn part of the network, implemented with a 1D DenseNet
architecture[23], acting as a deep feature extractor.

Depth. We extend the nade mechanism to the temporal dimension. To achieve
this, we replace the rnn with an additional nade hidden layer, responsible for
modelling the harmony autoregressively along the time dimension. The output of
the deep feature extractor determines the biases of all three nade layers (two
hidden and one visible); the first hidden layer (or _time_ hidden layer) is
used to determine the value of the second hidden layer (or _feature_ hidden
layer) and gets updated once per time step; the feature hidden layer is used
exactly as before and is updated at every sampling of a sub-label. We refer to
this approach as separable, since time and feature dimensions are modelled
separately (see Fig. 2). An alternative to this would be to flatten both
dimensions and model them with a single hidden layer. Through preliminary
experiments we verify that the former performs better and therefore only
consider the separable variant in the rest of the paper. The model is made
bidirectional by defining two parallel nades, one left-to-right and one right-
to-left, the logits of which are then averaged before sampling at inference
time.

Directed Root (_droot_) Label. We simplify the task of determining the _droot_
label by adding a constant value of $3$ to the logits of the calculated
_droot_ in order to direct the model towards the right choice (see Section
III-A for further details). The value of $3$ was chosen empirically by
noticing that typical logits for a confident prediction were ranging between
$-5$ and $+5$: $3$ thus represents a strong nudge in the right direction of
the same order of magnitude of all other effects in the model.

The proposed architecture is shown in Figures 1 and 2. The serenade has 175k
weights and can be trained on a single GPU in about one hour.

Figure 2:  Our proposed serenade model in the “infer” configuration. When
training, logits are not sampled and instead the targets are used to take
advantage of the teacher forcing paradigm. When using an oracle, instead,
either ground truth or samples are used based on the value of an oracle mask:
if the mask is active, take the ground truth; if inactive, sample from the
logits. In all three configurations, the biases change at every time step. The
final version of the model is bidirectional, which means that two identical
nade are used, one going from the past to the future and the other vice versa.
Their logits are then averaged before sampling.

###  II-C Human-in-the-loop Simulation

We simulate a human in the loop as an oracle that provides a sparse annotation
for certain sub-labels at given time steps (for brevity, we sometimes extend
the name oracle to the sparse annotation itself). The oracle can make
independent choices of time steps for different sub-labels. For example, one
can provide the chord root at frames 48 and 102 and the quality at frame 55.
In all cases, whenever the oracle is active, it is used both to overwrite the
output sampled by the model and also to update the hidden state of serenade by
means of teacher forcing. To help us in the evaluation of the oracle
experiments we define two new quantities.

Oracle cost. The _oracle cost_ is a measure of how many annotations we have
provided an oracle for. It is expressed as a percentage with respect to the
total number of annotations in the audio extract under analysis. For example,
providing the oracle for the key root at all time steps has a cost of 16.67%
because the key root is one of 6 sub-labels. If we were to provide it only
half the times the cost would drop to 8.33%.

Oracle roi. The second quantity is the _oracle roi_, or Return On Investment.
The roi is defined as the ratio between the change in accuracy due to the
oracle and the oracle cost:

$\textsc{roi}=\frac{\text{accuracy}_{\text{oracle}}-\text{accuracy}_{\text{original}}}{\text{oracle
cost}}$

In a model without autoregression, an oracle always has an roi between 0 and
1. It is 0 if all the labels provided were already correctly predicted by the
model, 1 in the opposite (ideal) case where the oracle was used only on wrong
predictions, and in between these extremes for all other situations. This is
no longer true for serenade. Due to its autoregressive nature, a correction at
one specific sub-label can be propagated to different sub-labels at the same
time step and also to other time steps. A possibility for roi larger than 1 is
therefore unlocked. This means that the model can make human corrections more
effective and, for example, considerably speed up the annotation of new
datasets. It is maybe not trivial to notice that, in theory, also negative
rois can happen, when the oracle is actually detrimental to the global quality
of the output. However, these cases are extremely unlikely and never occurred.

Figure 3:  Our simulated human-in-the-loop approach. We analyse chord quality and key root at the same time to show the effects across sub-labels, even at different time steps. We see that the confidence of the model in the key root is below the threshold 0.35. Providing an oracle when going below that threshold has a positive impact on the quality of the chord, which improves total accuracy. This is an example of roi $>$ 1\. See Section IV-B for more details.  TABLE I: Harmonic class labels used for training our models. The _droot_ values represent the semitone index relative to the key (for example the chord C:maj in the key of C major will have a _droot_ of 0, and G:maj in the same key will be 7.) The values -1 and N represent a musical passage without harmonic annotations (silence, for example). Pitches not mentioned in this table are mapped to their enharmonic equivalents. Type | Possible Values
---|---
Key root | {C, C#, D, D#, E, F, F#, G, G#, A, A#, B, N}
Key quality | {major, minor, N}
Chord root | {C, C#, D, D#, E, F, F#, G, G#, A, A#, B, N}
_Droot_ | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, -1}
Chord quality | {maj, min, dim, aug, 7, maj7, min7, 5, 1, sus4, N}
Bass number | {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, -1}

##  III Dataset preparation

We use two datasets as ground-truth annotations (with all the caveats
discussed in the introduction) for our experiments: the _Isophonics_
annotations for _The Beatles_ and a subset of the _Billboard_ dataset. Both
datasets contain time-aligned key and chord label reference annotations. In
the following sections, we briefly detail the contents of these datasets,
followed by a description of our data pre-processing approach. We follow the
commonly used chord label notation convention introduced by Harte et al. in
Harte _et al._ [24].

Billboard. This dataset was introduced by Burgoyne et al. in [2] and contains
chord label annotations for songs sampled from the _Billboard ‘Hot 100’_ music
charts. The annotations are the result of the consensus of expert annotations,
and are a staple dataset for computational harmony analysis and automatic
chord estimation of popular music. This dataset contains reference annotations
for 890 songs. In this paper we use a subset of 267 songs that are accessible
to us without copyright restrictions.

Isophonics – The Beatles. This dataset comprises 178 tracks from The Beatles
catalog with annotations relating to chords, keys, structural segmentations,
and beats/bars. These annotations were collected by the Centre for Digital
Music (C4DM) of Queen Mary University of London[25]. For the experiments in
this paper, we leverage the chords and key annotations.

###  III-A Data Pre-Processing

By combining the _Billboard_ and _Beatles_ datasets we obtain a training
dataset with 445 songs. We resample the audio at 22,050 Hz and create two
files for each song: chromagram features and harmony labels.

Audio Features: Chroma. We extract _NNLS Chroma_ features, introduced by Mauch
et al. in [26]. Chroma features capture the pitch-class content of harmony in
terms of the 12 pitch classes folded into a single octave. They are commonly
used for automatic chord estimation since they provide a good proxy for
capturing harmonic information [27]. In addition to the standard set of 12
chroma features analysing the entire spectrum of frequencies, we also produce
another set of 12 containing only the frequencies from the lower octaves. The
distance in milliseconds between two successive frames in the chromagram is
roughly 46 ms. The specific implementation closely follows the _Billboard_
dataset [28, Appx. B] for compatibility reasons.

Audio Features: Pitch Profiles. Chroma features are extremely localised in
time: each frame only describes a few tens of ms of audio. Some harmonic
concepts, such as the key, are usually defined over much larger time scales.
To better bridge this gap, we augment the data with global pitch profiles
computed from the average of each of the 12 main chroma features over the
entire excerpt of audio used for analysis (60 seconds). We then take the inner
product of these additional 12 values with each of the 24 pitch profiles
proposed by Temperley for key detection[29]. The result is then concatenated
to the extracted chroma features at each time step. By including it, we can
simultaneously provide the model with local harmony (chroma) and global
harmony (pitch profiles) information. Previous research in ace has shown that
providing other musical parameters can improve chord estimation results [30].

Labels: Harmony Annotations. For each track, we create a new reference
annotation that is aligned with the extracted chromagrams. In these
annotations, we represent the harmony with six sub-labels: key root (e.g. C),
key quality (e.g. major), chord root (e.g. A), _droot_ (or _directed distance
to root_ , which is the name that we give to the distance in semitones between
the chord root and key root), chord quality (e.g. minor), and bass number (the
distance in semitones between the bass note and the root of the chord). Table
I provides an overview of all harmonic class labels used in our experiments.

Labels: _Droot_. The _droot_ is uniquely determined by the chord root and key
root. Hence, it is redundant and could be removed from the set of predictions.
However, _droot_ and quality are related: for example, the base triad of a
chord written on the dominant is (almost) always major, while one built on the
subdominant is usually major in major keys and minor in minor keys. Therefore
jointly predicting both is likely to provide an overall net benefit to the
model performance. A small positive effect has been _The Beatles_ by our
experiments in Section IV-B.

Labels: Key Quality. The _Billboard_ dataset does not include information on
the key quality. To estimate it, we look at the predominant quality of tonic
chords: if they are mostly minor we assume it’s a minor key; vice versa, we
assume it’s major. This excludes other modes such as dorian, aeolian,
mixolydian, but we believe that the additional information provided by this
simple inferral scheme outvalues the noise introduced in the dataset. For the
_Beatles_ tracks, we use the ground-truth annotations provided by the dataset,
but exclude tracks were the key is modal (i.e. other than major or minor).

Labels: Chord Quality. To prevent an explosion of possible chord label
combinations and reduce the effect of infrequently occurring chord types, we
reduce our chord qualities to one of 11 classes: maj (60.8% of annotations),
min (15.1%), dim (0.6%), aug (0.3%), 7 (6.7%), maj7 (2.2%), min7 (6.2%), 5
(1.8%), 1 (1.4%), sus4 (2.8%), and N (1.9%). To do so, chord qualities maj6,
9, 11, 13, maj9, maj13, and sus2, totalling 3.1% of the annotations, are
mapped to maj (57.7%). Qualities min6, minmaj7, min9, min11, and min13,
totalling 0.9% of the annotations, are mapped to min (14.2%). Finally, we
combine dim, dim7, and hdim7.

