# Transfer Learning for CSI-based Positioning with Multi-environment Meta-
learning

Anastasios Foliadis12, Mario H. Castañeda Garcia1, Richard A. Stirling-
Gallacher1, Reiner S. Thomä2  1Munich Research Center, Huawei Technologies
Duesseldorf GmbH, Munich, Germany
2Electronic Measurements and Signal Processing, Technische Universität
Ilmenau, Ilmenau, Germany
{anastasios.foliadis, mario.castaneda<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Utilizing deep learning (DL) techniques for radio-based positioning of user
equipment (UE) through channel state information (CSI) fingerprints has
demonstrated significant potential. DL models can extract complex
characteristics from the CSI fingerprints of a particular environment and
accurately predict the position of a UE. Nonetheless, the effectiveness of the
DL model trained on CSI fingerprints is highly dependent on the particular
training environment, limiting the trained model’s applicability across
different environments. This paper proposes a novel DL model structure
consisting of two parts, where the first part aims at identifying features
that are independent from any specific environment, while the second part
combines those features in an environment specific way with the goal of
positioning. To train such a two-part model, we propose the multi-environment
meta-learning (MEML) approach for the first part to facilitate training across
various environments, while the second part of the model is trained solely on
data from a specific environment. Our findings indicate that employing the
MEML approach for initializing the weights of the DL model for a new unseen
environment significantly boosts the accuracy of UE positioning in the new
target environment as well the reliability of its uncertainty estimation. This
method outperforms traditional transfer learning methods, whether direct
transfer learning (DTL) between environments or completely training from
scratch with data from a new environment. The proposed approach is verified
with real measurements for both line-of-sight (LOS) and non-LOS (NLOS)
environments.

###### Index Terms:

Wireless Positioning, CSI, Fingerprinting, Transfer Learning, Meta-learning,
Deep Learning

##  I Introduction

Accurate user positioning is considered one of the crucial requirements for
next generation communication networks [1, 2, 3, 4]. The ability to localize
users is essential for enhancing the efficiency of applications such as
industrial internet of things (IIOT). One of the main enablers of high
accuracy radio positioning in a wireless network is the support of multiple
antennas and larger bandwidths which are made possible by current and future
communication networks.

User positioning in wireless networks has mainly been geometry driven [5]. The
channel state information (CSI) between a user equipment (UE) and base station
(BS) is first estimated and from that, some parameters are extracted such as
time of arrival (ToA) and angle of arrival (AoA). These parameters reflect in
some way the geometry of the environment and therefore can be used to estimate
the position of the user. More recently though, there has been an increased
focus on using deep learning (DL) to aid or perform positioning by leveraging
the ability to collect large amounts of data [6, 7, 8, 9, 10]. Positioning
methods that are based on DL are able to learn and extract the relevant
positioning information from the CSI even in scenario where the classical
geometry driven positioning approaches fail [11].

One of the most prominent user positioning methods that leverage the
capabilities of DL and NNs is called fingerprinting [12, 13, 14]. The idea
behind those methods is to train the NN to exploit the information that is
embedded in a multi-path channel between the UE and the BS which can be
considered a unique fingerprint of the user’s location. It is generally
divided into two phases. In the offline (or training) phase, a database of CSI
fingerprints, measured for different UE positions, along with the respective
position is created. This database is used to train a NN to estimate the
position. During the deployment phase, the trained NN is used for estimating a
UE’s position by mapping the CSI of the UE to an estimated position.

Positioning methods that are based on CSI fingerprints using DL approaches
have shown very promising results achieving sub-meter accuracy in some cases
[15, 16]. Although, despite their advantages against classical methods they
are limited in some scenarios. Namely, a NN trained with data from a
particular environment fails when employed into some other environment [17].
Similarly some change in the environment may cause the model to completely
fail because it was trained on data that are not up to date [18, 19].

In the context of DL the method of taking a DL-model that was trained on data
from some particular distribution, e.g., wireless propagation environment, and
applying it in some other distribution is called transfer learning [20]. The
paradigm under which the models are designed and trained in such a way to make
transfer learning more efficient is called meta-learning [21]. In [22], a
method called Model-Agnostic Meta-Learning (MAML) [23] was used to improve
generalizability when the training data is limited. The authors of [22] also
propose a variation called task biased MAML (TB-MAML) to further improve
transfer learning when the target environment is known.

In [24] a 2-part approach to conventional DL methods of radio-positioning was
proposed. This approach is inspired by classical positioning methods where
generally a 2-step approach is employed, i.e., environment independent
parameter estimation in the first step and environment dependent parameter
combination for position estimation in the second step. The main idea
presented in [24] is to train the first part of the DL model to extract
environment independent features of the environment, by training on data from
multiple source environments. This way, the environment independent part of
the DL model can be reused in a target environment and reduce the amount of
data required from that environment (i.e., transfer learning is improved).

