# On Romanization for Model Transfer Between Scripts
in Neural Machine Translation

Chantal Amrhein1 Rico Sennrich1,2
1Department of Computational Linguistics, University of Zurich
2School of Informatics, University of Edinburgh
<EMAIL_ADDRESS>
###### Abstract

Transfer learning is a popular strategy to improve the quality of low-resource
machine translation. For an optimal transfer of the embedding layer, the child
and parent model should share a substantial part of the vocabulary. This is
not the case when transferring to languages with a different script. We
explore the benefit of romanization in this scenario. Our results show that
romanization entails information loss and is thus not always superior to
simpler vocabulary transfer methods, but can improve the transfer between
related languages with different scripts. We compare two romanization tools
and find that they exhibit different degrees of information loss, which
affects translation quality. Finally, we extend romanization to the target
side, showing that this can be a successful strategy when coupled with a
simple deromanization model.

##  1 Introduction

Neural Machine Translation (NMT) has opened up new opportunities in transfer
learning from high-resource to low-resource language pairs (Zoph et al., 2016;
Kocmi and Bojar, 2018; Lakew et al., 2018). While transfer learning has shown
great promise, the transfer between languages with different scripts brings
additional challenges. For a successful transfer of the embedding layer, both
the parent and the child model should use the same or a partially overlapping
vocabulary (Aji et al., 2020). It is common to merge the two vocabularies by
aligning identical subwords and randomly assigning the remaining subwords from
the child vocabulary to positions in the parent vocabulary (Lakew et al.,
2018, 2019; Kocmi and Bojar, 2020).

This works well for transfer between languages that use the same script, but
if the child language is written in an unseen script, most vocabulary
positions are replaced by random subwords. This significantly reduces the
transfer from the embedding layer. Gheini and May (2019) argue that
romanization can improve transfer to languages with unseen scripts. However,
romanization can also introduce information loss that might hurt translation
quality. In our work, we study the usefulness of romanization for transfer
from many-to-many multilingual MT models to low-resource languages with
different scripts. Our contributions are the following:

  * -

We show that romanized MT is not generally optimal, but can improve transfer
between related languages that use different scripts.

  * -

We study information loss from different romanization tools and its effect on
MT quality.

  * -

We demonstrate that romanization on the target side can also be effective when
combined with a learned deromanization model.

##  2 Related Work

Initial work on transfer learning for NMT has assumed that the child language
is known in advance and that the parent and child model can use a shared
vocabulary (Nguyen and Chiang, 2017; Kocmi and Bojar, 2018). Lakew et al.
(2018) argue that this is not feasible in most real-life scenarios and propose
using a dynamic vocabulary. Most studies have since opted to replace unused
parts of the parent vocabulary with unseen subwords from the child vocabulary
(Lakew et al., 2019; Kocmi and Bojar, 2020); others use various methods to
align embedding spaces (Gu et al., 2018; Kim et al., 2019). Recently, Aji et
al. (2020) showed that transfer of the embedding layer is only beneficial if
there is an overlap between the parent and child vocabulary such that
embeddings for identical subwords can be aligned. Such alignments are very
rare if the child language uses an unseen script.

Gheini and May (2019) train a universal vocabulary on multiple languages by
romanizing languages written in a non-Latin script. Their many-to-one parent
model can be transferred to new source languages without exchanging the
vocabulary. In our work, we extend this idea to many-to-many translation
settings using subsequent deromanization of the output. We study the trade-off
between a greater vocabulary overlap and information loss as a result of
romanization. Based on experiments on a diverse set of low-resource languages,
we show that romanization is helpful for model transfer to related languages
with different scripts.

##  3 Romanization

Romanization describes the process of mapping characters in various scripts to
Latin script. This mapping is not always reversible. The goal is to
approximate the pronunciation of the text in the original script. However,
depending on the romanization tool, more or less information encoded in the
original script is lost. We compare two tools for mapping our translation
input to Latin script:

uroman111https://github.com/isi-nlp/uroman (Hermjakob et al., 2018) is a tool
for universal romanization that can romanize almost all character sets. It is
unidirectional; mappings from Latin script back to other scripts are not
available. uconv222https://linux.die.net/man/1/uconv is a command-line tool
similar to iconv that can be used for transliteration. It preserves more
information from the original script, which is expressed with diacritics.
uconv is bi-directional for a limited number of script pairs.

Below is an example of the same Chinese sentence romanized with uroman and
uconv:

她到塔皓湖去了
---
uroman: ta dao ta hao hu qu le
uconv: tā dào tǎ hào hú qù le
“She went to Lake Tahoe.”

The two tools exhibit different degrees of information loss. uroman ignores
tonal information and consequently collapses the representations of 塔 (Pinyin
tǎ; ‘tower’) and 她 (Pinyin tā; ‘she’). Romanization with uconv retains this
distinction but it still adds ambiguity and loses the distinction between 她
(Pinyin tā; ‘she’) and 他 (Pinyin tā; ‘he’), among others. While uconv exhibits
less information loss, its use of diacritics limits subword sharing between
languages. We measure character-level overlap between English and romanized
Arabic, Russian and Chinese with chrF scores (Popović, 2015) and find they are
much higher for uroman (9.6, 18.8 and 13.3) compared to uconv (6.8, 18.1 and
7.2 respectively).

##  4 Deromanization

Romanization is not necessarily reversible with simple rules due to
information loss. Therefore, previous work on romanized machine translation
has focused on source-side romanization only (Du and Way, 2017; Wang et al.,
2018; Aqlan et al., 2019; Briakou and Carpuat, 2019; Gheini and May, 2019). We
argue that romanization can also be applied on the target side, followed by an
additional deromanization step. This step can be performed by a character-
based Transformer (Vaswani et al., 2017) that takes data romanized with uroman
or uconv as input and is trained to map it back to the original script. We
provide more details on our deromanization systems in Appendix A.2.

##  5 Experimental Setup

###  5.1 Data

We use OPUS-100 (Zhang et al.,
2020)333https://github.com/EdinburghNLP/opus-100-corpus, an English-centric
dataset that includes parallel data for 100 languages. It provides up to 1
million sentence pairs for every X-EN language pair as well as 2,000 sentence
pairs for development and testing each. There is no overlap between any of the
data splits across any of the languages, i.e. every English sentence occurs
only once.

|  | script | related to | # sentence pairs
---|---|---|---|---
pretraining | Arabic (ar) | Arabic | he, mt | 1,000,000
languages | German (de) | Latin | yi | 1,000,000
French (fr) | Latin | - | 1,000,000
Russian (ru) | Cyrillic | sh | 1,000,000
| Chinese (zh) | Simpl. Han | - | 1,000,000
(a) | Amharic (am) | Ge’ez | - | 71,222
Marathi (mr) | Devanagari | - | 21,985
Tamil (ta) | Tamil | - | 198,927
(b) | Hebrew (he)* | Hebrew | ar | 50,000
Yiddish (yi) | Hebrew | de | 7,718
(c) | Maltese (mt)* | Latin | ar | 100,000
Serbo-Croatian (sh)* | Latin | ru | 98,421
Table 1: Overview of all languages, the script they are written in, other
languages in this set they are closely related to (considering lexical
similarity) and number of X$\leftrightarrow$EN sentence pairs. (*) means
artificial low-resource settings were created.

We pretrain our multilingual models on 5 high-resource languages that cover a
range of different scripts {AR, DE, FR, RU, ZH} $\leftrightarrow$ EN. For our
transfer learning experiments, we choose 7 additional languages that are
either:

  * (a)

Not closely related to any of the pretraining languages and written in an
unseen script, e.g. Marathi is not related to any of our pretraining languages
and written in Devanagari script.

  * (b)

Closely related to a pretraining language and written in an unseen script,
e.g. Yiddish is related to German and written in Hebrew script.

  * (c)

Written in Latin script but closely related to a pretraining language in non-
Latin script, e.g. Maltese is related to Arabic and written in Latin script.

Our selection of low-resource languages covers a wide range of language
families and training data sizes. Table 1 gives an overview of the selected
languages.