The chord quality mapping was made with music theory concerns in mind, trying
to aggregate chords that are similar to each other either in terms of pitch-
class content or of compositional usage. We recognize that no chord reduction
is optimal in all settings. One could even argue that all chord reductions are
wrong to some degree. However, mapping chord labels to a smaller number of
classes is a commonly used approach in automatic chord estimation to remedy
the combinatorial explotion of chord labels, and to facilitate evaluation and
model training [31].

In the following sections, we discuss our experiments and results. First we
evaluate our model in a way that is common in ace research in Section IV-A. In
these experiments, we use common ace metrics and compare our results with the
current state of the art. Then, in Section IV-B, we focus on the experiments
we have run to test our human-in-the-loop approach.

##  IV Results

###  IV-A Baseline Results on Automatic Chord Estimation

Fig. 4 shows the chord root prediction for 60 seconds of _“Let It Be”_ , by
The Beatles. Overall the predictions are accurate, except that most of them
arrive one or two frames later than the ground truth. This could either be due
to a limitation of our model or to a small alignment issue in the ground
truth.

Figure 4: The chord root prediction of serenade (heatmap, in greyscale)
follows the ground truth (horizontal red lines) quite closely.

The performance of our model compared to top performing entries in the mirex
evaluation[32] is shown in Table II. All metrics are computed using the python
package mir_eval[31]. As a caveat, the mirex evaluation dataset is not
publicly available so we are evaluating on a different subset of 61 songs
coming from the _Billboard_ v2.0. Our proposed model is the best performing in
terms of root detection. The majmin score is competitive and, adding a key
root oracle (see Section IV-B) it outperforms the baselines. The sevenths
score, however, shows room for improvement. We hypothesise the difference with
JLCX2 be due to the training dataset size or choice encoding for qualities,
which treats every common extension to the triads as a separate binary unit.
Such a scheme seems to be a good candidate for being applied also to our
serenade model and we are considering it for further experiments.

TABLE II:  Comparison between our model and others that were evaluated as part of the mirex competition. [wko] = with key oracle, as described in Experiment 1. *: the results for our model serenade are obtained on a different subset of the _Billboard_ dataset as the one used for mirex competitions is not publicly available. serenade has been trained on a smaller dataset than others, due to copyright reasons.  Model | root | majmin | sevenths
---|---|---|---
CLSYJ1 | 73.69 | 69.47 | 57.41
NMSD2 | 74.72 | 71.40 | 59.30
WL1 | 75.22 | 72.53 | 57.87
JLW2 | 75.40 | 72.77 | 57.49
CB4 | 76.29 | 72.01 | 59.15
KO2 | 76.52 | 73.82 | 60.21
SG1 | 77.92 | 72.41 | 58.26
JLCX2 | 79.38 | 77.92 | 64.22
KBK2 | 80.60 | 78.27 | 55.81
serenade* | 80.89 | 77.57 | 57.82
serenade* [wko] | 80.78 | 79.20 | 59.13

###  IV-B Testing the Human-in-the-Loop Approach Using Oracles

To test the capabilities of the model to adapt its predictions to human input
we develop a few experiments involving oracles. We run all the experiments on
a randomly selected test set made of 30 recording from the _Beatles_ dataset
and 61 from the _Billboard_ dataset which was set apart before training our
models. When comparing results to mirex, only the _Billboard_ data is used.

####  IV-B1 Oracles on Key Root and Key Quality

Key root and key quality are usually relatively inexpensive oracles to provide
in terms of human resources as they tend to be constant for long stretches of
time, with a majority of pop songs having only one key for the entire piece.

When providing these two oracles, we see that there is a global improvement on
the standard mir_eval metrics of majmin and sevenths, as shown in Table II.
Such a result is qualitatively impossible with standard shallow classifiers
instead of nade due to the lack of direct transfer of information between the
sub-labels. This result was already reported in [16].

####  IV-B2 Oracle on _Droot_

Our hypothesis is that the _droot_ can help in determining the quality of the
chord and we test this by adding the _droot_ oracle. We observe a small
performance improvement: the quality prediction accuracy over the test dataset
is 66.9% in the absence of any oracles and 67.2% with the _droot_ oracle.

A detailed oracle on the _droot_ is very expensive to make and amounts to
basically producing a full harmonic annotation by hand, making this avenue
impractical to follow for speeding up chord annotations. However, it validates
our choice of using expert knowledge to define the _droot_ in order to improve
the ace quality.

####  IV-B3 Oracles Where the Model is Wrong

Given the knowledge of the ground truth at all times, one can devise an oracle
that is active only on a subset of the wrong predictions. For example, when
providing the correct label for 10% of the wrong predictions on root, quality,
and bass number, we obtain an average roi of 1.18 over all tracks in the
dataset, while the roi for each individual audio excerpt ranges from 1 to 2.5.

This roi is not particularly large for practical purposes but the mere fact
that it is larger than 1 is a proof of concept that a model optimised to do so
can be devised and used for faster data labeling. Also, it is important to
notice that this scheme is not unique to the task of harmonic analysis and
could be generalised to other tasks, within and outside mir.

####  IV-B4 Oracles on Predictions with Low Model Confidence

In some real-world scenarios we may not have a priori knowledge of which
predictions are wrong. For the oracle to be practically useful in such cases,
we require a scheme to automatically determine when the annotator’s
intervention is most likely to be required.

We propose to simulate this scenario by providing an oracle whenever the model
outputs a prediction with a confidence smaller than a threshold $t$. We then
study the distribution over the test dataset of the rois we obtain as a
function of $t$ and plot the results as a heatmap in Fig. 5. We can see that
there is a roi $>$ 1 for thresholds roughly between 0.2 and 0.35. Below 0.2,
the oracle is not used because the model never outputs such low confidence
values. Above 0.35, the ROI $<$ 1 reveals that the oracle just confirms a
prediction that is already correct.

Figure 5:  The distribution of the rois for the oracle. The red line
represents the average of the distribution as a function of the confidence
threshold. The black shadow is the Kernel Density Estimate (KDE) of the
distribution, calculated with the python package Seaborn.

The assumption that the outputs with low probability are the ones that are
most likely to be wrong is verified: a threshold up to 0.35 always provides
roi larger than 1.

##  V Discussion and Conclusions

In this work, we have introduced serenade, a first approach to human-in-the-
loop automatic chord estimation. We showed that by leveraging the
autoregressive nature of our model, we can in-paint partial annotations from
an oracle to obtain higher-quality predictions overall. We defined two new
quantities to evaluate the results: an oracle cost and an oracle roi. Our
experiments show an average roi of 1.18, meaning that for every single oracle
annotation provided, 1.18 labels are corrected. This positive feedback, if
harnessed correctly, could eventually lead to a significant speedup in the
generation of new annotated datasets. For example, in an annotation task a
user would only have to provide a corrected key for a piece to improve ace
results overall.

We identified two main avenues for future work. Firstly, assessing the long-
term coherence of the model. For example, if an oracle provides a correct
annotation for the first appearance of the chorus, will the model be able to
apply that correction to all choruses? Recent work in generative models showed
promising long-term coherence modelling improvenent by employing hierarchical
architectures and representations [33, 34], which may also be beneficial to
serenade.

Secondly, improving the human-in-the-loop formalism. In its current form, the
oracle cost may not be a fair representation of the actual effort provided by
a human annotator. In particular, it treats frames independently and all
harmonic elements (such as key root, key quality, chord root, etc.) equally.
As a result the cost of annotating 10 frames is 10 times greater than
annotating one frame, whereas it would likely be ”cheap” for a human to
annotate segments of many contiguous frames. We expect that with a refined
oracle cost metric the benefit of such a human-in-the-loop approach may become
even more apparent.

##  VI Acknowledgements

This work was supported by UK Research and Innovation [grant number
EP/S022694/1] and Universal Music Group.