In this work we expand upon the work of [24] by including uncertainty
estimation during initial training of the first part of the DL model as well
as during training of the two-part model in target environment. We show that
regardless by including the uncertainty estimation during initial training the
transferring capabilities of the model are increased. Additionally, in
contrast to [24] where only line of sight (LOS) simulated propagation
environments were used, we verify our results using real CSI measurements from
both LOS and non-line of sight (NLOS) propagation environments. Lastly, we
propose to use gradual unfreezing[25] during target training in the target
environment. Gradual unfreezing is a technique which unfreezes layers of deep
neural network models from top to bottom during training. It is used to more
efficiently train the model for the target environment and further exploit the
information that is embedded in the environment independent first part of the
model, especially for small amount of target environment CSI samples.

##  II System Model

We consider an uplink setup with $N_{R}$ antennas at a BS and a single antenna
at the UE. The UE transmits a reference signal on $N_{C}$ subcarriers within
an orthogonal frequency division multiplexing (OFDM) symbol. The received
uplink signal is used to estimate the CSI between UE and the BS. The $i$-th
estimated channel over the $N_{C}$ subcarriers between the $k$-th receive
antenna and the UE can be described as:

$\boldsymbol{h}^{k}_{i}=[h_{0,i}^{k},h_{1,i}^{k},...,h_{N_{C}-1,i}^{k}]^{\text{T}}\in\mathbb{C}^{N_{C}}.$
(1)

Accordingly, the measured channel between UE and BS’s antennas that forms the
basis of the CSI fingerprint can be described as:

$\tilde{\boldsymbol{H}}_{i}=[\boldsymbol{h}^{0}_{i},\boldsymbol{h}^{1}_{i},...,\boldsymbol{h}^{N_{R}-1}_{i}]\in\mathbb{C}^{N_{R}\times
N_{C}}.$ (2)

Eventually, the CSI fingerprint which is used as input to the DL model is
determined by stacking the real and imaginary parts of the matrix
$\tilde{\boldsymbol{H}}_{i}$ resulting in a 3-dimensional matrix
$\boldsymbol{H}_{i}\in\mathbb{R}^{N_{R}\times N_{C}\times 2}$. Subsequently,
the CSI fingerprints $\boldsymbol{H}_{i}$ along with position labels
$\boldsymbol{p}_{i}=[x_{i},y_{i}]\in\mathbb{R}^{2}$ are stored and used to
train a DL-model for the user positioning task.

We also consider measurements from $N_{B}$ different environments, with each
environment associated with its own dataset
$\\{(\boldsymbol{H}_{n,i},\boldsymbol{p}_{n,i})\\}_{i=1}^{M_{N}}$ where
$M_{n}$ are the number of measurements of the $n$-th environment.
Subsequently, the CSI fingerprints of the $n$-th task $\boldsymbol{H}_{n,i}$
along with position labels
$\boldsymbol{p}_{n,i}=[x_{n,i},y_{n,i}]\in\mathbb{R}^{2}$ are used to train
the parameters $\epsilon_{n}$ of a DL model
$f_{\epsilon_{n}}(\boldsymbol{H}_{n,i})$ to give an estimate
$\tilde{\boldsymbol{p}}_{n,i}=[\tilde{x}_{n,i},\tilde{y}_{n,i}]\in\mathbb{R}^{2}$
of the $i$-th position of the user in the $n$-th environment based on the CSI
of the user.

Normally, for DL based positioning with CSI fingerprints the DL model $f_{n}$
is optimized with respect ot the mean squared error (MSE):

$\mathcal{L}_{\text{MSE}}(f_{\epsilon_{n}})=\mathcal{L}_{x,n}+\mathcal{L}_{y,n},$
(3)

where $\mathcal{L}_{x,n}=E[|\tilde{x}_{n,i}-x_{n,i}|^{2}]$,
$\mathcal{L}_{y,n}=E[|\tilde{y}_{n,i}-y_{n,i}|^{2}]$ and $E[\cdot]$ denotes
the mean value over the samples in the training set. However, using the MSE
criterion does not evaluate the uncertainty of the predictions.

In radio positioning using DL there exist two types of uncertainties. The data
dependent uncertainty which is called aleatoric uncertainty reflects the
uncertainty that a measurement has about the positioning task and the
epistemic uncertainty which reflects the uncertainty of the model[26]. An
example of a measurement with high aleatoric uncertainty is a CSI measurement
with low receive SNR. As this type of uncertainty is data-dependent, it can be
learned from the available data [26, 27]. In order to estimate it, the loss
that has to be minimized is the negative log-likelihood loss (NLL):

$\mathcal{L}_{\text{NLL}}(f_{\epsilon_{n}})=\frac{1}{2(\sigma_{n,i}^{x})^{2}}\mathcal{L}_{x,n}+\frac{1}{2(\sigma_{n,i}^{y})^{2}}\mathcal{L}_{y,n}+\log({\sigma_{n,i}^{x}}{\sigma_{n,i}^{y}}),$
(4)

Figure 1: Multi-environment training over $N_{B}$ environments Figure 2:
Training of target environment

where $\sigma_{n,i}^{x}$ and $\sigma_{n,i}^{y}$ are the aleatoric
uncertainties for the outputs $x_{n,i}$ and $y_{n,i}$ respectively. When
training a model using the NLL loss the output of the neural network has to be
modified to include the learned aleatoric uncertainty
$\boldsymbol{\sigma}_{n,i}=[\sigma_{n,i}^{x},\sigma_{n,i}^{y}]$. Essentially,
the aleatoric uncertainty reflects the inherent uncertainty that each
measurement has about the positioning task, e.g., due to noise. This is in
contrast to the epistemic uncertainty which reflects the uncertainty of the
model itself, e.g., due to low amount of training data. The reader is referred
to [26] for more information about the specific uncertainty types.