###  5.2 Model Descriptions

We use nematus444https://github.com/EdinburghNLP/nematus (Sennrich et al.,
2017) to train our models and SacreBLEU555BLEU+case.mixed+lang.XX-XX+numrefs.1
+smooth.exp+tok.13a+version.1.4.2 (Post, 2018) to evaluate them. We compute
statistical significance with paired bootstrap resampling (Koehn, 2004) using
a significance level of 0.05 (sampling 1,000 times with replacement from our
2,000 test sentences). Our subword vocabularies are computed with byte pair
encoding (Sennrich et al., 2016) using the SentencePiece implementation (Kudo
and Richardson, 2018). We use a character coverage of 0.9995 to ensure the
resulting models do not consist of mostly single characters.

Bilingual Baselines: We follow the recommended setup for low-resource
translation in Sennrich and Zhang (2019) to train our bilingual baselines for
the low-resource pairs (original script). For our bilingual low-resource
models, we use language-specific vocabularies of size 2,000.

Pretrained multilingual models: We pretrain three multilingual standard
Transformer Base machine translation models (Vaswani et al., 2017): One keeps
the original, non-Latin script for Arabic, Russian and Chinese (orig). The
others (uroman and uconv) apply the respective romanization to these parent
languages. We follow Johnson et al. (2017) for multilingual training by
prepending a target language indicator token to the source input. For our
pretrained models, we use a shared vocabulary of size 32,000. An overview of
our model hyperparameters is given in Appendix A.1.

Finetuning: We finetune our pretrained models independently for every low-
resource language X. For finetuning on a child X$\leftrightarrow$EN pair, we
use the same preprocessing as for the respective parent, i.e. we keep original
script, use uroman, or use uconv for romanization. We reuse 250,000 sentence
pairs from the original pretraining data and oversample the
X$\leftrightarrow$EN data for a total of around 650,000 parallel sentences for
finetuning. This corresponds roughly to a 3:2 ratio which helps to prevent
overfitting. We early stop on the respective X$\leftrightarrow$EN development
set. For finetuning, we use a constant learning rate of 0.001. The remaining
hyperparameters are identical to pretraining.

###  5.3 Vocabulary Transfer

For our transfer baseline without romanization, we merge our bilingual
baseline vocabulary with that of the parent model following previous work (Aji
et al., 2020; Kocmi and Bojar, 2020). First, we align subwords that occur in
both vocabularies. Next, we assign the remaining subwords from the bilingual
baseline vocabulary to random unused positions in the parent vocabulary. With
uroman, we can reuse the parent vocabulary as is. uconv, however, may produce
unseen diacritics, which can result in a small number of unseen subwords. If
that is the case, we perform the same vocabulary replacement for these
subwords as for the vocabulary with the original script.

##  6 Results

###  6.1 Does Romanization Hurt Translation?

To study the effects of information loss from romanization, we compare the
translation quality of our three pretrained multilingual models. To minimize
the impact of deromanization, we only discuss X$\rightarrow$EN directions for
languages with non-Latin scripts. The results are presented in Table 2.
Whether romanization hurts the translation quality depends largely on the
language pair. For example, for ZH$\rightarrow$EN, both romanization tools
perform worse than the model trained on original scripts. This is in line with
our previous discussion: Even though uconv keeps tonal information, there is
still more ambiguity compared to using Chinese characters. The model trained
with uconv romanization consistently outperforms uroman. This indicates that
it is more important to minimize information loss than to maximize subword
sharing.

| orig | uroman | uconv
---|---|---|---
ar-en | 37.4 | 36.3 | 37.4
ru-en | 33.3 | 33.5 | 34.1
zh-en | 39.5 | 37.0 | 39.2
Table 2: X$\rightarrow$EN BLEU scores (Papineni et al., 2002) of the
multilingual pretrained models trained on original scripts (orig), romanized
with uroman and uconv. Best systems (no other being statistically
significantly better) marked in bold.

An additional effect of using romanization, and thus being able to reuse the
subword segmentation model during transfer, is that compression rates are
worse than for dedicated segmentation models (see Table 3). The resulting
longer sequences with potentially suboptimal subword splits may also have a
negative influence on translation quality.

| orig | uroman (%) | uconv (%)
---|---|---|---
ar | 67.7 | \+ 2.2 | \+ 9.9
de | 97.8 | \- 0.5 | \- 0.8
fr | 131.7 | \- 0.4 | \- 0.6
ru | 91.5 | \+ 3.3 | \- 0.2
zh | 54.1 | \+ 98.9 | \+ 156.6
am | 113.0 | \+ 70.4 | \+ 83.1
he | 40.3 | \+ 17.6 | \+ 20.1
mr | 42.4 | \+ 36.8 | \+ 35.4
mt | 176.5 | \- 1.9 | \- 1.4
sh | 168.0 | \- 4.7 | \- 5.5
ta | 138.3 | \+ 20.1 | \+ 22.3
yi | 54.2 | \+ 12.4 | \+ 39.5
Table 3: Average number of subwords per sentence with original script data
(orig) and % relative change after romanization (uroman and uconv). Original
script data is segmented with a shared subword segmentation model for
{AR,DE,EN,FR,RU,ZH} and language-specific models for low-resource languages.
For uroman and uconv, all languages are segmented using a shared model for
{AR,DE,EN,FR,RU,ZH}, romanized with the respective tool.

###  6.2 Can We Restore the Original Script?

Table 4 compares our character-based Transformers to uconv’s built-in, rule-
based deromanization. Relying on uconv’s built-in deromanization is not
optimal. First, it does not support mappings back into all scripts. Second,
the performance of built-in uconv deromanization varies with the amount of
“script code-switching”, e.g. due to hyperlinks or email addresses. Character-
based Transformers can learn to handle mixed script and outperform uconv’s
built-in deromanization.

Our models can reconstruct the original script much better from uconv data
than from uroman. This is not surprising considering that uroman causes more
information loss and ambiguity. As a shallow measure of the ambiguity
introduced, we can compare the vocabulary size (before subword segmentation):
With romanization, the total number of types in our training sets decreases on
average by 10% for uconv and by 14% for uroman.

Preliminary experiments with artificial low-resource settings (Appendix B.1)
showed that additional training data can improve deromanization but it
performs well even with very small amounts of training data (10,000
sentences). This shows that our proposed character-based Transformer models
are powerful enough to learn a mapping back to the original script as much as
this is possible, given the increased ambiguity. This finding is supported by
concurrent work showing that character-based Transformers are well-suited to a
range of string transduction tasks (Wu et al., 2020).

| built-in | learned
---|---|---
| uconv | uconv | uroman
ar | 92.7 | 98.1 | 94.6
ru | 94.9 | 99.2 | 98.8
zh | - | 96.7 | 94.0
am | - | 99.7 | 97.8
he | 98.7 | 99.7 | 96.9
mr | 79.1 | 99.3 | 97.6
ta | 72.9 | 98.3 | 98.0
yi | 49.6 | 96.9 | 89.6
Table 4: chrF scores of the deromanization to the original script. Best
systems marked in bold.

###  6.3 Transfer to Low-Resource Languages

Table 5 shows the results from our experiments on transfer learning with
romanization. Romanizing non-Latin scripts is not always useful. For low-
resource languages that use an unseen script but are not related to any of the
pretraining languages (a), the performance degrades for uroman and is not
statistically significantly different for uconv. The extremely low BLEU score
for EN$\rightarrow$AM shows another problem with uroman romanization: uroman
ignores the Ethiopic word space character which increases the distance between
translation and reference.

However, for languages that are related to a pretraining language with a
different script (groups (b) and (c)), there is an added benefit of using
romanization. The statistically significant improvement of uconv over uroman
strengthens our claim that it is important to keep as much information as
possible from the original script when mapping to Latin script. Despite
potential information loss from romanization and error propagation from
deromanization, our results show that romanization has merit when applied to
related languages that can profit from a greater vocabulary overlap.

|  |  | transfer from
---|---|---|---
|  |  | multilingual parent
|  | base | orig | uroman | uconv
(a) | am-en | 14.4 | 16.2 | 16.5 | 16.0
en-am | 12.7 | 13.7 |  06.5 | 14.3
mr-en | 34.3 | 45.0 | 43.4 | 42.8
en-mr | 25.7 | 33.4 | 33.2 | 33.0
ta-en | 21.9 | 29.3 | 29.0 | 29.2
en-ta | 13.5 | 21.5 | 21.0 | 22.4
avg imp | - | \+ 6.1 | \+ 4.5 | \+ 5.9
(b) | yi-en |  06.9 | 22.5 | 24.9 | 28.9
en-yi |  09.5 | 12.0 | 20.7 | 19.7
he-en | 22.8 | 28.6 | 28.5 | 29.0
en-he | 21.1 | 24.5 | 25.2 | 26.6
avg imp | - | \+ 6.8 | \+ 9.8 | \+ 11.0
(c) | mt-en | 46.5 | 59.1 | 59.5 | 59.5
en-mt | 35.6 | 45.0 | 45.2 | 45.3
sh-en | 40.1 | 55.5 | 56.3 | 56.7
en-sh | 33.8 | 52.1 | 52.3 | 53.7
avg imp | - | \+ 13.9 | \+ 14.3 | \+ 14.8
Table 5: BLEU scores of the bilingual baselines (no transfer learning) and
finetuned models using original scripts (orig), romanized with uroman and
uconv. Average improvement over bilingual baseline is shown per group of
languages. Best systems (no other being statistically significantly better)
marked in bold.