## References

  * [1] J. Paulus, M. Müller, and A. Klapuri, “Audio-based music structure analysis,” _Proceedings of the 11th International Society for Music Information Retrieval Conference, ISMIR 2010_ , pp. 625–636, 2010.
  * [2] J. A. Burgoyne, J. Wild, and I. Fujinaga, “An expert ground truth set for audio chord recognition and music analysis.” in _ISMIR_ , vol. 11, 2011, pp. 633–638.
  * [3] T. De Clercq and D. Temperley, “A corpus analysis of rock harmony,” _Popular Music_ , vol. 30, no. 1, pp. 47–70, 2011.
  * [4] M. McVicar, R. Santos-Rodríguez, Y. Ni, and T. De Bie, “Automatic chord estimation from audio: A review of the state of the art,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 22, no. 2, pp. 556–575, 2014.
  * [5] E. J. Humphrey and J. P. Bello, “Four timely insights on automatic chord estimation.” in _ISMIR_ , vol. 10, 2015, pp. 673–679.
  * [6] B. McFee and J. P. Bello, “Structured training for large-vocabulary chord recognition.” in _ISMIR_ , 2017, pp. 188–194.
  * [7] T.-P. Chen and L. Su, “Harmony transformer: Incorporating chord segmentation into harmony recognition,” in _ISMIR_ , 2019.
  * [8] N. N. López, M. Gotham, and I. Fujinaga, “Augmentednet: A roman numeral analysis network with synthetic training examples and additional tonal tasks.” in _ISMIR_ , 2021, pp. 404–411.
  * [9] H. V. Koops, W. B. de Haas, J. A. Burgoyne, J. Bransen, A. Kent-Muller, and A. Volk, “Annotator subjectivity in harmony annotations of popular music,” _Journal of New Music Research_ , vol. 48, no. 3, pp. 232–252, 2019.
  * [10] Y. Ni, M. McVicar, R. Santos-Rodriguez, and T. De Bie, “Understanding effects of subjectivity in measuring chord estimation accuracy,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 21, no. 12, pp. 2607–2615, 2013.
  * [11] G. Micchi, M. Gotham, and M. Giraud, “Not all roads lead to rome: Pitch representation and model architecture for automatic harmonic analysis,” _Transactions of the International Society for Music Information Retrieval (TISMIR)_ , vol. 3, no. 1, pp. 42–54, 2020.
  * [12] H. Koops, W. de Haas, J. Bransen, A. Volk _et al._ , “Chord label personalization through deep learning of integrated harmonic interval-based representations,” in _Proc. of the Int. Workshop on Deep Learning and Music. Anchorage, US._ , 2017.
  * [13] A. Flexer, “On inter-rater agreement in audio music similarity.” in _ISMIR_ , 2014, pp. 245–250.
  * [14] K. Yamamoto, “Human-in-the-loop adaptation for interactive musical beat tracking.” in _ISMIR_ , 2021, pp. 794–801.
  * [15] B. Kim and B. Pardo, “A human-in-the-loop system for sound event detection and annotation,” _ACM Transactions on Interactive Intelligent Systems (TiiS)_ , vol. 8, no. 2, pp. 1–23, 2018.
  * [16] G. Micchi, K. Kosta, G. Medeot, and P. Chanquion, “A deep learning method for enforcing coherence in automatic chord recognition.” in _ISMIR_ , 2021, pp. 443–451.
  * [17] H. Larochelle and I. Murray, “The neural autoregressive distribution estimator,” in _Proceedings of the fourteenth international conference on artificial intelligence and statistics_. JMLR Workshop and Conference Proceedings, 2011, pp. 29–37.
  * [18] B. Uria, M.-A. Côté, K. Gregor, I. Murray, and H. Larochelle, “Neural autoregressive distribution estimation,” _The Journal of Machine Learning Research_ , vol. 17, no. 1, pp. 7184–7220, 2016.
  * [19] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, “Array programming with NumPy,” _Nature_ , vol. 585, no. 7825, pp. 357–362, 2020.
  * [20] R. J. Williams and D. Zipser, “A learning algorithm for continually running fully recurrent neural networks,” _Neural computation_ , vol. 1, no. 2, pp. 270–280, 1989.
  * [21] N. Boulanger-Lewandowski, Y. Bengio, and P. Vincent, “Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription,” _arXiv preprint arXiv:1206.6392_ , 2012.
  * [22] C.-Z. A. Huang, T. Cooijmans, A. Roberts, A. Courville, and D. Eck, “Counterpoint by convolution,” _arXiv preprint arXiv:1903.07227_ , 2019\.
  * [23] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 4700–4708.
  * [24] C. Harte, M. B. Sandler, S. A. Abdallah, and E. Gómez, “Symbolic representation of musical chords: A proposed syntax for text annotations.” in _ISMIR_ , vol. 5, 2005, pp. 66–71.
  * [25] M. Mauch, C. Cannam, M. Davies, S. Dixon, C. Harte, S. Kolozali, D. Tidhar, and M. Sandler, “Omras2 metadata project 2009,” in _Proc. of 10th International Conference on Music Information Retrieval_ , 2009.
  * [26] M. Mauch and S. Dixon, “Approximate note transcription for the improved identification of difficult chords.” in _ISMIR_ , 2010, pp. 135–140.
  * [27] E. Gómez, “Tonal description of music audio signals,” _Department of Information and Communication Technologies_ , 2006.
  * [28] J. A. Burgoyne, _Stochastic processes & database-driven musicology_. McGill University (Canada), 2012.
  * [29] D. Temperley, “What’s key for key? the krumhansl-schmuckler key-finding algorithm reconsidered,” _Music Perception_ , vol. 17, no. 1, pp. 65–100, 1999.
  * [30] M. Mauch, K. C. Noland, and S. Dixon, “Using musical structure to enhance automatic chord transcription.” in _ISMIR_ , 2009, pp. 231–236.
  * [31] C. Raffel, B. McFee, E. J. Humphrey, J. Salamon, O. Nieto, D. Liang, D. P. Ellis, and C. C. Raffel, “Mir_eval: A transparent implementation of common mir metrics.” in _ISMIR_ , 2014, pp. 367–372.
  * [32] J. S. Downie. (2023) Mirex home.
  * [33] I. Manco, E. Benetos, E. Quinton, and G. Fazekas, “Muscaps: Generating captions for music audio,” in _2021 International Joint Conference on Neural Networks (IJCNN)_ , 2021, pp. 1–8.
  * [34] A. Agostinelli, T. I. Denk, Z. Borsos, J. Engel, M. Verzetti, A. Caillon, Q. Huang, A. Jansen, A. Roberts, M. Tagliasacchi _et al._ , “Musiclm: Generating music from text,” _arXiv preprint arXiv:2301.11325_ , 2023.


11institutetext: INAF-Osservatorio Astronomico di Cagliari, via della Scienza
5, 09047 Selargius
11email<EMAIL_ADDRESS>22institutetext: National Radio Astronomy
Observatory, 1003 Lopezville Rd, Socorro, NM 87801 33institutetext: CSIRO,
Space and Astronomy, PO Box 1130, Bentley WA 6102, Australia 44institutetext:
CSIRO, Space and Astronomy, PO Box 76, Epping, NSW 1710, Australia
55institutetext: Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands 66institutetext: Ruhr
University Bochum, Faculty of Physics and Astronomy, Astronomical Institute,
44780 Bochum, Germany

# A depolarizing H i tidal tail in the western lobe of Fornax A

F. Loi 11 P. Serra 11 M. Murgia 11 F. Govoni 11 C. Anderson 22 G. Heald 33 D.
Kleiner 11 E. Lenc 44 V. Vacca 11 F. M. Maccagni 1155 R. J. Dettmar 66

(Received December 10, 2021; accepted ****, ****)

Recent MeerKAT neutral hydrogen (H i) observations of Fornax A reveal tidal
material intersecting in projection the western lobe of this radio galaxy. We
found a spatial coincidence between the northern H i tail and a depolarized
structure observed for the first time with the Australian Square Kilometre
Array Pathfinder (ASKAP) at 1.2 GHz. We analyzed the properties of the
rotation measure (RM) image obtained with ASKAP data at the location of the H
i tail and in its neighborhood. We modeled the observed RM structure function
to investigate the magnetic field power spectrum at the location of the H i
tail and in a nearby control region. We found that the observed RM, in the
control region and in a region enclosing the H i tail, cannot be due to the
intracluster Faraday screen caused by the Fornax cluster. An intragroup
magnetized medium with a central magnetic field strength of 18.5 $\rm\muup$G
can explain the control region RM, but it is clear that there is an excess in
correspondence with the H i tail region. We evaluated several scenarios in
which the H i tail is either in the lobe foreground or embedded in the lobe.
We determined a magnetic field strength on the order of $\sim$9.5$-$11
$\muup$G in the H i tail, a value consistent with constraints derived from
narrowband H$\alpha$ imaging of the ionized gas. The spatial coincidence
between H i tail and depolarization analyzed in this paper could be the first
observed evidence of a magnetic field that either has passed through a radio
galaxy lobe or has survived the lobe expansion.

###### Key Words.:

Galaxies: individual: Fornax A – galaxies: magnetic fields – galaxies: ISM –
galaxies: intergalactic medium – polarization – radio lines: galaxies

##  1 Introduction

In the last decade broadband spectropolarimetric observations have shown a
pronounced network of patches and/or filamentary structures associated with
the lobes of some radio galaxies (Brienza et al., 2021; Fanaroff et al., 2021;
Maccagni et al., 2020; Ramatsoku et al., 2020; Anderson et al., 2018a, b),
both in total intensity and polarization. Detailed studies of these systems
also reveal the presence of “low-p” patches, which are regions with low total
intensity and polarization emission, depolarization structures, and rotation
measure (RM) enhancements (Anderson et al., 2018b; Guidetti et al., 2011,
2012). Even though the depolarization observed for radio galaxies has usually
been ascribed to foreground Faraday screens (Dreher et al., 1987; Laing et
al., 2008; Guidetti et al., 2010), such as the intracluster medium of
clusters, it is clear now that the interaction between the lobes and the
environment can play a key role. Shock waves (Carilli et al., 1988; Hardcastle
et al., 2012; Banfield et al., 2019), hydrodynamical or magneto-hydrodynamical
instabilities, and shocked interstellar medium (Hardcastle, 2003) can cause
the observed depolarized structures and RM enhancements.
Fornax A is a spectacular example of a radio galaxy showing lobes with a
filamentary structure and depolarized patches. Thanks to its proximity ($\rm
D_{L}$ = 20.8 $\pm$ 0.5 Mpc; Cantiello et al., 2013), these features were
already evident based on the VLA data published by Fomalont et al. (1989). The
radio galaxy is the brightest member of a galaxy group located southwest of
the center of the Fornax galaxy cluster, at a projected separation of $\sim$
1.3 Mpc, and it is likely falling towards the cluster central regions
(Drinkwater et al., 2001). The radio lobes of Fornax A were generated during
several intermittent episodes of active galactic nucleus (AGN) activity, as
recently demonstrated by Maccagni et al. (2020), and extend over a linear size
of $\sim$290 kpc. The two lobes do not show hot spots and are not connected to
the AGN with visible jets (S-shaped radio jets are observed within 6 kpc from
the host galaxy center; Geldzahler & Fomalont, 1978, 1984; Maccagni et al.,
2020). Therefore, the source cannot be classified as a Fanaroff-Riley type I
or type II radio galaxy.
Anderson et al. (2018b) study in detail the polarization properties of the two
lobes using the Australia Telescope Compact Array (ATCA) between 1.28 and 3.1
GHz with a resolution of 20″$\times$30″. These images show several low-p
patches across the eastern lobe from the northeast to southwest, and several
regions in the western lobe as well. Here in particular, two low-p patches
correspond to regions where the depolarization is due to known foreground
objects: NGC 1310 (Fomalont et al., 1989; Schulman & Fomalont, 1992) and the
Ant, which is associated with an extragalactic cloud of ionized hydrogen with
a recessional velocity similar to that of Fornax A (Bland-Hawthorn et al.,
1995). In the remaining regions, line-of-sight magnetic field reversals have
been observed, and therefore Anderson et al. (2018b) interpret these low-p
patches as being due to the interaction with the environment: magnetized
thermal plasma is advected from the interstellar medium of NGC 1316, from the
peculiar early-type galaxy hosting Fornax A, or from the intracluster or
intragroup medium along the line of sight (or close to it) generating RM
gradients and low emission because of the magnetic field orientation.
The galaxy NGC 1316 is the result of a 10:1 merger that occurred $\sim$2 Gyr
ago between a massive early-type galaxy and a gas-rich late-type galaxy.
During the merger, stellar and gaseous tails and loops formed and are now
visible in the outskirts of the stellar body (Schweizer, 1980; Mackie &
Fabbiano, 1998; Goudfrooij et al., 2001; Serra et al., 2019). The H i content
of NGC 1316 and its group have been studied by several authors. Very recently,
Kleiner et al. (2021) observed the NGC 1316 group with the MeerKAT radio
telescope reaching a column density sensitivity of 1.4 $\times 10^{19}$ atoms
cm-2. These authors estimate that, during the merger, 7$-$11$\cdot$108 M⊙ of H
i was ejected to a large radius by tidal forces, and is now detected as clouds
and tails in the intragroup medium.