We propose the Multi Environment meta-Learning (MEML) approach which is shown
in Fig. 1. The basic idea is to consider a two-part structure for the DL
model, where the first part is trained with data from all $N_{B}$ environments
while the second part is trained only with data specific to one environment.
This way, the first part $\phi_{\theta}(\cdot)$ is shared across all $N_{B}$
environments and the second part $g_{\epsilon_{n}}(\cdot)$ is environment
specific and trained on data only from the $n$-th environment. Essentially we
define model $f_{\theta,\epsilon_{n}}(\boldsymbol{H}_{n,i})$ such that
$f_{\theta,\epsilon_{n}}(\boldsymbol{H}_{n,i})=g_{\epsilon_{n}}(\phi_{\theta}(\boldsymbol{H}_{n,i}))$.
By employing this type of training, the parameters of the common part $\theta$
are the same regardless of the environment, i.e., they are environment
independent while the parameters $\epsilon_{n}$ depend on the $n$-th
environment. The training is carried out by minimizing the sum of the losses
for each environment:

$\min_{\theta,\epsilon_{n}}\sum_{n}\mathcal{L}(f_{\theta,\epsilon_{n}}).$ (5)

When training with the MSE loss, the uncertainty estimate
$\boldsymbol{\sigma}_{n,i}$ is not included in the model. It is important to
point out that the aim is to make the function $\phi_{\theta}(\cdot)$
environment independent, i.e., as the function that extracts position relevant
parameters from the CSI. Although the function $\phi_{\theta}(\cdot)$ is
common across different environments, the features $\boldsymbol{z}_{n,i}$ are
specific to each environment. This is because the distribution of CSI
fingerprints $\boldsymbol{H}_{n,i}$, varies with the $n$-th environment. The
environment dependent features $\boldsymbol{z}_{n}$ are then combined in an
environment specific way using the $g_{\epsilon_{n}}(\cdot)$ function.

Ultimately, the goal of the training of the common part is to acquire an
environment independent function which can later be used in a new target
environment to aid transfer learning, as shown in Fig 2. The trained part can
enable a more efficient minimization of the loss in the new unseen environment
$t$ with dataset
$\\{(\boldsymbol{H}_{t,i},\boldsymbol{p}_{t,i})\\}_{i=1}^{M_{t}}$:

$\min_{\theta,\epsilon_{t}}\mathcal{L}(f_{\theta,\epsilon_{t}}),$ (6)

where
$f_{\theta,\epsilon_{t}}=g_{\epsilon_{t}}(\phi_{\theta}(\boldsymbol{H}_{t,i}))$
and $g_{\epsilon_{t}}(\cdot)$ is initialized with random weights before
training, while $\phi_{\theta}(\cdot)$ is initialized with the weights
obtained with the MEML training in the source environments. In our simulations
we also consider both MSE loss (3) and the negative log likelihood loss (4)
for the target environment $t$.

Additionally, we propose to use a gradual unfreezing method [25] of training
when transferring the environment independent function to mitigate
catastrophic forgetting [17]. Since the weights of the function
$\phi_{\theta}(\cdot)$ are already trained, we freeze those layers when
transferring to the target environment $t$, therefore only the weights of the
function $g_{\epsilon_{t}}(\cdot)$ are determined with training in the new
environment. After this initial step, when there is no further improvement in
the performance of the overall DL-model the weights of the first part are
unfrozen and the learning rate is reduced. The training continues again until
it reaches its performance limit. This method of training ensures that the
first part of the model will preserve the knowledge that it has acquired from
the source environments while allowing some adaptation to the new environment.

For the models which are trained using the NLL loss we propose to use the
continuous ranked probability score (CRPS) metric [28] along with the mean
error distance between ground truth and estimation. Since these models
estimate not only the position but also the uncertainty of the estimation we
are interested in a metric that takes into account the reliability of the
uncertainty. In other words, a model should be able to identify the cases
where the position estimation error is high. The CRPS metric provides a way to
quantify the reliability of the uncertainty estimates by comparing the ground
truth value to a probability distribution (i.e., the position estimation and
it uncertainty). A description of the CRPS metric along with an example that
showcases its usefulness is presented in Appendix V.

##  III Simulation Setup

###  III-A Database Description

To evaluate our proposed approach we use the Dichasus channel measurements
described in [29]. The measurements were collected with a single-antenna UE
transmitting an OFDM signal in the uplink with a bandwidth of 50 MHz and with
a pilot sent at every tenth subcarrier out of 1024 subcarriers, i.e.,
$N_{C}=103$. The carrier frequency is $f_{c}=1.272\>\text{GHz}$. The ground
truth positions are measured with a tachymeter robotic total station, a very
precise instrument that tracks the robot’s antenna with a laser with at least
centimeter-level accuracy.