##  7 Conclusion

We analyzed the value of romanization for transferring multilingual models to
low-resource languages with different scripts. While we cannot recommend
romanization as the default strategy for multilingual models and transfer
learning across scripts because of the information loss inherent to it, we
find that it benefits transfer between related languages that use different
scripts. The uconv romanization tool outperforms uroman because it preserves
more information encoded in the original script and consequently causes less
information loss. Furthermore, we demonstrated that romanization can also be
successful on the target side if followed by an additional, learned
deromanization step. We hope that our results provide valuable insights for
future work in transfer learning and practical applications for low-resource
languages with unseen scripts.

## Acknowledgements

We thank our colleagues Anne, Annette, Duygu, Jannis, Mathias, Noëmi and the
anonymous reviewers for their helpful feedback. This work was funded by the
Swiss National Science Foundation (project MUTAMUR; no. 176727).

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##  Appendix A Model Details

###  A.1 Multilingual Pretrained Models

We train multilingual Transformer Base machine translation models (Vaswani et
al., 2017) with 6 encoder layers, 6 decoder layers, 8 heads, an embedding and
hidden state dimension of 512 and a feed-forward network dimension of 2048. We
regularize our models with a dropout of 0.1 for the embeddings, the residual
connections, in the feed-forward sub-layers and for the attention weights.
Furthermore, we apply exponential smoothing of 0.0001 and label smoothing of
0.1. We tie both our encoder and decoder input embeddings as well as the
decoder input and output embeddings (Press and Wolf, 2017). All of our
multilingual machine translation models are trained with a maximum token
length of 200 and a vocabulary of size 32,000.

For optimization, we use Adam (Kingma and Ba, 2015) with standard
hyperparameters and a learning rate of 0.0001. We follow the Transformer
learning schedule described in (Vaswani et al., 2017) with a linear warmup
over 4,000 steps. Our token batch size is set to 16,348 and we train on 4
NVIDIA Tesla V100 GPUs. All models were trained using the implementation
provided in nematus (Sennrich et al., 2017) using early stopping on a
development set with patience 5.

###  A.2 Character-Based Deromanization

We train character-based Transformer Base machine translation models (Vaswani
et al., 2017). To achieve character-level deromanization, we do not make any
changes to the architecture. We simply change the input format such that every
character is separated by spaces. The original space characters are replaced
by another character that does not occur in the training data ($\diameter$).
The following example shows the parallel training data for learned
deromanization:

uroman source: C H t o $\diameter$ t a m $\diameter$ d a l s h e ?
---
uconv source: Č t o $\diameter$ t a m $\diameter$ d a l ’ š e ?

Ч т о $\diameter$ т а м $\diameter$ д а л ь ш е ? “What’s next?”

We use a maximum sequence length of 1,200 since character-level sequences are
much longer than subword-level sequences. Our vocabularies are made up of all
characters that occur in the respective training data. All other parameters
are set as for multilingual pretraining described in Appendix A.1.

##  Appendix B Supplementary Results

###  B.1 Effect of Data Size on Deromanization

Figure 1: chrF scores of deromanization models trained on 1%, 10% and 100% of
the total data (corresponding to 10,000, 100,000 and 1,000,000 parallel
sentences). Results compare romanization with uroman and uconv for Arabic,
Russian and Chinese.

| uroman
---|---
| 1% | 10% | 100%
en-ar | 20.3 | 21.6 | 21.7
en-ru | 26.5 | 27.9 | 28.5
en-zh | 38.8 | 41.6 | 41.9

| uconv
---|---
| 1% | 10% | 100%
en-ar | 21.2 | 21.8 | 21.7
en-ru | 27.9 | 28.2 | 29.3
en-zh | 40.2 | 41.8 | 41.9

Table 6: EN$\rightarrow$X BLEU scores of the multilingual pretrained models
after deromanization. Deromanization models were trained on 1%, 10% and 100%
of the total data (corresponding to 10,000, 100,000 and 1,000,000). Best
systems (no other being statistically significantly better) marked in bold.

Figure 1 shows the influence of the training data size on the chrF score
between the deromanized test set and the original script test set. Additional
data can improve deromanization models, especially for languages such as
Chinese, where a mapping back to the original script is difficult to learn due
to the information loss from romanization.

We analyze how deromanization quality affects the BLEU score of deromanized
translations. This is shown in Table 6. We find that the deromanization models
for uroman are more affected by an extreme low-resource setting. For uconv,
deromanization models trained on smaller data sets show less performance loss
compared to using full data. It is notable that training uconv deromanization
models only on 100,000 sentences has almost no effect on the BLEU score for
EN$\rightarrow$AR and EN$\rightarrow$ZH. For EN$\rightarrow$RU, there is a
loss of 1.1 BLEU points compared to training on 100% of the data. Looking at
the deromanization outputs for EN$\rightarrow$RU, we found that deromanization
models trained on less data could not handle “script code-switching” as well
as the models trained on full data.

While these results show that additional training material can improve
deromanization, they do not mean that romanization on the target side cannot
be used in low-resource machine translation settings. First, our results in
Section 6.3 have shown that romanization on the target side can bring
improvements even if deromanization models cannot perfectly reconstruct the
original script. Second, it will often be possible to find additional
monolingual data to improve deromanization models.


# Discovery of Dipolar Chromospheres in Two White Dwarfs††thanks: Dedicated to
Prof. Tom Marsh (1961–2022)

J. Farihi1, J. J. Hermes2, S. P. Littlefair3, I. D. Howarth1, N. Walters1, S.
G. Parsons3
1Department of Physics and Astronomy, University College London, London WC1E
6BT, UK
2Department of Astronomy & Institute for Astrophysical Research, Boston
University, Boston 02215, USA
3Department of Physics and Astronomy, University of Sheffield, Sheffield S3
7RH, UK
E-mail<EMAIL_ADDRESS>
###### Abstract

This paper reports the ULTRACAM discovery of dipolar surface spots in two cool
magnetic white dwarfs with Balmer emission lines, while a third system
exhibits a single spot, similar to the prototype GD 356. The light curves are
modeled with simple, circular, isothermal dark spots, yielding relatively
large regions with minimum angular radii of $20\degr$. For those stars with
two light curve minima, the dual spots are likely observed at high inclination
(or colatitude), however, identical and antipodal spots cannot simultaneously
reproduce both the distinct minima depths and the phases of the light curve
maxima. The amplitudes of the multi-band photometric variability reported here
are all several times larger than that observed in the prototype GD 356;
nevertheless, all DAHe stars with available data appear to have light curve
amplitudes that increase toward the blue in correlated ratios. This behavior
is consistent with cool spots that produce higher contrasts at shorter
wavelengths, with remarkably similar spectral properties given the diversity
of magnetic field strengths and rotation rates. These findings support the
interpretation that some magnetic white dwarfs generate intrinsic
chromospheres as they cool, and that no external source is responsible for the
observed temperature inversion. Spectroscopic time-series data for DAHe stars
is paramount for further characterization, where it is important to obtain
well-sampled data, and consider wavelength shifts, equivalent widths, and
spectropolarimetry.