Figure 1: ASKAP polarized image (defined as the peak of the Faraday dispersion
function) and, on the right, a zoomed image centered on the southern part of
the H i tail. The beam size is shown in the bottom left corner of the image;
its FWHM corresponds to 12″. In both images VST H$\alpha$ and MeerKAT H i
contours are overlaid in yellow and blue, respectively. The H$\alpha$ contour
starts at 2$\rm\cdot 10^{-18}erg\cdot cm^{-2}\cdot s^{-1}\cdot arcsec^{-2}$,
while the H i contour represent a column density of
N${}_{H\,\textsc{i}}$=1.4$\times$1019 atoms cm-2. The black dashed regions and
arrows indicates the position of NGC 1310, of the Ant, and the region labeled
“K” in Anderson et al. (2018b). Figure 2: ASKAP RM image (top), defined as the
Faraday depth at the peak of the Faraday dispersion function, and its
uncertainty (bottom). The beam size is shown in the bottom left corner of the
image; its FWHM corresponds to 12″. In both images VST H$\alpha$ and MeerKAT H
i contours are overlaid in yellow and blue, respectively. The H$\alpha$
contour starts at 2$\rm\cdot 10^{-18}erg\cdot cm^{-2}\cdot s^{-1}\cdot
arcsec^{-2}$, while the H i contour represent a column density of
NHI=1.4$\times$1019 atoms cm-2.

In this work we present the first observed spatial coincidence between neutral
hydrogen tidal material detected by MeerKAT and a depolarization pattern
observed with the Australian Square Kilometre Array Pathfinder (ASKAP) in the
western lobe of Fornax A. We analyze the properties of the RM along the H i
tail and in a similar region to characterize the observed association.
In Section 2 we present the data used throughout the paper; in Section 3 we
determine the RM properties in the H i tail region and in a control region
with the same dimensions and orientation; in Section 4 we present the 2D
numerical simulations performed in order to determine the power spectrum
properties of the magnetic fields; in Section 5 we use 3D numerical
simulations to interpret the data; and in Section 6 we use H$\alpha$
observations to obtain upper limits on the thermal plasma density. Finally, we
summarize the results and draw our conclusions in Section 7. Throughout this
paper we assume a luminosity distance of (20.8$\pm$0.5) Mpc to NGC 1316
(Cantiello et al., 2013; Hatt et al., 2018). At this distance, 1 arcmin
corresponds to 5.8 kpc.

##  2 Data

The data analyzed in this work consists of the H i image obtained with MeerKAT
and already published by Kleiner et al. (2021), together with the polarized
image and RM image obtained with ASKAP during commissioning tests of the
telescope in view of the Early Science program which covered the Fornax
cluster (Anderson et al., 2021). The commissioning observations presented here
correspond to a different scheduling block with respect to the Fornax cluster
observations by Anderson et al. (2021). As we describe in the following, the
settings are not the same. Nevertheless, the data reduction and the imaging
were carried out in a very similar way. Here, we briefly describe the
observational setup, the data reduction, and the imaging of these data sets.

###  2.1 MeerKAT H i data

The MeerKAT H i images were produced from two commissioning observations
conducted in June 2018. The observations were both performed with the SKARAB
correlator in 4k mode, which consists of 4096 channels in full polarization in
the frequency range 856-1712 MHz with a resolution of 209 kHz (equivalent to
44.1 km s-1 for H i at the distance of the Fornax cluster). The two
observations had 36 and 62 antennas connected to the correlator; the target
was observed for 8 h and 7 h, respectively. The full data reduction process is
described in Kleiner et al. (2021) and we refer to that paper for further
details.

Relevant to this work is the Northern H i tail (TN) that overlaps in
projection with the Fornax A western lobe. TN was first detected by Serra et
al. (2019), where they showed that NGC 1316 underwent a 10:1 merger 1 – 3 Gyr
ago, and tidal forces ejected H i in the intragroup medium out to large radii
($\sim$ 150 kpc) in the form of two tails. The northern H i tail is made of
tidally stripped material once located within the spiral galaxy involved in
the merger that formed NGC 1316. Kleiner et al. (2021) detected additional H i
components of the H i tails that doubled the mass and length of TN. In this
work we explore how the H i of TN may be interacting with the Fornax A western
lobe and influencing its polarization properties.

###  2.2 ASKAP polarized data and RM

We observed Fornax A during commissioning tests of the ASKAP radio telescope
(DeBoer et al., 2009; Johnston et al., 2007; Schinckel & Bock, 2016) on April
3, 2019. We observed a single pointing (SBID 8377, start time 2019-04-03
01:48:39.435877) for ten hours using a single formed beam (McConnell et al.,
2016), covering a 288 MHz frequency range spanning 800$-$1088 MHz, achieving a
measured band-averaged sensitivity of 24 $\muup$Jy beam-1 at 10″ per Stokes
parameter, and recording the full set of polarization products.
The data reduction and imaging were performed as described in Anderson et al.
(2021) and we refer to that work for more details. The only difference with
respect to that work consists in the spatial resolution used to smooth the
final images (12″$\times$12″), which is the spatial resolution of our lowest
frequency channel. As in Anderson et al. (2021), the Q and U datacubes with
dimensions RA, Dec, $\rm\lambda^{2}$ were used to compute the Faraday
Dispersion Spectrum (FDS) over the range from -200 to +200 rad m-2 using RM
synthesis111https://github.com/brentjens/rm-synthesis, version 1.0-rc4 (Burn,
1966; Brentjens & de Bruyn, 2005). The result is a complex-valued FDS datacube
with dimensions RA, Dec, and $\phi$. We generated maps of the peak polarized
intensity (peak-$P$) and Faraday depth at the peak of the FDS (hereafter RM,
for historical reasons) across the field from the FDS cube using Miriad’s
moment function (Sault et al., 1995).

Figure 1 shows the polarized intensity of Fornax A with blue and yellow
contours respectively representing the MeerKAT H i emission and the VLT Survey
Telescope (VST) H$\alpha$ emission mapped with narrowband deep photometric
observations taken with the VST-OmegaCAM (Kleiner et al., 2021). The Fornax A
western lobe has a filamentary shape with the well-known low-p patches
(Fomalont et al., 1989; Anderson et al., 2018b) and new features revealed by
the sensitivity and resolution of the ASKAP image. The zoomed-in image in the
right panel of Figure 1 shows the H i tail. We note that the polarization
properties are significantly different along the H i tail with respect to
close-by regions. In particular, we can see the presence of a small-scale
($\leq 2$ kpc) depolarized structure along the H i tail. This pattern seems to
extend beyond the tail towards the south, connecting the H i tail with the
low-p patch labeled “K” in Anderson et al. (2018b), which we indicate with a
dashed box in the figure. The region called the Ant is also highlighted in
Figure 1 (dashed circle). Here, as reported in the Introduction, a cloud of
ionized hydrogen has been detected and it is acting as a Faraday screen.

Figure 3: RM images and properties of the H i tail and in a nearby region.
Left: Magnified RM image with VST H$\alpha$ contours (yellow) and MeerKAT H i
contours (blue). The black and magenta boxes indicate the H i tail and control
regions considered in our analysis. Middle: Distribution of the RM values in
the H i tail (black) and in the control region (magenta). Right: RM structure
functions evaluated in the H i tail (black) and in the control region
(magenta).

The RM image is shown in Figure 2 (top image). We subtracted from the image a
constant value of 6 rad/m2 due to the Galactic foreground. Figure 2 also shows
the RM uncertainty (bottom image), computed according to Eq. 13 in Sotomayor-
Beltran et al. (2013):

$\rm\Delta\phi(l)=\frac{1}{S/N}\frac{\sqrt{3}}{\lambda_{2}^{2}-\lambda_{1}^{2}},$
(1)

where S/N is the signal-to-noise ratio, while $\rm\lambda_{1}$ and
$\rm\lambda_{2}$ are the minimum and maximum wavelength of the bandwidth. The
RM image presents regions with a high RM scatter. These are found in
correspondence of the low-p patches, as already pointed out by Anderson et al.
(2018b). However, also in this case, we observe a new feature along the H i
tail which had not been detected probably because of the very high resolution
achieved with the ASKAP image. Here we can see an enhancement of the RM
scatter on small scales. Similar patterns can be observed in other locations
in correspondence of either H i emission or H$\alpha$ detections. In the next
section we characterize the RM in the tail region with the aim of
understanding whether the properties along the H i tail are different with
respect to the nearby regions. For the sake of simplicity, in this paper we
focus only on the H i tail since the tidal nature of this emission is well
understood compared with the less certain origin of the nearby H i clouds
(Kleiner et al., 2021).

##  3 Characteristic of the RM in the H i tail region

We compared the properties of the RM in a rectangular region which encloses
the H i emission and in a nearby box with the same dimensions and orientation.
We chose this control region according to two criteria. First, it has to be as
close as possible to the H i tail to meet similar physical conditions; second,
it should not contain detected H i or H$\alpha$ emission. We want to
understand the role of the H i tail as depolarizing screen, and need to
exclude any additional components when comparing it with the lobe.
The two regions are shown in the left image of Figure 3 as black and magenta
boxes for the H i tail and the control region, respectively. In the central
panel of this figure, we also show the distribution of the RM values in the
two regions. The mean, median, and the standard deviation of the RM are equal
to -20 rad/m2, -17 rad/m2, and 20 rad/m2 for the H i tail region and -13
rad/m2, -13 rad/m2, and 9 rad/m2 for the control. The average RM over the
entire western lobe is equal to -7 rad/m2. We recall that the galactic RM has
been subtracted from the RM image.
We also evaluated the RM structure function which is defined as the average
mean square difference in RM values between pixels at a given distance:

$\rm S(dr)=<[RM(r)-RM(r+dr)]^{2}>.$ (2)

The uncertainty associated with the RM structure function has been computed by
propagating the RM errors:

$\rm\Delta S(dr)=\frac{2\sqrt{2S(dr)}}{\sqrt{N}}\Delta\phi(l).$ (3)

The results are shown in the right plot of Figure 3 for the H i tail (in
black) and for the control region (in magenta). The error bars are within the
points.
It is clear that the two RM structure functions have a different
normalization. As more clearly seen in the next section, the shape and the
slope are also different in the two regions. Assuming a uniform density of the
thermal gas, this likely indicates a different geometry and strength of the
underlying magnetic fields.