$\displaystyle{-12.5}$$\displaystyle{-10.0}$$\displaystyle{-7.5}$$\displaystyle{-5.0}$$\displaystyle{-2.5}$$\displaystyle{0.0}$$\displaystyle{2.5}$x
coordinate
(m)$\displaystyle{-14}$$\displaystyle{-12}$$\displaystyle{-10}$$\displaystyle{-8}$$\displaystyle{-6}$$\displaystyle{-4}$$\displaystyle{-2}$y
coordinate (m)ABCDBSsBlockerPossible UE positions

Figure 3: Layout of considered source environment. DICHASUS Industrial
environment [29]

The source environments are based on data that were collected at four antenna
arrays distributed on the corners of an industrial area shown in Fig. 3, where
it shown the area where the UE can lie as well as the 4 BS. Each of the
antenna arrays consists of a $4\times 2$ uniform rectangular array (URA) with
vertical and horizontal antenna spacing of half a wavelength. We consider the
wireless propagation environment between the UE and each of the BSs as the 4
source environments. Even though this is a single warehouse were all 4 BSs are
located, the propagation environment between UE and each of the BSs differs
and therefore they can be considered separate environments from the
perspective of the BS. This can be verified by training each a model with data
from one BS and then testing using data from another BS which results in a
huge decrease in performance. Furthermore, it can be seen from the figure that
the channels between each possible UE location and every BS include both LOS
and NLOS locations.

For the target environments we consider 2 different datasets based on
measurements with one BS in a lab room included in Dichasus measurements [29].
One of the datasets includes only LOS measurements while the other includes
both LOS and NLOS due to the placement of a blocker which results in a NLOS
between some of the UE’s positions and the BS. The measurement area for the
LOS environment is show in Fig. 4 while for the NLOS environment is shown in
Fig. 5. The measurements for both datasets were collected using one URA with
$8\times 4$ elements at the BS and a single-antenna UE. To align the
measurements with those from the source environments, we consider only the
measurements from a $4\times 2$ subarray within the $8\times 4$ array of the
BS in the lab room.

In our simulations we compare the different options that we believe would make
sense in a real deployment scenario which are shown in Fig. 6. Namely, the
goal would be to use the trained model and reuse the environment independent
part in a target scenario, which in our case are either the LOS or the NLOS
target environments. Additionally, depending on the requirements of the
deployment one may choose to train the model in the target environment either
using the MSE loss or the NLL loss. These options concern with the target
environment exclusively therefore we keep them constant and we compare the
various training approaches based on source environment settings. Namely we
keep constant the target environment (LOS or NLOS) and the target loss (MSE or
LLS) and we compare the impact of source loss (MSE or NLL) and the employed
transfer learning strategy. The different strategies are the MEML scheme with
$N_{B}=4$ source environments and direct transfer learning (DTL) which
involves training a model on one environment and re-training in the target
environment. MEML for $N_{B}=1$ is the same as DTL. Additionally, we include
the option of no initial training as baseline, i.e., when training the model
from scratch in the target environment.

$\displaystyle{-6}$$\displaystyle{-5}$$\displaystyle{-4}$$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$$\displaystyle{0}$x
coordinate
(m)$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle{3}$y
coordinate (m)BSPossible UE positions

Figure 4: Layout of considered LOS target environment. DICHASUS Indoor LOS lab
room [29] Figure 5: Layout of considered NLOS target environment. DICHASUS
Indoor NLOS lab room [29] Figure 6: Overview of different simulation options

###  III-B Neural Network and training

For the NN we consider a model that uses convolutional neural networks (CNNs)
to form the basis for the pooling block [30]. The considered DL model is shown
in Fig. 7 while the configuration of the considered Pooling Block is shown in
Fig. 8. The pooling block aims to downsample the input to reduce the
computational complexity in the network, while minimizing the information
loss, which has also been previously used in the context of AI radio
positioning [30]. Two pooling blocks are placed one after the other and are
followed by 3 dense layers with 128 neurons each and finally a dense layer
with 2 or 4 neurons for the output, depending on whether the MSE or the NLL
loss are used for training. The number of parameters per model are $185\>922$,
with $152\>640$ parameters in the first part (environment independent) and
$33\>282$ in the second part (environment dependent).

The number of training CSI samples for the source and target environments are
shown in Table I. For the source environment the number of samples in the
table refers to the number of CSI measurements for each BS. Since we don’t
consider a deployment scenario in the source environment we don’t use any test
samples in that environment. For both target environments we incrementally
increase the amount of training and validation samples we use from 20 percent
to 100 percent of all available samples shown in Table I, while the number of
test samples remains constant.