###### keywords:

stars: evolution— stars: magnetic field— white dwarfs

##  1 Introduction

The origin of magnetism in white dwarf stars is an outstanding astrophysical
puzzle more than a half century old, but recent and ongoing developments are
now shedding light on this fundamental, and still poorly understood aspect of
stellar evolution. The first signatures of white dwarf magnetism resulted from
the detection of circular polarization in spectra that were quasi-featureless
or with unidentified absorption bands (Kemp et al., 1970; Angel & Landstreet,
1971; Landstreet & Angel, 1971) that were later understood to be shifted
hydrogen and (neutral) helium lines, mostly consistent with centered or offset
dipole field geometries (Kemic, 1974; Garstang, 1977; Wickramasinghe & Martin,
1979). A summary of magnetic white dwarf research over the first several
decades can be found in two published reviews (Wickramasinghe & Ferrario,
2000; Ferrario et al., 2015).

One of the key developments was the recognition that magnetic white dwarfs are
nearly exclusively found as isolated stars, or in cataclysmic variables
(Liebert et al., 2005). This empirical finding led to the hypothesis that
fields are generated during common envelope evolution (Tout et al., 2008;
Nordhaus et al., 2011; Belloni & Schreiber, 2020), a process that may function
effectively for stars, brown dwarfs, and giant planets that are engulfed
during the post-main sequence (Farihi et al., 2011; Kissin & Thompson, 2015;
Guidarelli et al., 2019). And while fast-spinning and massive magnetic white
dwarfs are known, and thus consistent with a stellar merger origin (Ferrario
et al., 1997b; García-Berro et al., 2012; Kilic et al., 2021; Williams et al.,
2022), it is also clear that magnetism, high remnant mass, and rapid rotation
are far from tightly correlated (Ferrario & Wickramasinghe, 2005; Brinkworth
et al., 2013).

It has been suspected for decades that cooler white dwarfs are more often
found to be magnetic (Liebert & Sion, 1979; Liebert et al., 2003). However,
luminosity and sensitivity biases exist, where the coolest white dwarfs
essentially require metal pollution to detect Zeeman splitting (Kawka &
Vennes, 2014; Hollands et al., 2015; Bagnulo & Landstreet, 2019). Despite
these uncertainties, the possibility that magnetic fields first emerge in cool
and isolated white dwarfs is intriguing, as substantial cooling is necessary
for core crystallization, which has been hypothesized to be a source of an
internal dynamo powered by the liquid-solid phase separation at the core
boundary (Isern et al., 2017). In this scenario, magnetic field generation is
decoupled from external sources of mass and angular momentum, but
nevertheless, all else being equal, more rapidly rotating remnants should have
stronger fields.

In a pioneering effort to overcome the aforementioned biases, and determine
the actual frequency of magnetism as a function of white dwarf
characteristics, Bagnulo & Landstreet (2021) carried out a nearly complete
census of ($N\approx 150$) white dwarfs within 20 pc. This volume-limited
survey used sensitive circular spectropolarimetry and resulted in the first
unbiased study of white dwarf magnetism, where the principal findings can be
summarized as follows.

  1. 1.

All spectral classes have similar incidences of magnetism, regardless of
atmospheric composition.

  2. 2.

The field strength distribution is uniform over four orders of magnitude from
40 kG to 300 MG.

  3. 3.

Magnetism is detected more frequently in white dwarfs with higher than average
mass.

  4. 4.

White dwarfs with cooling ages younger than 0.5 Gyr – prior to core
crystallization – are rarely magnetic.

  5. 5.

There is no evidence of field strength decay over time.

It is within this background of recent developments that emerged the
relatively new and small class of DAHe white dwarfs (D: degenerate, A: Balmer
lines strongest, H: magnetic line splitting, e: emission). The prototype is GD
356, an isolated $T_{\rm eff}\approx 7500$ K star with Balmer emission lines
split in a $B\approx 13$ MG field. There are deep, multi-wavelength, non-
detections that yield stringent upper limits on an X-ray corona, ongoing
accretion, and low-mass companions (Greenstein & McCarthy, 1985; Ferrario et
al., 1997a; Weisskopf et al., 2007). This apparently single white dwarf has a
1.927 h rotation period, based on a nearly sinusoidal light curve that is well
modeled by single dark spot, whose size is consistent with that of the
magnetic and heated region (Ferrario et al., 1997a; Brinkworth et al., 2004).
These enigmatic properties led to the hypothesis that, analogous to the
Jupiter-Io system, the relatively cool white dwarf surface could be heated by
Ohmic dissipation of a current loop set up by the orbital motion of a
conducting planet (Li et al., 1998; Wickramasinghe et al., 2010); referred to
as the unipolar inductor model.

GD 356111Previously thought to have a helium-rich atmosphere (Bergeron et al.,
2001; Limoges et al., 2015). was the only known DAHe white dwarf for 35 years,
until 2020 when second and third cases were reported (Reding et al., 2020;
Gänsicke et al., 2020). In addition to their shared spectral morphology and
strong magnetism implied from Zeeman splitting, these three cool white dwarfs
with emission lines all share commonalities with some magnetic white dwarfs:
relatively rapid rotation, masses only slightly above average, and no evidence
for low-mass stellar or substellar (detached) companions. A detailed time-
series study of the prototype has shown that (i) the spin period is stable
over two decades, with no other independent frequency signals as would be
expected from a unipolar inductor, (ii) the emission line strength oscillates
in anti-phase with the broad-band stellar brightness, and (iii) so far, DAHe
stars share a tightly correlated set of effective temperatures and
luminosities (Gänsicke et al., 2020; Walters et al., 2021). This clustering is
potentially related to core crystallization and magnetic field diffusion
toward the stellar surface (Ginzburg et al., 2022).

This paper reports detailed light curves for three DAHe white dwarfs: the
second known example, SDSS J125230.93$-$023417.7 (Reding et al. 2020;
hereafter SDSS J1252), and two recently identified members of this class, LP
705-64 and WD J143019.29$-$562358.3 (Reding et al. 2023; hereafter WD J1430).
Two of the three stars reveal light curves with asymmetric dimming events that
are $180\degr$ out-of-phase, and thus consistent with dipolar star spots.
These data are inconsistent with a unipolar inductor model, and instead
support the generation of intrinsic chromospheres in some isolated, magnetic
white dwarfs. The observations and data are discussed in Section 2, the time-
series analysis is presented in Section 3, followed by a summary and
discussion.

##  2 Observations

This study focuses on light curves and the resulting periodicities of three
DAHe white dwarfs, using both ground- and space-based photometric monitoring
as described below.

###  2.1 Target properties and selection

SDSS J1252 is the second discovered example of a DAHe white dwarf, reported to
have emission lines split in a $B\approx 5$ MG field, and with a sinusoidal
light curve dominated by a period of 317.3 s (Reding et al., 2020). The fast
rotation of this star makes it an attractive target for high-cadence
photometric monitoring from the ground, with a goal to obtain a detailed light
curve. LP 705-64 and WD J1430 are two newer members of the DAHe spectral class
with indications from TESS data that their full spin cycles could each be
readily covered in a single night of ground-based photometry (Reding et al.,
2023). The initial observational goals were similar to those achieved by
Walters et al. (2021), to establish robust ephemerides against which future
period changes might be investigated (e.g. within a unipolar inductor and
orbiting planet model), and to constrain the nature of the emitting and
magnetic regions.

Table 1: Chronological summary of ULTRACAM observing runs. Target | Observing | $t_{\rm exp}$ | Coverage | Filters
---|---|---|---|---
| Dates | (s) | (min) |
SDSS J1252 | 2021 Apr 06 | 10.05 | 57 | $ugr$
| 2021 Apr 08 | 10.35 | 86 | $ugr$
| 2021 Aug 20 | 10.05 | 24 | $ugr$
LP 705-64 | 2021 Aug 17 | 8.05 | 73 | $ugi$
| 2021 Aug 19 | 8.05 | 77 | $ugr$
| 2021 Aug 20 | 8.05 | 146 | $ugr$
WD J1430 | 2022 Apr 26 | 6.06 | 277 | $ugi$
| 2022 Jun 05 | 6.33 | 176 | $ugi$

###  2.2 ULTRACAM observations

Figure 1: Approximately 1 h of ULTRACAM $g$-band light curves for SDSS J1252,
each taken on a different night. The data are plotted as observed in sequence,
each light curve normalized, offset vertically by $\pm 0.1$, and shifted
horizontally to exhibit the same photometric phase. Visual inspection reveals
that adjacent minima are unequal in depth, which is also true but more subtle
for adjacent maxima. These real-time observations were the first indication
that SDSS J1252 has two star spots, 180°out-of-phase, and the true rotation
period is twice as long as the 317.3 s previously reported by Reding et al.
(2020).