##  4 Modeling the magnetic field power spectrum

Numerical simulations can be used to constrain the properties of magnetic
fields starting from spectropolarimetric radio observations. Using the FARADAY
tool (Murgia et al., 2004), we want to determine the power spectrum properties
of the magnetic field associated with the observed RM and its strength with 2D
and 3D simulations, respectively.
The RM power spectrum is proportional to the magnetic field power spectrum.
Assuming a particular magnetic field power spectrum model, we can therefore
derive the RM power spectrum and compute the RM structure function. By
comparing the model RM structure function with the observed one, we can
determine the magnetic field power spectrum repeating this procedure until we
find the best-fit model. To reach our goal we used the least squares method.
We followed this procedure assuming a random Gaussian magnetic field power
spectrum characterized by a minimum and a maximum scale of fluctuations,
namely $\rm\Lambda_{min}$ and $\rm\Lambda_{max}$, a normalization and a slope
n: $\rm|B_{k}|^{2}\propto k^{-n}$. The magnetic field power spectrum presents
degeneracy between its parameters. We decided to fix the minimum and maximum
scale of fluctuations to 0.6 kpc and 46 kpc respectively for both the H i tail
region and the control region. Such scales correspond to half of the beam size
of the RM image and to the length of the box used to evaluate the RM structure
functions, respectively. We simulated the noise in the model images by adding
a Gaussian noise with rms equals to the maximum uncertainty in the H i tail
and control region, which corresponds to 4 rad/m2. We performed 50
realizations of each different combination of slope and normalization of the
magnetic field power spectrum, varying the slope between 1 and 2.4 with a step
of 0.2, and the normalization from 50 to 500 $\muup$G Mpc3 in steps of 25
$\muup$G Mpc3.

Figure 4: RM structure functions evaluated in the H i tail (black) and in the
control region (magenta). The gray lines are, for each RM structure function,
50 simulated functions generated from magnetic field power-law power spectrum
models with parameters reported on the left.

Figure 4 shows the results of our analysis. The gray lines are respectively
the 50 different realizations of the best-fit magnetic field power spectra
with parameters reported in the top and middle left corner for the H i tail
and the control region, respectively. The dashed black and magenta lines
represent the average simulated RM structure function for the H i tail and
control region, respectively. We found a magnetic field power spectrum with a
slope of n=1.6 and a total power of 375 $\muup$G Mpc3 in the H i tail region,
and a slope of n=2 and a total power of 75 $\muup$G Mpc3 in the control
region. The RM auto-correlation length is $\rm\Lambda_{RM}$=1.9 kpc and 3.3
kpc for the H i tail and the control region, respectively. With this power
spectra we obtain magnetic fields with auto-correlation length
$\rm\Lambda_{c}$ equal to 1.3 kpc and 1.9 kpc for the H i tail and the control
region, respectively.
The different properties of the two magnetic field power spectra would suggest
that the two regions host different magnetic fields under the assumption of a
uniform density of the thermal gas.

##  5 RM interpretation

In the following paragraphs we discuss and analyze several Faraday screen
models which are due to the intracluster medium of the Fornax galaxy cluster
(Sect. 5.1), to the intragroup medium of the Fornax A group (Sect. 5.2), and
to a Faraday screen generated in the lobe surface and within the H i tail
(Sect. 5.3). We also investigate the possibility of a fluctuating density of
the thermal plasma within the H i tail (Sect. 5.4) and the physical
implications of the H i tail being inside the lobe (Sect. 5.5). These models
are used to find the best representation of the data in the control region and
in the H i tail region too.
To reach our goal we first evaluated the magnetic field strength needed to
cause the observed depolarization with a simple analytic approach. This gives
us an order of magnitude estimate which we improved with 3D numerical
simulation: starting from the magnetic field power spectrum models, we can
assume a magnetic field and a thermal plasma distribution and derive the
corresponding RM image. Again, we can compute the RM structure function from
the RM simulated image and we can in principle determine the structure and the
strength of the magnetic field by changing the model parameters and comparing
with the observed RM structure function. It is worth noting that also in this
case there is degeneracy between parameters. In particular, the RM depends on
the magnetic field strength and morphology, on the thermal plasma density, and
on the integral path length. The choice of the value of these parameters
strictly depends on the interpretation of the data.

###  5.1 Faraday screen from the Fornax intracluster medium

We derived an estimate of the RM at the western lobe location assuming a
magnetic field tangled on a single scale $\rm\Lambda_{c}$, randomly oriented
from cell to cell (see Appendix A of Loi et al., 2019). The result is a
Gaussian-like distribution of the RM with zero mean and dispersion
$\rm\sigma_{RM}$ (Lawler & Dennison, 1982; Felten, 1996). Assuming a
$\rm\beta$-model profile for the thermal plasma

$\rm n=n_{0}\left(1+\left(\frac{r}{r_{c}}\right)^{2}\right)^{-3\beta/2},$ (4)

with central density $\rm n_{0}$ and core radius $\rm r_{c}$, and a magnetic
field strength radial profile which follows the thermal plasma profile

$\rm B=B_{0}\left(\frac{n}{n_{0}}\right)^{\eta},$ (5)

the RM standard deviation is a function of the projected distance according to
Dolag et al. (2001):

$\rm\sigma_{RM}(r)=KB_{0}\Lambda_{c}^{0.5}n_{0}r_{c}^{\frac{1}{2}}\frac{1}{(1+\frac{r^{2}}{r_{c}^{2}})^{\frac{6\beta(1+\eta)-1}{4}}}\sqrt{\frac{\Gamma[3\beta(1+\eta)-\frac{1}{2}]}{\Gamma[3\beta(1+\eta)]}}.$
(6)

The factor K is a constant that depends on the integral path that we consider
to go from the Fornax A surface lobe to a distance of 1 Mpc. We assumed the
Fornax A western lobe to be spherical with a radius of 80 kpc and that its
center and the Fornax cluster center are at the same distance from us (i.e.,
they lie on the same plane perpendicular to the line of sight). Therefore,
with the cluster center as the origin of our reference frame, we started the
integral path from 80 kpc. While we do not know the characteristics of the
Fornax intracluster magnetic field, the thermal plasma distribution was
constrained by Chen et al. (2007) assuming the beta profile with a core radius
of $\rm r_{c}$=173 kpc, $\rm\beta$=0.804, and a central density $\rm
n_{0}=9\cdot 10^{-4}$ cm-3. With these values at the Fornax A location (i.e.,
at a distance of r=1.3 Mpc from the cluster center) we should observe RM
values on the order of $\rm\sigma_{RM}\approx$0.01$\rm\cdot B_{0}$ rad/m2
assuming a $\rm\Lambda_{c}$=1.98 kpc (see Section 2) and an index
$\rm\eta$=0.5 (see Govoni et al., 2017). In the framework of this model to
observe a $\rm\sigma_{RM}=9\,rad/m^{2}$, as in the control region, we would
need a central magnetic field strength of 900 $\muup$G. It is clear that the
RM at the Fornax A location cannot be due to the intracluster medium of the
Fornax cluster since magnetic field strengths in clusters are at most on the
order of tens of $\muup$G (Vacca et al., 2012).

###  5.2 Faraday screen from the Fornax A intragroup

We considered the RM due to the Fornax A group to explain the observed RM in
the control region. We assumed the $\rm\beta$-profile for the thermal plasma
by Babyk et al. (2018) with $r_{c}$=0.21 kpc, $\rm\beta$=0.43, and n0=11.76
cm-3 (adopting a mean mass of 0.63 times the proton mass). Using Eq. 6, an
integral path starting from the lobe surface up to 1 Mpc of distance,
$\rm\eta$=0.5, and $\rm\Lambda_{c}$=1.98 kpc, we found
$\rm\sigma_{RM}\approx$0.3$\cdot$ B0 rad/m2. To explain the observed
$\rm\sigma_{RM}$ in the control region, where this quantity equals 9 rad/m2,
we would need a central magnetic field strength of B${}_{0}\approx$30
$\muup$G.
We ran 3D numerical simulations to obtain a better estimate of the intragroup
magnetic field from the RM structure functions. In this case we created a
simulated RM image starting from the best-fit magnetic field power spectra
described in the previous section, which determine the geometry of the fields.
We then assumed the $\rm\beta$-model by Babyk et al. (2018), a magnetic field
profile following the plasma density distribution with $\rm\eta$=0.5 (Govoni
et al., 2017), and we performed several simulations changing the central
magnetic field strength between 1 and 60 $\muup$G with steps of 0.5 $\muup$G
to vary the RM normalization. For each value of the magnetic field strength,
we performed 50 realizations. We determined B0 with the least squares method
and this resulted in 50 and 18.5 $\muup$G in the case of the H i tail and
control region, respectively.

The results are shown in Figure 5.

Figure 5: RM structure functions evaluated in the H i tail (black) and in the
control region (magenta). The gray lines are, for each RM structure, 50
simulated functions generated assuming a Faraday screen due to the Fornax A
intragroup with magnetic field central strength reported on the left.

The RM structure function model for the control region follows the observed
trend, while the adopted modeling cannot trace the behavior of the H i region
and in particular the decrease starting at 5 kpc and then the rapid increment
at $\sim$25 kpc. The hypothesis of a RM due to an intragroup Faraday screen
cannot explain simultaneously the RM observed in the H i tail and in the
control region. In other words, the variation of the thermal plasma density
from the H i tail to the control region predicted by Babyk et al. (2018) is
insufficient to obtain consistent values for the central magnetic field
strength. This means that at the H i tail region we are detecting an
additional Faraday screen whose nature could be due to the H i tidal material
itself.