Additionally, we employ a K-Fold cross-validation method with $k=5$ folds to
reduce bias and provide a more accurate estimate of the model’s ability to
generalize [20]. In this approach, the dataset is partitioned into $k$ equally
sized, mutually exclusive subsets. The model is then trained $k$ times, each
time using $k-1$ as the training set and the remaining fold as the validation
set. This process ensures that each data point is used exactly once for
validation. The results from each fold are averaged to produce a single
estimation. K-fold cross-validation is beneficial in providing a reliable
measure of model accuracy and generalizability, as it utilizes all available
data for both training and validation.

As an evaluation metric we consider the mean error (ME) of the UE positioning
which is given by the euclidean distance between the estimated position and
the true position of the UE in the test set. For the models that learn the
aleatoric uncertainty through the NLL loss, we use the CRPS metric [28](see
Appendix V). The results presented in the next section show the average value
of the 5 folds.

TABLE I: Available Samples for each scenario Samples |  Source env. |  LOS target env. |  NLOS target env.
---|---|---|---
Training | $60\>000$ | $22\>234$ | $16\>928$
Validation | $6000$ | $5558$ | $4234$
Testing | N/A | $13\>131$ | $5001$
Figure 7: Considered DL model Figure 8: Pooling Block

##  IV Simulation Results

We test the effectiveness of the MEML approach by learning the function
$\phi_{\theta}(\cdot)$ with data from the source environment shown in Fig. 3
and using it to initialize the layers of a new DL models which are trained on
data from the LOS and NLOS target environments (Fig. 4 and 5 respectively). We
compare the results for different types of source loss (i.e., MSE or NLL).
Additionally, we compare the different transfer learning strategies. Namely we
compare the MEML approach to direct transfer learning (DTL) whereby a model
$f_{n}$ is trained on one environment only and then retrained on the target
environments and also to the case where a model is simply trained from scratch
with data from the target environments. We consider those comparisons for each
target environment type (LOS or NLOS) and for each target loss type (MSE or
NLL) as shown in Fig. 6. Lastly, for the best option of each of the target
environments and target environments loss types, we compare fine-tuning with
gradual unfreezing the model.

Firstly, we transfer the trained $\phi_{\theta}(\cdot)$ model to the LOS
environment using the MSE loss as target loss. The results are shown in Fig.
9. In the figure we see the mean error achieved in the target environment test
set for each source training loss and each transfer learning strategy.
Regardless of the number of training samples the MEML approach outperforms
training the model from scratch or DTL. We also see that there is a slight
benefit to training the MEML model using the NLL loss in the source
environments, especially for low amount of target training samples. The reason
for that is that the model learns better to weigh the important information of
each environment during training with data from the source environments.

Next, we compare the transfer learning capabilities of different
configurations for the LOS target environment when using the NLL loss as
target loss function. The results for the mean error in the LOS test set are
shown in Fig. 10. Similarly to when using the MSE loss function as target
loss, we see that the MEML method achieves better performance regardless of
the target training samples. Since when training using the NLL loss function
in the target environment we estimate the uncertainty of each position
estimate, in Fig. 11 we compare the reliability of the uncertainty estimates
of different configurations using the CRPS metric. We see that we can reach a
similar conclusion, namely that using the MEML not only benefits the position
estimate when transferring to a new environment, but also provides a more
reliable uncertainty estimation.

Now we present the same set of results for the NLOS target environment. In
fig. 12 we see the mean error at the test set of the NLOS environment for the
different source loss options and transfer learning strategies. In contrast to
the LOS target environment we see that DTL may be worse than training the
model from scratch in the NLOS target environment when the number of training
samples is small. The reason for that is that for small amount of samples the
model is not able to adapt to the target environment and leverage the
knowledge from the source environments. On the other hand, we again see the
benefit of the MEML approach when transferring to a new environment regardless
of the number of samples, and again there is a slight benefit to using the NLL
loss in the source environment.

$\displaystyle{5000}$$\displaystyle{10000}$$\displaystyle{15000}$$\displaystyle{20000}$$\displaystyle{25000}$Target
train
samples$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$$\displaystyle{0.8}$$\displaystyle{0.9}$$\displaystyle{1.0}$$\displaystyle{1.1}$Mean
error (m)Target loss: MSE, Target env.: LOSScratchSource loss: MSE, 1 source
env.Source loss: NLL, 1 source env.Source loss: MSE, 4 source env.Source loss:
NLL, 4 source env.

Figure 9: Mean Error when transferring to LOS env. using the MSE loss

$\displaystyle{5000}$$\displaystyle{10000}$$\displaystyle{15000}$$\displaystyle{20000}$$\displaystyle{25000}$Target
train
samples$\displaystyle{0.2}$$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$Mean
Error (m)Target loss: NLL, Target env.: LOSScratchSource loss: MSE, 1 source
env.Source loss: NLL, 1 source env.Source loss: MSE, 4 source env.Source loss:
NLL, 4 source env.

Figure 10: Mean Error when transferring to LOS env. using the NLL loss

$\displaystyle{5000}$$\displaystyle{10000}$$\displaystyle{15000}$$\displaystyle{20000}$$\displaystyle{25000}$Target
train
samples$\displaystyle{0.10}$$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$CRPSTarget
loss: NLL, Target env.: LOSScratchSource loss: MSE, 1 source env.Source loss:
NLL, 1 source env.Source loss: MSE, 4 source env.Source loss: NLL, 4 source
env.