All three stars were observed with ULTRACAM, a frame-transfer CCD imaging
camera (24 ms dead time between exposures; Dhillon et al. 2007) that is
permanently mounted on the 3.6 m NTT telescope at the La Silla Observatory in
Chile. The instrument has three independent channels that enable the use of
independent filters simultaneously, and data were taken with filters similar
to standard $u$, $g$, and one of $r$ or $i$ bandpasses, but with higher
throughput. In each case, the blue channel was co-added every three frames to
improve the effective signal-to-noise on the target. The observation details,
including exposure times (same as the cadences for ULTRACAM), and durations of
the resulting light curve segments, are summarized in Table 1.

Images were corrected for bias and flat fielded with normalized sky images
obtained during evening twilight (taken in a continuous spiral to remove
stars). Differential brightnesses were measured relative to field stars with
dedicated software222https://github.com/HiPERCAM/hipercam. using photometric
apertures that were typically scaled to $2\times$ the mean full width at half
maximum of the stellar profiles for each exposure. The sky annuli were fixed
to span the region 8.75–15.75 arcsec from the stars, where a clipped mean was
used to determine the background. For all stars in all observations, the same
sets of comparison stars were used to generate light curves, consisting of two
or three stars in the $gri$ frames, and one to two stars in the $u$-band
images.

Light curves were constructed by dividing the science target flux by the sum
of the comparison star fluxes, and normalizing the result. Measurement errors
were propagated from the aperture photometry, by summing in quadrature the
fractional flux errors of all stars measured for a given light curve. All
ULTRACAM times were converted to Barycentric Julian Day (BJD) using
Barycentric Dynamical Time (TDB), following Eastman et al. (2010).

###  2.3 TESS data

Data for each of the three DAHe targets are available from TESS (Ricker et
al., 2015), and were downloaded from the MAST archive, where the pdcsap
processed light curves were retained for analysis. Time stamps were were
corrected to BJD = TESS_BJD + 2457000.

LP 705-64 (= TIC 136884288) was observed in Sector 30, while data were
collected for WD J1430 (= TIC 1039012860) within Sector 38, and for SDSS J1252
(= TIC 953086708) during Sector 46. All three stars have 120 s cadence
observations. These data were further cleaned of NaN flux entries, but with no
other processing based on data quality flags, yielding light curves that
retained between 80 and 90 per cent of their pdcsap array values. Lastly,
outliers beyond $\pm 5\upsigma$ of the local time average (or phase average)
flux were removed, which were fewer than five points in total for each source.

It is worth noting that these data are not all equally useful in subsequent
analysis. The following TESS benchmarks summarize their relative quality: SDSS
J1252 has $G=17.5$ mag, a mean flux of $19.2\pm 5.3$ e- s-1 (28 per cent
scatter); LP 705-64 has $G=16.9$ mag, a mean flux of $38.0\pm 5.6$ e- s-1 (15
per cent scatter), while WD J1430 has $G=17.4$ mag, a mean flux of $9.6\pm
5.4$ e- s-1 (73 per cent scatter), and lies within the Galactic plane.

##  3 Time-series analysis and results

All light curves were analyzed using period04 (Lenz & Breger, 2005), where a
Lomb-Scargle periodogram was constructed using ULTRACAM data, TESS pdcsap
light curves, or a combination of the two datasets, with a goal to identify
that which produces the most precise ephemerides for each target. Monte Carlo
simulations, run within period04, were used to determine errors in frequency
and phase for the strongest periodogram peak for each star and set of light
curves, then propagated to determine the error in $T_{0}$ corresponding to
photometric minimum. The frequency and phase were allowed to vary
independently during the simulations to determine errors, which were typically
repeated 1000 times.

###  3.1 SDSS J1252

For SDSS J1252, there are sufficient ULTRACAM data to uniquely determine the
photometric period and provide an improved ephemeris. Light curves cover more
than 30 epochs of its previously reported 317.3 s periodicity (frequency 272.3
d-1, Reding et al. 2020), with an observational baseline of 136 d, spanning
several $10^{4}$ cycles at this frequency. In Figure 1 are shown the first and
second $g$-band light curves obtained for this white dwarf, from which can be
discerned that there are two distinct set of minima (and maxima), each
manifesting every 317.3 s, and thus revealing an actual photometric period of
634.6 s.

The ULTRACAM $g+r$ co-added light curves were analyzed using data from all
three observing runs. The resulting best periodogram is plotted in Figure 2,
where the strongest peak is identical to the frequency reported in the
discovery paper; however, there is a second outstanding signal near 408 d-1.
This secondary peak is not as well-determined as the 272.3 d-1 signal, but
these two frequencies appear to have a near an exact ratio of 3:2.
Additionally, the periodogram also reveals a weak-amplitude peak at roughly
816 d-1, and the ratio of these three frequencies is 6:3:2.

Figure 2: Periodogram of SDSS J1252 based on three nights of ULTRACAM data
using co-added, $g+r$-band light curves, with amplitudes plotted in grey. The
data were bootstrapped 10 000 times to determine the amplitude above which
represents a 0.1 per cent chance a signal is spurious. This false alarm
amplitude of 0.007734 is delineated by the green dotted line, where only the
strongest peak is higher. However, the two frequencies with the largest
periodogram peaks have a near exact ratio of 3:2, and a weaker third peak is
consistent with a frequency that is an integer multiple of both lower
frequencies. For a fixed rate of stellar rotation, and despite a lack of
significant periodogram power (amplitude), this result indicates the
fundamental frequency is 136.150 d-1. This is half of the frequency with the
largest periodogram signal, and is consistent with two distinct, out-of-phase
star spots tracing the observed light curve morphology. The first six
harmonics of the fundamental are marked with blue triangles, where only those
showing noteworthy amplitude are numbered.

These periodicities are far shorter than any possible range of orbital signals
originating from non-degenerate companions, as the lowest-frequency
periodogram signal, at 272.3 d-1, corresponds to a Keplerian orbit near
$7\,R_{\star}$ (seven white dwarf radii), deep within the nominal Roche limit.
Only a compact object could survive at this orbital distance, such as those in
close but detached, double white dwarf binaries. And while there are a few
such systems known to have orbital periods comparable to the frequencies
exhibited by SDSS J1252, their light curves reveal ellipsoidal modulation
owing to tidal distortions in the primary white dwarfs (Kilic et al., 2011;
Brown et al., 2011; Burdge et al., 2019). Furthermore, these rare, deformed
degenerates are all helium-core white dwarfs less massive than 0.3 M⊙, and
thus significantly more prone to tidal distortion than SDSS J1252 and the DAHe
stars, which are considerably more compact (Walters et al., 2021; Reding et
al., 2020; Gänsicke et al., 2020).

Therefore, the light curve and resulting periodogram of SDSS J1252 are
interpreted as arising from a single star. It is reasonable to assume the
$T_{\rm eff}\approx 8000$ K white dwarf has a fixed spin period (no
differential rotation), and no stellar pulsations, as it is far from the
hydrogen atmosphere instability strip (Romero et al., 2022). The observed
signals are then interpreted as the 2nd, 3rd, and 6th harmonics of the stellar
rotation frequency. The periodogram signals and their amplitudes reflect the
fitting of sinusoids to the light curve, where the two highest have a 3:2
ratio in order to generate both the principal flux variation at 272.3 d-1, and
the alternating minima via the interference with the 408 d-1 frequency.

The revised stellar rotation period and associated uncertainty were determined
by dividing the 2nd harmonic frequency by two, yielding 136.15032(2) d-1
(equivalent to a period of 634.59273(9) s). Although essentially no amplitude
(or power) is seen in the periodogram at the frequency inferred to be the
fundamental, this is an expected consequence of the light curve morphology and
sinusoidal fitting (VanderPlas, 2018).

A similar analysis was attempted using the TESS light curve, both on its own
and in combination with ULTRACAM data. While the TESS 120 s cadence is
$2.6\times$ faster than the peak periodogram frequency for SDSS J1252, and
thus above the Nyquist rate, the data quality are relatively poor (Section
2.3). No time-series analysis utilizing TESS led to any improvement in
frequency or phase precision, and therefore all calculations for SDSS J1252
are based solely on the ULTRACAM observations.