Considering that $\rm\sigma_{RM}$=9 rad/m2 in the control region, to reach
$\rm\sigma_{RM}$=20 rad/m2 at the H i location we would need an extra
$\rm\sigma_{RM}$ of 18 rad/m2 from the tail itself. Assuming a single-scale
magnetic field within the H i tail, BHT, which extends for a length L=11.04
kpc, namely the box width of Figure 3, we know that this contribution can be
parametrized as

$\rm\sigma_{RM}=812\cdot\frac{B_{HT}}{\sqrt{3}}n_{e}\sqrt{\Lambda_{C}}\sqrt{L},$
(7)

where ne is the thermal density within the H i tail, and $\rm\Lambda_{c}$ is
the magnetic field auto-correlation length. According to the modeling of the
magnetic field power spectra in Section 4, for the H i region $\rm\Lambda_{c}$
is equal to 1.30 kpc. Using this formula, we would need a magnetic field of
$\rm B_{HT}\approx(0.010/n_{e})\,\muup G$. If we consider a thermal density
similar to the Fornax A value, which is on the order of 4$\cdot$10-3cm-3,
B${}_{\rm HT}\approx$2.5 $\muup$G in the H i tail. The Fornax A intragroup
medium modeling suggests a magnetic field of $\sim$0.35 $\muup$G at the
control region position.
Whatever the magnetized plasma generating the RM observed in the western lobe
of Fornax A, it seems clear that there is an additional Faraday screen in
correspondence of the H i tail.

###  5.3 A Fornax A shell lobe and a Faraday screen H i tail

As suggested by Anderson et al. (2018b), the RM gradients observed in the
low-p patches of the Fornax A lobes could be due to the advection of
magnetized thermal plasma on the lobe surface. We can therefore explore a
simple scenario in which the observed RM in correspondence of the control
region is entirely due to a magnetized plasma on the surface of the lobe. The
thermal plasma density is assumed to be constant, while the magnetic field
fluctuates according to the models in Section 4 with a given average strength
B0.

We started simulating the RM structure function due to the lobe surface. As
already mentioned, the Faraday RM shows degeneracy in three parameters: the
magnetic field strength, the thermal plasma density, and the path length. This
means that we cannot derive an estimate for the magnetic field at the control
region, but what is important here is to model the lobe RM in order to derive
information about the H i tail magnetic field. We considered a path length of
11.04 kpc, which is the width of the control region, and we assume that the
thermal plasma density is constant within this path length, while the magnetic
field fluctuates around a given value B0. Assuming this picture, we explored
the parameter space between B${}_{0}\cdot$n0=1$\cdot 10^{-3}\,\muup$G cm-3 and
B${}_{0}\cdot$n0=60$\cdot 10^{-3}\,\muup$G cm-3 in steps of
B${}_{0}\cdot$n0=0.5$\cdot 10^{-3}\,\muup$G cm-3. By minimizing the
$\rm\chi^{2}$, we determined B${}_{0}\cdot$n0=3.5$\cdot 10^{-3}\,\muup$G cm-3
for the control region, as can be seen in Figure 6. This modeling also seems
to reproduce quite well the observed RM structure function.

Figure 6: RM structure functions evaluated in the H i tail (black) and in the
control region (magenta). The gray lines are best-fit simulated RM structures
obtained integrating over a path length of 11.04 kpc assuming constant thermal
plasma densities n0 and magnetic fields showing fluctuations around an average
strength B0. The best-fit parameters are reported on the left.

If the emission detected in the direction of the H i tail region is outside of
the lobe in the foreground, then the observed RM is the sum of the Faraday
rotation due to the lobe surface (RMlobe) plus the Faraday rotation due to the
H i tail (RMtail). The Faraday dispersion function, in the absence of Faraday
complexity within the two contributions, will show a single polarized peak.
If, on the contrary, the H i tail is within the lobe then we expect to find
two Faraday components in the Faraday dispersion function: the polarized
signals emitted at different path lengths in the lobe will suffer from the
Faraday rotation due to the lobe surface (RMlobe) or a combination of this
term plus the one due to the H i tail (RMlobe+RMtail). It is clear that this
scenario will lead to a Faraday complexity in the direction of the H i tail
region, an aspect that has been already evidenced by Anderson et al. (2018b),
even if it is not clear whether this involves only regions where ionized
and/or molecular gas have been detected. Assuming that the Faraday complexity
consists of two polarized peaks crossing the two different Faraday screens
(i.e., in the absence of internal depolarization either within the lobe
surface or in the H i tail), we might wonder which polarized peak and
corresponding RM were selected to create the RM image (which technically
corresponds to the maximum polarized peak in the Faraday dispersion function):
is it the component due to the sum RMlobe+RMtail or simply to the RMlobe?
Since the RM at the H i location is dramatically different with respect to the
control region RM, we think that in this case the observed RM at the H i tail
region is the sum of RMlobe+RMtail.
Keeping all of these considerations in mind, we simulated two different
scenarios in the H i tail direction. We first assumed that the RM is entirely
due to the H i tail Faraday screen. This will lead to an upper limit on the
magnetic field estimate since we are assuming that only this field is
responsible for the Faraday rotation. In this case we obtained
B${}_{0}\cdot$n0=9$\cdot 10^{-3}$ $\muup$G cm-3 as shown in Figure 6.

Figure 7: RM structure functions evaluated in the H i tail (black) and in the
control region (magenta). The gray lines near the control region are the best-
fit simulated RM structures reported in Figure 6. Those near the H i tail
region are best-fit models for the H i tail excess obtained considering that
the observed RM at the H i position is due to the intrinsic H i tail
contribution plus the RM model of the control region. The best-fit parameters
are reported on the left.

The second scenario considers the Faraday rotation in the H i tail region as
the sum of the contributions of the H i tail itself and the Fornax A lobe
surface that we already modeled. In this scenario the H i tail could be either
in the lobe foreground or right behind the lobe surface. Between the H i tail
region and the lobe surface no polarized signal is emitted, or if it is, that
polarized signal is weaker than that emitted behind the H i tail. We derived
the H i tail RM properties by simulating a RM over a path length of 11.04 kpc,
varying the B${}_{0}\cdot$n0 parameter, and we add this term to the RM model
of the control region. We obtain B${}_{0}\cdot$n0=8.5$\cdot 10^{-3}$ $\muup$G
cm-3, as shown in Figure 7.

We note that it is unlikely that a depolarization internal to the H i tail is
taking place, which would require the H i tail to be emitting a linearly
polarized signal, which needs relativistic particles. These particles can be
found in H i regions if there is star formation. However, star formation is
expected with column density higher than the values observed along the HI tail
(see Kleiner et al., 2021) and ionized gas should also be there, but we do not
detect H$\alpha$ emission along the HI tail. Therefore, we do not expect
linear polarized emission from the HI tail itself nor internal depolarization.

###  5.4 Density fluctuations in the H i tail

In the previous section we modeled the RM structure function in the H i tail
region assuming that the corresponding Faraday screen volume is filled with a
thermal plasma with constant density. This yielded a flat RM structure
function (see Figure 7), which cannot follow the rapid increment starting at
$\sim$25 kpc. Here we try to relax the assumption of a constant density and,
as we show later, we find a better match between the observed and the
simulated RM structure functions.
To reach our goal, we assumed that the density has a power-law power spectrum
fluctuating between a minimum and maximum scale. For the sake of simplicity,
we assumed the same slope of the magnetic field power spectrum: n=1.6. We note
that the results strongly depend on the minimum scale because with the chosen
slope the energy is distributed on small scales. We reproduced the H i tail
excess as in the previous section: we created a Faraday screen with a magnetic
field fluctuating according to the model in Section 4, and assuming a
different average strength and different configuration of the thermal plasma
density and structure; this corresponds to the H i tail RM; the lobe
contribution is modeled in the same way used in the previous section. The sum
of the contributions is compared with the RM structure function to find the
best-fit model of the thermal density power spectrum. We explored several
configurations with $\rm\Lambda_{min}$=0.6 kpc and $\rm\Lambda_{max}$=2.4 kpc
to create a small-scale fluctuating structure: $\rm\Lambda_{min}$=0.6 kpc and
$\rm\Lambda_{max}$=46 kpc; $\rm\Lambda_{min}$=26 kpc, 46 kpc, and 92 kpc with
$\rm\Lambda_{max}$=200 kpc to simulate a structure fluctuating on large
scales.
We obtained a simulation that shows an increment at large distances with
$\rm\Lambda_{min}$=46 kpc, $\rm\Lambda_{max}$=200 kpc, and B0n0=16.5$\cdot
10^{-3}$ $\muup$G cm-3. Fifty realizations of this model are shown in Figure 8
with thermal plasma density parameters reported in the top left corner.

Figure 8: RM structure functions evaluated in the H i tail (black) and in the
control region (magenta). The gray lines near the H i tail region are best-fit
models for the H i tail excess assuming that within the H i tail the power
spectrum of the thermal density is a power law with parameter reported on the
left.

We recall that the H i tail analyzed in this work is the southern part of a
larger structure, labeled TN in Kleiner et al. (2021), which extends for about
200 kpc.

###  5.5 HI tail embedded inside the Fornax A lobe, equilibrium conditions

The exact location of the H i tail along the line of sight cannot be
determined with high accuracy. However, as we already pointed out, since we
can clearly associate the depolarizing effect with the H i tail, this material
should be either in the Fornax A lobe foreground or embedded in the lobe
itself. Considering the large volume occupied by the lobe (approximately a
sphere with a diameter of 25″$\sim$150 kpc) and the fact that the base of the
tidal tail must be near the stellar body of NGC 1316, we consider it more
likely that the HI-detected part of the tail is within the lobe.
Here, we assumed that the H i tail is within the Fornax A western lobe. This
allowed us to obtain an estimate of the H i tail magnetic field based on some
physical considerations.
If the H i tail is embedded in the lobe, we can reasonably assert that the
external pressure exerted by the radio lobe ($\rm P_{nth}$) is balanced by the
H i tail pressure, which is due to the gas ($\rm P_{gas}$=nkT) and to the
embedded magnetic field ($\rm P_{B}=$B${}^{2}/8\pi$):

$\rm P_{nth}=P_{gas}+P_{B}.$ (8)

Therefore, for a given nonthermal pressure and temperature, the magnetic field
strength depends on the square root of the density. Following this idea, we
can derive an estimate of the magnetic field strength in the tail assuming a
value for the radio lobe pressure from literature and typical values for the
temperature and gas density. Maccagni et al. (2020) determined the radio lobe
pressure according to the equipartition condition, which gives a lower limit
on this quantity ($\rm P_{nth}\geq 4.9\cdot 10^{-11}$ erg cm-3). The gas in
the H i tail should have a temperature between 102 and 104 K. We show in
Figure 9 how the magnetic field strength varies with respect to the gas
density in this range of temperatures.