Figure 11: CRPS when transferring to LOS env. using the NLL loss

In Fig. 13 and 14 we see the mean error and the CRPS metric when transferring
to NLOS target environment using the NLL target loss function. It is
interesting to see in this case that the MEML method when trained using the
NLL source loss shows a clear benefit when transferring with small amount of
target training samples. Interestingly, for small amount of training samples
the MEML method trained with NLL loss function is the only one that
outperforms training the model at the source environment from scratch.

Lastly, we evaluate the gradual unfreezing method of training, and compare it
to fine tuning for all MEML configurations.The biggest benefit of gradual
unfreezing comes for smaller amount of training samples since with small
amount of samples the risk of overfitting is higher. Gradual unfreezing avoids
overfitting and catastrophic forgetting of the knowledge that the first part
has acquired from the source environments. Therefore we only show the results
when using 20% of the total training samples from each target environment,
shown in Table I, i.e., 5558 for the LOS environment and 4232 for the NLOS
environment.

$\displaystyle{5000}$$\displaystyle{7500}$$\displaystyle{10000}$$\displaystyle{12500}$$\displaystyle{15000}$$\displaystyle{17500}$$\displaystyle{20000}$Target
train
samples$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$$\displaystyle{0.35}$$\displaystyle{0.40}$$\displaystyle{0.45}$$\displaystyle{0.50}$$\displaystyle{0.55}$Mean
Error (m)Target loss: MSE, Target env.: NLOSScratchSource loss: MSE, 1 source
env.Source loss: NLL, 1 source env.Source loss: MSE, 4 source env.Source loss:
NLL, 4 source env.

Figure 12: Mean Error when transferring to NLOS env. using the MSE loss

$\displaystyle{5000}$$\displaystyle{7500}$$\displaystyle{10000}$$\displaystyle{12500}$$\displaystyle{15000}$$\displaystyle{17500}$$\displaystyle{20000}$Target
train
samples$\displaystyle{0.2}$$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$Mean
Error (m)Target loss: NLL, Target env.: NLOSScratchSource loss: MSE, 1 source
env.Source loss: NLL, 1 source env.Source loss: MSE, 4 source env.Source loss:
NLL, 4 source env.

Figure 13: Mean Error when transferring to NLOS env. using the NLL loss

$\displaystyle{5000}$$\displaystyle{7500}$$\displaystyle{10000}$$\displaystyle{12500}$$\displaystyle{15000}$$\displaystyle{17500}$$\displaystyle{20000}$Target
train
samples$\displaystyle{0.10}$$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$CRPSTarget
loss: NLL, Target env.: NLOSScratchSource loss: MSE, 1 source env.Source loss:
NLL, 1 source env.Source loss: MSE, 4 source env.Source loss: NLL, 4 source
env.

Figure 14: CRPS when transferring to NLOS env. using the NLL loss

In table II we see the transferring capabilities of the MEML methods for both
MSE and NLL source training functions and for fine tuning and gradual
unfreezing. We see that the ME when using gradual unfreezing decreases
compared to all other transfer learning strategies. Specifically, when using
the MEML method trained using the NLL loss function there is a 30 percent
decrease in the positioning error compared to just training the model from
scratch in the target environment. We can reach similar conclusions when
looking the table III which compares the different transfer learning
configurations when training using NLL loss function in the LOS target
environment. The performance gain for both mean error and CRPS is more than 35
percent when using gradual unfreezing compared to training the model from
scratch in the target environment. Another way to look at the benefit of
gradual unfreezing is to compare it with the model which is trained from
scratch in Figures 9, 10 and 11. We see the model which is trained using
gradual unfreezing and the MEML transfer strategy needs only around one third
of the data to achieve the same performance.

Next, we see the effect of gradual unfreezing when training in the NLOS target
environment. In table IV we see the transferring capabilities of different
MEML methods and compare them to training a model from scratch. In contrast to
the LOS target environment here the MEML method trained using the MSE source
loss function and transferred by gradual unfreezing outperforms the MEML
method trained using the NLL source loss function albeit only slightly. The
benefit of MEML with gradual unfreezing is 20 percent compared to training a
model from scratch. Lastly we show the results when training using the NLL
loss function in the target environment. Both ME and CRPS show the benefit of
combining MEML with gradual unfreezing achieving more than 25 percent
improvement in performance compared to training the model from scratch.

As mentioned before, gradual unfreezing provides a big benefit for when a
small amount of target training samples for both LOS and NLOS target
environments. Therefore, we believe it can be beneficial in cases where
limited amount of samples is available.