###  3.2 LP 705-64 and WD J1430

TESS data were the initial means of identifying the stellar rotation rates in
these two DAHe white dwarfs (Reding et al., 2023). However, similar to as
observed in SDSS J1252, the ULTRACAM light curve of LP 705-64 exhibits two
unequal minima in a single cycle, and thus the period determined by TESS
represents one half its spin period (see Figure 3). For this source, the
ULTRACAM data alone do not span a sufficient number of cycles to determine the
photometric period with precision comparable to TESS. A significant
improvement in the TESS ephemeris is achieved using the combination of
ULTRACAM $g+r$ co-added light curves and TESS, resulting in a periodogram with
a single peak at 39.65325(3) d-1 [cf. 39.653(2) d-1; Reding et al. 2023], and
a corresponding higher precision in phase. However, the true spin period must
be calculated from this frequency by recognizing it is the 2nd harmonic of the
fundamental, which is 19.82662(1) d-1.

For WD J1430, the ULTRACAM light curves reveal a single maximum and minimum
with one of the largest amplitudes observed to date for a DAHe white dwarf
(5.8 per cent in the $g$ band). Similar to LP 705-64, there are insufficient
ULTRACAM data from which to derive a precise ephemeris for this source, and
thus the combination of ULTRACAM and TESS Sector 38 pdcsap data were utilized
for this goal. Initially, the analysis of these combined datasets improved the
precision of the periodogram frequency, but resulted in phase errors that were
larger than those based on TESS alone. Subsequently, these TESS data were re-
scaled (see Section 3.4) to more closely match those of the co-added
$g+i$-band ULTRACAM data, and the resulting analysis marginally improved the
uncertainty in phase. Ultimately for this star, the best constraints were
achieved by adding a third set of light curves into the periodogram analysis,
using full-frame data from TESS Sector 11, where fluxes were extracted based
on PSF-subtracted images following Han & Brandt (2023).

Figure 3: Normalized and phase-folded ULTRACAM light curves for all three
stars, where the blue points are $u$ band, the green points are $g$ band, and
the red points are $r$ or $i$ band. The time-series data have been folded on
the periods listed at the top of each panel, and have been re-sampled onto
regular grids. There are 80 phase bins for LP 705-64 and WD J1430 in all three
filters, while for SDSS J1252 there are 60 phase bins for $g$ and $r$, but
only 20 for the $u$ band (Section 3.3). These light curves highlight the
asymmetric, anti-phase modulation from two starspots in the case of both SDSS
J1252 and LP 705-64.

###  3.3 Light curve morphologies, ephemerides, spectral phases

Based on the preceding analysis, and to reveal light curve structures more
precisely as a function of phase, the ULTRACAM multi-band data were phase-
folded and binned using a weighted average onto regular grids. The resulting
light curves are shown in Figure 3, where there are 80 phase bins for LP
705-64 and WD J1430 in all three channels. In the case of SDSS J1252, the
short spin period indicates that a single ULTRACAM frame has a phase width of
0.016 in the green and red channels, but 0.048 in the blue channel (owing to
three co-adds). For this reason, the light curves of SDSS J1252 were re-
sampled into 60 phase bins in $g$ and $r$, but only 20 bins in $u$ band.

The folded light curves for SDSS J1252 and LP 705-64 both exhibit alternating
minima that are indicative of two distinct star spots $180\degr$ out-of-phase
during rotation. While this behavior is not novel among magnetic white dwarfs,
there appear to be only a few documented examples of white dwarf light curves
where dipolar spots are suggested or required (Hermes et al., 2017; Kilic et
al., 2019; Pshirkov et al., 2020). In contrast, the majority of magnetic white
dwarf light curves seem to be broadly consistent with sinusoidal (single spot)
morphologies (Brinkworth et al., 2013), including the prototype DAHe star GD
356 (Walters et al., 2021). However, it should be noted that incomplete phase
coverage and modest photometric precision can inhibit the detection of subtle
light curve features (e.g. the discovery light curve of SDSS J1252, and the
TESS light curve of LP 705-64; Reding et al. 2020, 2023).

To calculate accurate ephemerides based on the best precision achieved here,
$T_{0}$ was chosen from an ULTRACAM light curve located nearest to the middle
of the temporal coverage for each star, and where a feature could be
unambiguously identified as a true photometric minimum. The periodogram
analysis of the preceding sections then results in the following best
ephemerides for all three DAHe white dwarfs, where zero phase corresponds to
actual photometric minimum, and the periods are accurate and precise
determinations of their spins:

BJDTDB (SDSS J1252) | = 2459313.809921(6) | \+ 0.007344823(1) $E$
---|---|---
BJDTDB (LP 705-64) | = 2459444.92339(6) | \+ 0.05043723(3) $E$
BJDTDB (WD J1430) | = 2459696.8239(3) | \+ 0.05999529(3) $E$

As mentioned earlier, the ephemeris of SDSS J1252 is based solely on ULTRACAM,
whereas those of LP 705-64 and WD J1430 are based on the combination of TESS
and ULTRACAM. From these ephemerides, forward and backward extrapolations can
be made and compared with published, time-varying spectra of LP 705-64 and WD
J1430, but in the case of SDSS J1252 there is insufficient time resolution to
compare its spectroscopic variations in phase with photometry (Reding et al.,
2020, 2023).

Starting with the more straightforward case of WD J1430 which exhibits a
single spot, the photometric minimum (phase 0) occurred at BJD${}_{\rm
TDB}=2459049.056\pm 0.001$ nearest the two epochs of the published
spectroscopy. This notably falls close to halfway in time between the two
spectra plotted and described by Reding et al. (2023) as ’emission’ and
’absorption’. Specifically, and taking the reported epochs of observation at
face value, these spectra correspond to photometric phases $0.720\pm 0.007$
and $0.221\pm 0.007$, respectively, and thus both occur close to the average
stellar flux. While these two spectral phases are reported as potentially
representing a maximum and minimum magnetic field strength, this
interpretation seems uncertain, especially if other spectral phases exhibit
weaker emission or absorption, where Zeeman splitting is not a robust
diagnostic.

Superficially interpreting these spectral phases of WD J1430 as the highest
and lowest field strength would be somewhat inverse to that observed for the
prototype DAHe GD 356, where there are multiple, full-phase coverage
observations using both spectroscopy and spectropolarimetry. For this well-
studied case, the magnetic field variations, both from the observed parallel
component and using Zeeman splitting, display a peak and trough near phases
0.3 and 0.8, respectively, from photometric minimum (Walters et al., 2021).
For WD J1430, existing data may not probe the magnetic field with sufficient
sensitivity or phase coverage, and hence these comparative results should be
considered preliminary at best.

In the case of LP 705-64, the situation is more complex. Depending on the spot
sizes, one might expect two minima and maxima in both equivalent width and
magnetic field variations, one pair associated with each spot. However, there
are only two epochs of spectroscopy plotted and described by Reding et al.
(2023), and here again a comparison must be considered not only preliminary,
but possibly inapt for the aforementioned reasons. Again taking the published
epochs at face value, and where the the deeper of the two light curve minima
is zero phase, the spectrum shown with the broader Zeeman splitting
corresponds to photometric phase $0.048\pm 0.001$. The two reported spectral
epochs were chosen as to be separated by exactly one half spin cycle (Reding
et al., 2023), so that further interpretation would reflect the selection.

While the updated photometric ephemeris is sufficient to predict precise spin
phases for spectroscopic observations of LP 705-64, their potential
correlation is not yet straightforward. It has not yet been demonstrated that
high and low Zeeman splitting might be in-phase with photometric extrema (cf.
GD 356 Walters et al. 2021). The two published spectra may not represent
precise peak behavior, and there may be some uncertainty in the epoch dates
reported. Measurements of both equivalent width and magnetic field strength at
all rotational phases would eliminate these ambiguities. The sparse set of
published spectroscopic measurements of DAHe white dwarfs, currently prevents
a more robust correlation of photometric and spectroscopic phases.

Table 2: Multi-wavelength variability amplitudes $A_{\uplambda}$ in per cent
flux.