Figure 9: Magnetic field vs gas density according to the equilibrium condition
of Eq. 8 for three different values of the gas temperature.

Since the H i tail is made of material originally belonging to a spiral
galaxy, we expect to have typical gas densities on the order of 10-2 cm-3 in
the H i tail. Even assuming a denser gas, we estimated a magnetic field
strength of around 9.5$-$11 $\muup$G, weakly affected by the gas temperature.
It is worth noting that this value should be considered a lower limit since
the radio lobe pressure was determined under the equipartition condition.
Considering the H i excess determined in Section 5.3, we derived a thermal
plasma density of $\sim$8$-$9$\cdot 10^{-4}$ cm-3 in the H i tail, or
$\sim$1.5$-$1.7$\cdot 10^{-3}$ cm-3 if the thermal gas is fluctuating as
described in Section 5.4. In the next section we derive an upper limit on the
thermal density from H$\alpha$ observations.

##  6 H$\alpha$ constraints on thermal particles and magnetic field estimates

We can tentatively give an estimate of the H i tail thermal plasma density
from H$\alpha$ observations using the Eq. 4 in Barger et al. 2013:

$\rm
L_{H+}=2.75\left(\frac{T}{10^{4}K}\right)^{0.924}\left(\frac{I_{H\alpha}}{R}\right)\left(\frac{n_{e}}{cm^{-3}}\right)^{-2}\rm
pc.$ (9)

The dimension of the cloud $\rm L_{H+}$ can be determined from the width of
the H i tail, under the assumption of a cylindrical structure. This gives $\rm
L_{H+}$ = 11.04 kpc. Kleiner et al. (2021) did not detect H$\alpha$ along the
tail and we can use a 3$\rm\sigma$ threshold ($\rm\sigma=2\cdot
10^{-18}erg\cdot cm^{-2}\cdot s^{-1}\cdot arcsec^{-2}$) of their observations
as an upper limit for the H$\rm\alpha$ brightness in this region, IHα.
In a reasonable range of gas temperatures for an H$\alpha$ cloud
($10^{3}-10^{4}$ K) the thermal electron density is between 2$\cdot 10^{-3}$
and 6$\cdot 10^{-3}$ cm-3. This means that the thermal electron density in the
H i tail is below 6$\cdot 10^{-3}$ cm-3.
This value is compatible with the findings of the previous sections: if the H
i tail is embedded in the Fornax A western lobe, it is hosting a magnetic
field with a strength of $\sim$9.5$-$11 $\muup$G and a thermal plasma density
of $\sim$8$-$9$\cdot 10^{-4}$ cm-3 constant within the tail, or
$\sim$1.5$-$1.7$\cdot 10^{-3}$ cm-3 if it fluctuates. In both cases the
thermal plasma density is below the detection threshold of the H$\alpha$
observations by Kleiner et al. (2021).

##  7 Summary and conclusions

In this work we analyzed the spatial coincidence between H i tidal material
and a depolarized structure in the western lobe of Fornax A. Our goal was to
understand if the H i tail is acting as depolarizing Faraday screen. Our
results are summarized in Table 1 and discussed in what follows.

Faraday screen from the Fornax cluster intracluster medium |  |  |  |
---|---|---|---|---
Region | ne(r) | ne power-law | B(r) | B power-law | L | B0 | Plausible
|  | power spectrum |  | power spectrum | [kpc] | [$\muup$G] |
Control region |  $\beta-$model | no |  $\eta-$model | n=2 | 1000 | 900 | no
| (Chen et al., 2007) |  | with $\eta$=0.5 |  |  |  |
H i tail region |  $\beta-$model | no |  $\eta-$model | n=1.6 | 1000 | 2000 | no
| (Chen et al., 2007) |  | with $\eta$=0.5 |  |  |  |
Faraday screen from the Fornax A intra-group |  |  |  |
Region | ne(r) | ne power-law | B(r) | B power-law | L | B0 | Plausible
|  | power spectrum |  | power spectrum | [kpc] | [$\muup$G] |
Control region |  $\beta-$model | no |  $\eta-$model | n=2 | 1000 | 18.5 | yes
| (Babyk et al., 2018) |  | with $\eta$=0.5 |  |  |  |
H i tail region |  $\beta-$model | no |  $\eta-$model | n=1.6 | 1000 | 50 | no
| (Babyk et al., 2018) |  | with $\eta$=0.5 |  |  |  |
Faraday screen assuming the advection scenario |  |  |  |
Region | ne(r) | ne power-law | B(r) | B power-law | L | B0n0 | Plausible
|  | power spectrum |  | power spectrum | [kpc] | [$\muup$G cm-3] |
Control region | constant | no | constant | n=2 | 11.04 | 3.5$\cdot 10^{-3}$ | yes
H i tail region | constant | no | constant | n=1.6 | 11.04 | 9.0$\cdot 10^{-3}$ | yes
H i tail excess | constant | no | constant | n=1.6 | 11.04 | 8.5$\cdot 10^{-3}$ | yes
H i tail excess | constant | n=1.6 | constant | n=1.6 | 11.04 | 16.5$\cdot 10^{-3}$ | yes
|  |  $\rm\Lambda_{min}=46$ kpc |  |  |  |  |
|  |  $\rm\Lambda_{min}=200$ kpc |  |  |  |  |
Pressure equilibrium |  |  |  |
Region | Gas density | Temperature |  | Nonthermal pressure |  | B0 | Plausible
| [cm-3] | [K] |  | [erg cm-3] |  | [$\muup$G] |
H i tail | 10-2 | 10${}^{2}-10^{4}$ |  | 4.9 $\cdot 10^{-11}$ |  | 9.5$-$11 | yes
Table 1: Summary of the results. We specify the scales of fluctuation of the
density power spectra since they are different with respect to those of the
magnetic field (see text for further details).

We modeled the magnetic field power spectra along the H i tail and in a
control region with the same dimension and orientation by making a 2D
simulation of the RM structure function. Keeping the minimum and maximum scale
of fluctuation fixed to $\rm\Lambda_{min}$=0.6 kpc and $\rm\Lambda_{max}$=46
kpc, we found power spectra with different normalizations and slopes. In
particular, the control region showed a magnetic field power spectrum with
slope n=2, normalization equal to 75 $\rm\muup$G Mpc-3, and a RM auto-
correlation length of $\rm\Lambda_{RM}$=3.3 kpc, while the H i tail region has
a slope of n=1.6, a normalization of 350 $\rm\muup$G Mpc-3, and a RM auto-
correlation length of $\rm\Lambda_{RM}$=1.9 kpc.
We then used an analytical approach and 3D simulations to understand what
physical scenario is producing the observed RM structures. We found that it is
unlikely that the observed RM is due to the intracluster medium of the Fornax
cluster since we would need a very strong magnetic field (i.e., on the order
of thousands of $\muup$G). A modeling of the intragroup medium of the Fornax A
group could explain the RM observed in the control region if the central
magnetic field strength were 18.5 $\rm\muup$G. However, such modeling cannot
explain simultaneously the RM of the control region and that observed at the H
i tail location. Therefore, it seems clear that the H i tail itself is acting
as a Faraday screen.
We also investigated the possibility that the RM in the control region is due
to the advection of magnetized plasma on the lobe surface of Fornax A, as
suggested by Anderson et al. (2018b). Assuming a magnetic field fluctuating
around an average strength (as described in Section 4) and a constant and
uniform thermal density, and by integrating the Faraday screen over a length
of 11.04 kpc (i.e., the H i tail width), we found B${}_{0}\cdot$n0=3.5$\cdot
10^{-3}$ $\muup$G cm-3 in the control region. For the H i tail, we considered
two scenarios, first where the RM is entirely due to the H i tail, which gives
an upper limit of B${}_{0}\cdot$n0=9$\cdot 10^{-3}$ $\muup$G cm-3, and second
where the RM is due to the sum of the lobe RM and the H i tail RM. The second
scenario applies if the H i tail is in the lobe foreground or within the lobe,
very close to its surface. In this case, we obtained
B${}_{0}\cdot$n0=8.5$\cdot 10^{-3}$ $\muup$G cm-3 in the H i tail region. We
noted that the rapid increment of the H i tail region RM structure function at
large spatial scales cannot be traced by this modeling. Therefore, we assumed
that the thermal plasma has a power-law power spectrum with index n=1.6. We
found that only a thermal plasma fluctuating between a minimum and a maximum
scale of 46 kpc and 200 kpc, respectively, can reproduce the increase of the
structure function at distances larger than 25 kpc. In this case the parameter
B${}_{0}\cdot$n0=16.5$\cdot 10^{-3}$ $\muup$G cm-3. We noted that there is
degeneracy between the minimum scale of fluctuation and the slope of the
thermal plasma power spectrum. We do not have further constraints on the
thermal plasma distribution in the H i tail since H$\alpha$ is not detected at
this location or in the lobe surface. An exact reconstruction of the
distribution of the thermal density is beyond the scope of this paper.
Following the idea that the H i tail is embedded in the Fornax A lobe, we
derived an estimate of the magnetic field in the H i tail considering the
equilibrium condition where the nonthermal pressure exerted by the radio lobe
is balanced by the H i tail pressure, which is due to the gas and to the
magnetic field. From this equation we obtained a magnetic field strength of
$\sim$9.5$-$11 $\muup$G for a gas temperature between 102 and 104 K. These
values represent an upper limit since the nonthermal pressure of the radio
lobes was derived under equipartition condition.
We used H$\alpha$ observations to find an upper limit on the thermal plasma
distribution in the H i tail and in the control region. The result is a
thermal plasma density below 6$\cdot 10^{-3}$ cm-3.
From the 3D simulation of the RM structure function associated with the H i
tail, we obtained a magnetic field strength times the thermal plasma
distribution of B${}_{0}\cdot$n0=8.5$\cdot 10^{-3}$ $\muup$G cm-3 assuming a
constant thermal density, and B${}_{0}\cdot$n0=16.5$\cdot 10^{-3}$ $\muup$G
cm-3 if the thermal density is fluctuating. From the equilibrium condition we
know that the H i tail host a magnetic field of $\sim$9.5$-$11 $\muup$G.
Combining all these findings we obtained a thermal plasma distribution of
$\sim$8$-$9$\cdot 10^{-4}$ cm-3 or $\sim$1.5$-$1.8$\cdot 10^{-3}$ cm-3 cm-3
for a constant and a fluctuating thermal density, respectively. Both values
are compatible with the upper limit from H$\alpha$ observations.