##  V Conclusion

For positioning with CSI fingerprints, we propose a DL model consisting of two
parts, where the first part is trained to be environment independent while the
second part of the model depends on the specific environment. The proposed
MEML scheme is used to train a DL-model on multiple source environments and
the first part of the model, which is environment independent, is used to
initialize the weights of a model for a new unseen environment. We investigate
the performance of the MEML scheme when it is trained using the MSE loss or
the NLL loss and we compare it to the direct transfer learning (DTL) case,
i.e., training a model in some environment and fine tuning it in another
environment, and also to the case of training a model from scratch in the new
environment. We show, that regardless of the number of training samples, the
target environment loss function (MSE or NLL) and type of target environment
(LOS or NLOS) the MEML method outperforms the other methods. The evaluation is
done by means of mean euclidian error between position estimation and ground
truth and by use of CRPS metric which takes into account the reliability of
the uncertainty estimates of the model. We further propose to use a gradual
unfreezing scheme when training the MEML model in the target environment to
further improve its positioning accuracy and we achieve a benefit of at least
25 percent in all tested target environment configurations. We also show that
by combining gradual unfreezing with the MEML scheme we can reduce the amount
of samples used for training in a target environment by one third with no
performance reduction.

TABLE II: TL capabilities to LOS env. using MSE loss and 5558 Training Samples Source Loss |  TL type |  Target Training |  ME (m)
---|---|---|---
N/A | Scratch | Fine-tuning | 0.5345
MSE | MEML | Fine-tuning | 0.4292
NLL | MEML | Fine-tuning | 0.4074
MSE | MEML | Gradual Unfreezing | 0.3994
NLL | MEML | Gradual Unfreezing | 0.3796
TABLE III: TL capabilities to LOS env. using NLL loss and 5558 Training Samples Source Loss |  TL type |  Target Training |  ME (m) |  CRPS
---|---|---|---|---
N/A | Scratch | Fine-tuning | 0.6349 | 0.278
MSE | MEML | Fine-tuning | 0.5074 | 0.2627
NLL | MEML | Fine-tuning | 0.4716 | 0.2251
MSE | MEML | Gradual Unfreezing | 0.4114 | 0.1963
NLL | MEML | Gradual Unfreezing | 0.3925 | 0.1769
TABLE IV: TL capabilities to NLOS env. using MSE loss and 4323 Training Samples Source Loss |  TL type |  Target Training |  ME (m)
---|---|---|---
N/A | Scratch | Fine-tuning | 0.5288
MSE | MEML | Fine-tuning | 0.4803
NLL | MEML | Fine-tuning | 0.4603
MSE | MEML | Gradual Unfreezing | 0.4168
NLL | MEML | Gradual Unfreezing | 0.4194
TABLE V: TL capabilities to NLOS env. using NLL loss and 4323 Training Samples Source Loss |  TL type |  Target Training |  ME (m) |  CRPS
---|---|---|---|---
N/A | Scratch | Fine-tuning | 0.5954 | 0.2752
MSE | MEML | Fine-tuning | 0.6595 | 0.3795
NLL | MEML | Fine-tuning | 0.5494 | 0.2332
MSE | MEML | Gradual Unfreezing | 0.4479 | 0.2016
NLL | MEML | Gradual Unfreezing | 0.4364 | 0.1996

[Continuous ranked probability score (CRPS)] The Continuous Ranked Probability
Score (CRPS) is a statistical metric that compares distributional predictions
to ground-truth values[28]. Since the models trained with negative log-
likelihood loss predict a variance as well, our intention is to not only
achieve high positioning accuracy, but also achieve reliable uncertainty
estimations. The estimated uncertainty should be directly the knowledge that
the model has about this particular measurement. Since the uncertainty is
learned implicitly from the data, there is no ground truth to compare to.
Intuitively, if the position estimation error of the model is high, this
should be reflected with a corresponding high uncertainty. On the other hand,
a low positioning error should correspond to low uncertainty. The CRPS
provides a way to quantify this intuition by comparing a probability
distribution (i.e., the output of the model that includes the uncertainty) to
a ground truth. It is defined as [28]:

$\text{CRPS}(F,y)=\int(F(x)-\mathbbm{1}_{x\geq y})^{2}dx,$ (7)

where $F(x)$ is the cummulative distribution function (CDF) of the probability
distribution we want to evaluate, $y$ is the ground truth and:

$\mathbbm{1}_{x\geq y}=\begin{cases}1&\text{for }x\geq y\\\ 0&\text{for
}x<y\end{cases}$ (8)

is an indicator function. It can be shown see that the CRPS is an extension of
the mean absolute error but for distributions[28]. A lower CRPS is desired.

TABLE VI: Toy example ME |  ME (m)
---|---
Model 1 | 0.3117
Model 2 | 1.39405
TABLE VII: Toy example CRPS |  Estimated uncertainty $\boldsymbol{\Sigma}_{1}$ |  Estimated uncertainty $\boldsymbol{\Sigma}_{2}$
---|---|---
Model 1 | 0.13282 | 0.34417
Model 2 | 0.80749 | 0.59402

When the model’s prediction is a normal distribution
$\mathcal{N}(\mu,\sigma^{2})$ (as we assume in our case), then Eq. (7) is
modified to:

$\text{CRPS}(\mu,\sigma^{2},y)=\sigma[\omega(2\Phi(\omega)-1)+2\phi(\omega)-\pi^{-1/2}],$
(9)

where $\omega=(y-\mu)/\sigma$.