SDSS J1252 | LP 705-64 | WD J1430 | GD 356
---|---|---|---
$u$: $5.72\pm 0.23$ |  $u$: $4.48\pm 0.17$ |  $u$: $6.62\pm 0.14$ |  $u$: $1.50\pm 0.06$
$g$: $4.97\pm 0.06$ |  $g$: $3.92\pm 0.06$ |  $g$: $5.78\pm 0.04$ |  $g$: $1.22\pm 0.04$
$r$: $2.99\pm 0.06$ |  $r$: $2.38\pm 0.05$ | … |  $V$+$R$: $0.81\pm 0.02$
… | … |  $i$: $3.84\pm 0.07$ | …
$T$: $2.05\pm 0.30$ |  $T$: $1.60\pm 0.17$ | … |  $T$: $0.62\pm 0.02$

Note. The $u$-band amplitude for GD 356 was obtained from a light curve taken
at the WHT using PF-QHY on 2020 Jun 21. $A_{T}$ is the flux variation
amplitude in the TESS band, where is no corresponding entry for WD J1430.
While both SDSS J1252 and WD J1430 are similarly faint ($G\approx 17.5$ mag)
and thus near the limit of what TESS can observe, the latter source has a
target-to-total aperture flux of only 0.03 (cf. 0.89 for SDSS J1252, 0.56 for
LP 705-64, and 0.87 for GD 356; Walters et al. 2021). Thus the TESS amplitude
for WD J1430 is likely unreliable.

###  3.4 Multi-band light curve amplitudes

Figure 4: The multi-band photometric variability amplitudes of the three DAHe
white dwarfs with ULTRACAM light curves, together with similar measurements
for the prototype GD 356 (Walters et al., 2021). The adopted central
wavelengths are 3600, 4700, 6200, and 7500 Å for $ugri$ (Fukugita et al.,
1996; Gunn et al., 2006), and 7900 Å for TESS (Ricker et al., 2015). All data
were analyzed in a uniform manner for this plot and the Table 2 values, using
period04 as described in Section 3.4. The light curve amplitudes are
normalized to the $g$-band value for a given star, with a horizontal offset of
$\pm 100$ Å applied to separate the data points, and errors given for the
sinusoidal fits to individual bandpass data.

To better understand the nature of the spots and their associated magnetic
regions, multi-band light curves for DAHe white dwarfs were used to calculate
the photometric amplitude for each star in each observed bandpass. For the
three stars observed by ULTRACAM as well as GD 356, this was done by taking
only the strongest signal in the best periodogram for each star (Figures 2 and
3), and determining the sinusoidal amplitude for each light curve in each
bandpass at that frequency. In this way, all light curve amplitudes are
evaluated by their strongest sinusoidal components, including those stars with
little or no periodogram power at their true rotation frequency. Light curve
amplitudes and uncertainties were determined using period04, using fixed
frequencies and Monte Carlo simulations.

Table 2 lists the multi-band, photometric amplitudes for the four DAHe white
dwarfs with available data, the three stars reported here and the prototype GD
356 (Walters et al., 2021). It is interesting to note that the amplitude of
photometric variation is relatively small in GD 356 (on the order of 1 per
cent; Brinkworth et al. 2004), compared to the newer DAHe stars with several
percent variations in their light curves. Although there is not yet any
published multi-band photometry of SDSS J1219, its light curve amplitude is
around 3 per cent in the $B$ band (Gänsicke et al. 2020; roughly halfway
between the $u$ and $g$ filters), and thus at least double that of GD 356 at
similar wavelengths. It is also possible that WD J161634.36+541011.51 (Manser
et al. 2023; hereafter WD J1616) has a photometric amplitude comparable to the
strongest found here. These are likely the results of observational bias,
which enhances their detection as variables in surveys such as Gaia and ZTF
(e.g. Guidry et al. 2021).

It should be noted that WD J1430 is positioned in a crowded field at Galactic
latitude $\upbeta<4$ degr, and the TESS fluxes are dominated by other sources
in the photometric aperture (pipeline keyword crowdsap = 0.032). This pipeline
metric implies that only 3.2 per cent of the flux in the extracted aperture is
likely from the white dwarf, and subsequently the extracted flux has been
dramatically reduced to recover a more accurate stellar brightness in the
pdcsap light curve (Stumpe et al., 2012). While the ULTRACAM observations
independently confirm the stellar spin frequency identified by periodogram
analysis of the TESS light curve, the pipeline fluxes are simply too noisy
(see Section 2.3) and likely offset significantly from the true mean flux.
Thus, no reliable variability amplitude can be deduced for the TESS bandpass
(see footnote to Table 2).

The relative strengths of the photometric variations in DAHe stars appear to
follow a trend as a function of wavelength, with increasing amplitudes towards
the blue. Figure 4 plots the strengths of the multi-band variability for the
four white dwarfs, where the photometric amplitude for each bandpass is
plotted relative to the $g$ band for each star. Three of the four stars have
data in $ugr$ (or similar) bandpasses, and all three exhibit a relatively
tight correlation in their amplitude ratios as a function of these three
wavelength ranges. Three stars have reliable TESS amplitudes where again the
same behavior is evident, and suggesting a phenomenon associated with this
emerging spectral family. Based on the narrow range of $T_{\rm eff}$ among
DAHe white dwarfs, Figure 4 implies their spots have similar spectral
properties. This indication is remarkable given the range of DAHe rotation
periods and especially magnetic field strengths.

To date, only relatively weak Balmer features have been detected blueward of
H$\upbeta$ in any DAHe star, and yet the photometric variability remains
strongest at shorter wavelengths. This is consistent with the previous finding
that the photometric variability arises from changes in the stellar continuum,
and not from fluctuations in the Balmer emission lines (Walters et al., 2021).

In $2002-2003$, the $V$-band (5500 Å) light curve amplitude of the GD 356 was
recorded as 0.2 per cent (Brinkworth et al., 2004). But in 2020 the
photometric variations observed using an SDSS $g$-band filter (4700 Å), and a
$V+R$ filter (6200 Å) were found to be $4\times$ $-$ $6\times$ higher (Table
2). It thus seems possible that the starspot has evolved during this time
frame; however, all six emission features within H$\upalpha$ and H$\upbeta$
seem consistent over at least 35 years (Greenstein & McCarthy, 1985; Ferrario
et al., 1997a; Walters et al., 2021).

###  3.5 Simple spot modeling

Table 3: Spot modeling parameter ranges and step sizes. Parameter | Range | Step size
---|---|---
$i$ | $10\degr-90\degr$ | $10\degr$
$\uptheta$ | $10\degr-90\degr$ | $2\degr$
$\upalpha_{1}$ | $20\degr-80\degr$ | $5\degr$
$\upalpha_{2}$ | $5\degr-20\degr$ | $5\degr$
$f_{1}(T)$ | 0.99-0.48 | 0.01
$f_{2}(T)$ | 0.99-0.10 | 0.01
Figure 5: Illustrative spot models fitted to ULTRACAM light curves. The upper
panels plot the $g+r$-band data for LP 705-64, which exhibits the more extreme
depth change between its two minima (cf. SDSS J1252 in Figure 3), and for
which the simple models calculated here are moderately deficient. In the top
panel are shown the RMS difference minimum models over a broad range of
(fixed) spot sizes, all of which result in the highest inclination (or
colatitude). The reason these maximum inclination models fit the data best is
illustrated, by contrast, in the middle panel, where a representative model at
lower inclinations is shown for a (fixed) spot radius of $40\degr$. In these
middle panel models, the resulting secondary minima are decreasing in depth,
but this causes an increasing shift in the phase positions of the predicted
flux maxima (toward phase 0.5). The bottom panel plots the $g+i$-band light
curve of WD J1430 with analogous models that demonstrate a spot radius smaller
than around $20\degr$ is insufficient to reproduce the observed variations.

A basic set of spot models and corresponding light curves were constructed to
better constrain the observed stellar surfaces as a function of rotation, with
a particular motivation towards those stars where two spots appear to be
necessary. Each white dwarf was treated as a $T_{\rm eff}=8000$ K blackbody,
with one or two circular, isothermal spots whose single temperature is
controlled by a scaling factor $f(T)$. Where required by the light curve
morphology, two identical spots were placed on the surface at antipodal
points. The other model parameters are the inclination ($i$) of the stellar
rotation axis to the observer, the spot colatitude ($\uptheta$), and the spot
angular radius ($\upalpha$). It is commonly acknowledged that such models are
potentially degenerate when the values of $i$ and $\uptheta$ are interchanged
(e.g. Wynn & King 1992).