From this analysis is clear that we have found an additional Faraday screen
due to the H i tail. We therefore conclude that the tail, which is the remnant
of a galaxy merger that occurred $\sim 2$ Gyr ago and is made of material
originally belonging to a Milky Way-like galaxy, is either driving a multi-
phase magnetized gaseous medium through the radio lobe of Fornax A or that it
survived the lobe expansion. This could be the first observed evidence of such
phenomena.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement no. 679627; project name FORNAX). FL acknowledges financial
support from the Italian Minister for Research and Education (MIUR), project
FARE, project code R16PR59747, project name FORNAX-B. FL acknowledges
financial support from the Italian Ministry of University and Research $-$
Project Proposal CIR01$\\_$00010\. VV and MM acknowledge support from INAF
mainstream project ”Galaxy Clusters Science with LOFAR” 1.05.01.86.05. Part of
the data published here have been reduced using the CARACal pipeline,
partially supported by ERC Starting grant number 679627 ”FORNAX”, MAECI Grant
Number ZA18GR02, Department of Science and Innovation$-$NRF Grant Number
113121 as part of the ISARP Joint Research Scheme, and BMBF project 05A17PC2
for D$-$MeerKAT. Information about CARACal can be obtained online under the
URL: https://caracal.readthedocs.io. We also thank our anonymous referee for
their constructive feedback which has strengthened this work.

## References

  * Anderson et al. (2021) Anderson, C. S., Heald, G. H., Eilek, J. A., et al. 2021, PASA, 38, e020. doi:10.1017/pasa.2021.4
  * Anderson et al. (2018a) Anderson, C., Heald, G., O’Sullivan, S., et al. 2018, Galaxies, 6, 127
  * Anderson et al. (2018b) Anderson, C. S., Gaensler, B. M., Heald, G. H., et al. 2018, ApJ, 855, 41. doi:10.3847/1538-4357/aaaec0
  * Babyk et al. (2018) Babyk, I. V., McNamara, B. R., Nulsen, P. E. J., et al. 2018, ApJ, 857, 32. doi:10.3847/1538-4357/aab3c9
  * Banfield et al. (2019) Banfield, J. K., O’Sullivan, S. P., Wieringa, M. H., et al. 2019, MNRAS, 482, 5250. doi:10.1093/mnras/sty3108
  * Bland-Hawthorn et al. (1995) Bland-Hawthorn, J., Ekers, R. D., van Breugel, W., et al. 1995, ApJ, 447, L77. doi:10.1086/309575
  * Brentjens & de Bruyn (2005) Brentjens, M. A. & de Bruyn, A. G. 2005, A&A, 441, 1217. doi:10.1051/0004-6361:20052990
  * Brienza et al. (2021) Brienza, M., Shimwell, T. W., de Gasperin, F., et al. 2021, Nature Astronomy. doi:10.1038/s41550-021-01491-0
  * Burn (1966) Burn, B. J. 1966, MNRAS, 133, 67. doi:10.1093/mnras/133.1.67
  * Cantiello et al. (2013) Cantiello, M., Grado, A., Blakeslee, J. P., et al. 2013, A&A, 552, A106. doi:10.1051/0004-6361/201220756
  * Carilli et al. (1988) Carilli, C. L., Perley, R. A., & Dreher, J. H. 1988, ApJ, 334, L73. doi:10.1086/185315
  * Chen et al. (2007) Chen, Y., Reiprich, T. H., Böhringer, H., et al. 2007, A&A, 466, 805. doi:10.1051/0004-6361:20066471
  * DeBoer et al. (2009) DeBoer, D. R., Gough, R. G., Bunton, J. D., et al. 2009, IEEE Proceedings, 97, 1507. doi:10.1109/JPROC.2009.2016516
  * Dreher et al. (1987) Dreher, J. W., Carilli, C. L., & Perley, R. A. 1987, ApJ, 316, 611. doi:10.1086/165229
  * Dolag et al. (2001) Dolag, K., Schindler, S., Govoni, F., et al. 2001, A&A, 378, 777. doi:10.1051/0004-6361:20011219
  * Drinkwater et al. (2001) Drinkwater, M. J., Gregg, M. D., Holman, B. A., et al. 2001, MNRAS, 326, 1076. doi:10.1046/j.1365-8711.2001.04646.x
  * Fanaroff et al. (2021) Fanaroff, B., Lal, D. V., Venturi, T., et al. 2021, arXiv:2105.11695
  * Felten (1996) Felten, J. E. 1996, Clusters, Lensing, and the Future of the Universe, 88, 271
  * Fomalont et al. (1989) Fomalont, E. B., Ebneter, K. A., van Breugel, W. J. M., et al. 1989, ApJ, 346, L17. doi:10.1086/185568
  * Geldzahler & Fomalont (1978) Geldzahler, B. J. & Fomalont, E. B. 1978, AJ, 83, 1047. doi:10.1086/112289
  * Geldzahler & Fomalont (1984) Geldzahler, B. J. & Fomalont, E. B. 1984, AJ, 89, 1650. doi:10.1086/113668
  * Goudfrooij et al. (2001) Goudfrooij, P., Alonso, M. V., Maraston, C., et al. 2001, MNRAS, 328, 237. doi:10.1046/j.1365-8711.2001.04860.x
  * Govoni et al. (2017) Govoni, F., Murgia, M., Vacca, V., et al. 2017, A&A, 603, A122. doi:10.1051/0004-6361/201630349
  * Guidetti et al. (2010) Guidetti, D., Laing, R. A., Murgia, M., et al. 2010, A&A, 514, A50. doi:10.1051/0004-6361/200913872
  * Guidetti et al. (2011) Guidetti, D., Laing, R. A., Bridle, A. H., et al. 2011, MNRAS, 413, 2525. doi:10.1111/j.1365-2966.2011.18321.x
  * Guidetti et al. (2012) Guidetti, D., Laing, R. A., Croston, J. H., et al. 2012, MNRAS, 423, 1335. doi:10.1111/j.1365-2966.2012.20961.x
  * Kleiner et al. (2021) Kleiner, D., Serra, P., Maccagni, F. M., et al. 2021, A&A, 648, A32
  * Hardcastle (2003) Hardcastle, M. J. 2003, MNRAS, 339, 360. doi:10.1046/j.1365-8711.2003.06102.x
  * Hardcastle et al. (2012) Hardcastle, M. J., Massaro, F., Harris, D. E., et al. 2012, MNRAS, 424, 1774. doi:10.1111/j.1365-2966.2012.21247.x
  * Hatt et al. (2018) Hatt, D., Freedman, W. L., Madore, B. F., et al. 2018, ApJ, 866, 145. doi:10.3847/1538-4357/aadfe8
  * Johnston et al. (2007) Johnston, S., Bailes, M., Bartel, N., et al. 2007, PASA, 24, 174. doi:10.1071/AS07033
  * Laing et al. (2008) Laing, R. A., Bridle, A. H., Parma, P., et al. 2008, MNRAS, 391, 521. doi:10.1111/j.1365-2966.2008.13895.x
  * Lawler & Dennison (1982) Lawler, J. M. & Dennison, B. 1982, ApJ, 252, 81. doi:10.1086/159536
  * Loi et al. (2019) Loi, F., Murgia, M., Govoni, F., et al. 2019, MNRAS, 490, 4841. doi:10.1093/mnras/stz2699
  * Maccagni et al. (2020) Maccagni, F. M., Murgia, M., Serra, P., et al. 2020, A&A, 634, A9. doi:10.1051/0004-6361/201936867
  * Mackie & Fabbiano (1998) Mackie, G. & Fabbiano, G. 1998, AJ, 115, 514. doi:10.1086/300203
  * McConnell et al. (2016) McConnell, D., Allison, J. R., Bannister, K., et al. 2016, PASA, 33, e042. doi:10.1017/pasa.2016.37
  * Murgia et al. (2004) Murgia, M., Govoni, F., Feretti, L., et al. 2004, A&A, 424, 429. doi:10.1051/0004-6361:20040191
  * Norris et al. (2021) Norris, R. P., Marvil, J., Collier, J. D., et al. 2021, arXiv:2108.00569
  * Ramatsoku et al. (2020) Ramatsoku, M., Murgia, M., Vacca, V., et al. 2020, A&A, 636, L1. doi:10.1051/0004-6361/202037800
  * Sault et al. (1995) Sault, R. J., Teuben, P. J., & Wright, M. C. H. 1995, Astronomical Data Analysis Software and Systems IV, 77, 433
  * Schinckel & Bock (2016) Schinckel, A. E. T. & Bock, D. C.-J. 2016, Proc. SPIE, 9906, 99062A. doi:10.1117/12.2233920
  * Schweizer (1980) Schweizer, F. 1980, ApJ, 237, 303. doi:10.1086/157870
  * Schulman & Fomalont (1992) Schulman, E. & Fomalont, E. B. 1992, AJ, 103, 1138. doi:10.1086/116131
  * Serra et al. (2019) Serra, P., Maccagni, F. M., Kleiner, D., et al. 2019, A&A, 628, A122. doi:10.1051/0004-6361/201936114
  * Sotomayor-Beltran et al. (2013) Sotomayor-Beltran, C., Sobey, C., Hessels, J. W. T., et al. 2013, A&A, 552, A58. doi:10.1051/0004-6361/201220728
  * Vacca et al. (2012) Vacca, V., Murgia, M., Govoni, F., et al. 2012, A&A, 540, A38. doi:10.1051/0004-6361/201116622