For each star, a grid of models was generated with the following parameter
ranges and step sizes as given in Table 3. For small angular radii
($\upalpha<20\degr$) the spot temperature range was expanded because in order
to reproduce a fixed photometric amplitude, the smallest spots must be
darkest. The root mean square (RMS) difference between the model and observed
fluxes in a given bandpass as a function of spin phase was computed and used
to identify a best-fit model for each $\upalpha$, although in practice there
is nearly always a range of models that yield similarly satisfactory results.

While the modeling is relatively simple, a few basic results emerge. Small
spots with $\upalpha\lesssim 15\degr$ cannot generate a sufficiently large
photometric amplitude for any of the three white dwarfs with ULTRACAM data,
but otherwise the spot size is mostly unconstrained. Beyond this small angular
size threshold, the combination of adjustable geometry and spot temperature
permits sufficient model flexibility to achieve comparably good fits within a
range of the other three parameters. Nevertheless, the spots must be at least
modestly large; in terms of solid angle, covering several per cent or more of
one hemisphere. In the case of WD J1430, which has the largest photometric
variability amplitude, only spots with $\upalpha\geq 20\degr$ can reproduce
the observed flux changes.

However, the models with the smallest RMS differences for the light curves
with two minima exhibit clear shortcomings. The resulting inclinations (or
colatitudes) tend toward $90\degr$, and consequently the model fits have
equally deep, light curve minima. These fitted parameters are driven in the
direction of maximum inclination (or colatitude) because, while the lower $i$
(or $\uptheta$) solutions can reproduce a more shallow secondary minimum as
observed, this particular shape exhibits light curve maxima whose phase
positions are shifted towards the secondary minimum. Examples of these
modeling outcomes are illustrated in Figure 5.

Despite these limitations, the modeling demonstrates that the basic geometry
of antipodal spots is essentially correct. However, in the context of the
simple model assumptions, it is unlikely that there is a centered but tilted,
symmetric dipolar arrangement for the dual-spotted stars. Instead, the spots
may not be circular, and where two opposing spots are necessary each may be
distinct in size, shape, or temperature; alternatively, the spots may be in a
dipolar configuration that is offset from the rotational center of the star.

##  4 Discussion and Summary

The discovery of DAHe white dwarfs whose light curves require two spots in a
basic dipolar configuration, now totaling at least three systems (Manser et
al., 2023), is a modest breakthrough in their characterization. It raises the
immediate question of whether all DAHe stars have dual spots, which manifest
as light curves with either one or two minima, depending on the viewing angle
and spot orientation. As of this publication, there are now just over two
dozen known DAHe stars, but where only six have robustly measured light curves
(Reding et al., 2020; Gänsicke et al., 2020; Reding et al., 2023; Manser et
al., 2023). There is a seventh DAHe candidate with a well-measured TESS light
curve, SDSS J041246.85+754942.26, but which currently lacks any type of
magnetic field indication or upper limit (Tremblay et al., 2020; Walters et
al., 2021).

Of these seven objects, three of their light curves exhibit two photometric
minima, and four are consistent with a single minimum. With such small numbers
and weak constraints on spot properties, a statistical assessment of the
inferred viewing geometry is not possible, but the data to date are likely
consistent with all class members having dipolar magnetic and spotted regions.
If the simple modeling performed here is any indication, it may be that
magnetic (spot) axes must be highly inclined towards the viewer (or
equivalently have similar colatitudes) for both spots to transit, i.e. have
ingress and egress as opposed to being partly visible at all times.

The detection of photometric variability in GD 356 yielded limited results on
its spot properties, where the size of the temperature-contrast surface was
assumed to be identical to that of the magnetic region inferred from modeling
of spectropolarimetry, around $40\degr$ (Brinkworth et al., 2004). Otherwise
the modeling followed the same assumptions as those described in Section 3.5,
and the mostly sinusoidal light curve was ultimately fitted to two sets of
models, one with a dark spot near the rotational pole (low $\uptheta$) viewed
at high inclination, and a second viewed near the axis of rotation (low $i$),
but with high colatitude; an example of the degeneracy between $i$ and
$\uptheta$. In the case that GD 356 has antipodal spots, in the former
scenario the secondary spot can remain hidden from the observer at all spin
phases, and in the latter scenario, it is possible for both spots to be partly
visible at all times. If the prototype does indeed have two spots, the
previous photometric modeling would disfavor the latter orientation, as it
would result in some light curve impact on from both spots.

As with the DAHe prototype, it is tempting to co-identify the sizable spots
with their magnetic and chromospherically active regions (Ferrario et al.,
1997a; Brinkworth et al., 2004). In one such picture, the spots are dark and
magnetic regions underlying the chromospheric activity, so that the Balmer
emission lines are at maximum brightness when the stellar continuum yields
photometric minimum (Walters et al., 2021). This behavior may also be seen in
SDSS J1212 and SDSS 1219 (Reding et al., 2020; Gänsicke et al., 2020), but
insufficient phase coverage and sampling prevent any certainty at present, and
equivalent widths have not been determined for those stars. The results here
for LP 705-64 and WD J1430 are currently ambiguous for similar reasons, and
owing to additional complications.

Interestingly, time-series spectroscopy for SDSS J1252, WD J1430, and WD J1616
suggest that their emission lines may effectively disappear at some phases
(Reding et al., 2020, 2023; Manser et al., 2023), presumably when a spot or
spots (and associated magnetic region) are out of view or have minimum
visibility. However, as discussed in Section 3.3, this is likely an
oversimplified picture; with the exception of GD 356, there is a distinct lack
of magnetic field determinations across the entire spin phases of DAHe white
dwarfs, and Zeeman splitting may be an ineffective tool for weak or transient
emission features.

At present, empirical metrics associated with published, DAHe time-series
spectroscopy are sparse, and it would be ideal for observers to provide both
equivalent widths and central wavelengths for emission features over at least
one full cycle with sufficient sampling. In contrast to magnetic field
strength estimates, which may not be possible to measure at all spin phases
via Zeeman splitting if magnetic regions rotate in and out of view, only the
phase behavior of equivalent width has been robustly characterized, and only
in the prototype (Walters et al., 2021). It is thus essential that full
spectroscopic phase coverage of DAHe white dwarfs is carried out with these
measurements in mind, and where spectropolarimetry will be more sensitive to
magnetic field strength, particularly when emission or absorption features are
weak.

The dual-spotted nature of at least three DAHe white dwarfs has direct bearing
on the hypothesis that a heated region can be caused by star-planet
interactions such that a current loop is dissipated in one region of the star
(e.g. the unipolar inductor Li et al. 1998; Wickramasinghe et al. 2010). If
such planetary interactions are in fact taking place within the strong
magnetospheres of DAHe white dwarfs, they are unlike the interactions that
lead to unipolar, Jupiter-Io footprint mechanisms (Goldreich & Lynden-Bell,
1969). Going forward, models that require the presence of closely orbiting and
interacting planets, which at present lack no empirical support in
observations of DAHe stars, should require strong evidence to be re-
considered. Given the lack of additional periodic signals and the compelling
evidence of DAHe white dwarf clustering in the HR diagram (Walters et al.,
2021; Reding et al., 2023; Manser et al., 2023), an intrinsic mechanism is the
most likely source for the spotted regions and chromospheric activity.

## Acknowledgements

J. Farihi is grateful to the Laboratory for Atmospheric and Space Physics at
the University of Colorado Boulder, and the Kavli Institute for Theoretical
Physics at the University of California, Santa Barbara, for hosting during
extended visits, and to J. S. Pineda for an illuminating discussion on the
nature of chromospheres in sun-like and low-mass stars. The authors
acknowledge the European Southern Observatory for the award of telescope time
via program 105.209J. J. Farihi acknowledges support from STFC grant
ST/R000476/1 and National Science Foundation Grant No. NSF PHY-1748958. S. P.
Littlefair acknowledges the support of the STFC grant ST/V000853/1. N. Walters
has been supported by a UK STFC studentship hosted by the UCL Centre for
Doctoral Training in Data Intensive Science. S. G. Parsons acknowledges the
support of a STFC Ernest Rutherford Fellowship. For the purpose of open
access, the authors have applied a creative commons attribution (CC BY)
license to any author accepted manuscript version arising. This paper includes
data collected by the TESS mission, which is funded by the NASA Explorer
Program.

## Data Availability

ULTRACAM data are available on reasonable request to the instrument team,
while TESS data are available through the Mikulski Archive for Space
Telescopes.

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