Next we will present a toy example which illustrates the usefulness of the
CRPS metric.

###  -A CRPS Toy Example

We assume that we have build 2 DL-models that give the positions
$\boldsymbol{p}_{\text{LOS}}$ of the $i$-the sample from the LOS environment
in Fig. 4. Due to the nature of positioning using fingerprints, each model
would have different uncertainty for each position, but for simplicity we
assume the the uncertainty of the model is the same regardless of the input
sample. These underlying expression for each model which includes the
uncertainty is described as:

  * •

Model 1:
$\boldsymbol{p}_{1,i}=\boldsymbol{p}_{\text{LOS},i}+\boldsymbol{\eta}_{1},\,\boldsymbol{\eta}_{1}\sim\mathcal{N}([0,0],0.1\boldsymbol{I}_{2})$

  * •

Model 2:
$\boldsymbol{p}_{2,i}=\boldsymbol{p}_{\text{LOS},i}+\boldsymbol{\eta}_{2},\,\boldsymbol{\eta}_{2}\sim\mathcal{N}([0,0],2\boldsymbol{I}_{2})$,

which is the ground truth plus some random noise variable.
$\boldsymbol{I}_{2}$ denotes the identity matrix of dimension $2\times 2$. The
underlying process that describes model would depend on the amount of samples
used during training, the configuration of the model (number of layers and
nodes) and other hyperparameters such as learning rate.

We train each of the models to give an estimate of its own uncertainty, which
in reality may differ than the real underlying uncertainty of the model.
Intuitively, we prefer the first model to the second one since it is more
accurate which is also expressed by the mean error (ME) of each model shown in
Table VI. On the other hand, if we have only the second model available, we
want the model to accurately estimate its uncertainty, i.e., not to be
overconfident, in order to let the application know that the estimates may not
be as accurate. This intuition can be formalized using the CRPS metric.

For the first model let’s assume that there are two options for its own
uncertainty estimation:

  * •

Low estimated uncertainty: $\boldsymbol{\Sigma}_{1}=0.1\boldsymbol{I}_{2}$

  * •

High estimated uncertainty: $\boldsymbol{\Sigma}_{2}=2\boldsymbol{I}_{2}$

If the reported uncertainty of the model is $\boldsymbol{\Sigma}_{1}$ then we
can see that the model accurately estimates its own uncertainty but when the
reported uncertainty is $\boldsymbol{\Sigma}_{2}$ then the model is under-
confident. The CRPS metric for the model 1, which is shown in table VII,
increases when the model is underconfident which reflects the decreased
performance.

We make the same assumption of the estimated uncertainty for the 2nd model.
Namely we assume that the model estimates its own uncertainty as
$\boldsymbol{\Sigma}_{1}$ or $\boldsymbol{\Sigma}_{2}$. In contrast to the
first model, when the second model estimates its uncertainty as
$\boldsymbol{\Sigma}_{2}$ then it is accurate and when the model’s estimated
uncertainty is $\boldsymbol{\Sigma}_{1}$ then it means that the model is
overconfident. The CRPS metric for the second model with the 2 different
uncertainty estimations is shown in Table VII.

Out of those 4 cases, shown in Table VII, we see that the CRPS metric
indicates that the first model is better regardless of its estimated
uncertainty since it is more accurate, which aligns with our intuition.
Additionally, if the first model is under confident it has higher CRPS which
is an indication that the model is either underconfident or over confident.
The CRPS is decreased when the uncertainty is estimated accurately . For the
second model, the CRPS score is lower when the model is able to accurately
estimate its own uncertainty and it increases if the estimated uncertainty
diverges from the actual underlying uncertainty of the model.

In this Appendix we showed the usefulness of using the CRPS metric for
positioning when we want to estimate the reliability of uncertainty estimates
in addition to positioning accuracy.

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# White dwarf - main sequence star collisions
from wide triples in the field

Erez Michaely1, Michael M. Shara2
1Astronomy Department, University of Maryland, College Park, MD 20742
2Department of Astrophysics, American Museum of Natural History, Central Park
West at 79th Street, New York, NY 10024, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Multiple star systems interact strongly with Galactic field stars when the
outer semi-major axis of a triple or multiple star is $>10^{3}$ AU. Stable
triples composed of two white-dwarfs (WD) and a low mass main sequence (MS)
star in a wide outer orbit can thus be destabilized by gravitational
interactions with random field stars. Such interactions excite the
eccentricity of the distant third star sufficiently so that it begins to
interact significantly with the inner binary. When this occurs the triple
undergoes multiple binary-single resonant encounters. These encounters may
result either in a collision between the nondegenerate component and a WD, or
the breakup of the triple into a compact binary and a third object which is
ejected. The compact binary can be either a MS-WD pair which survives, or
collides, or a double WD which may inspiral through gravitational wave
emission. We calculate the collision rate between a MS and WD star, and the
merger rate of double WDs. Additionally, we describe the prospects of
detectability of such a collision, which may resemble a sub-luminous SN event.

###### keywords:

Keyword – Keyword – Keyword

††pubyear: 2020††pagerange: White dwarf - main sequence star collisions  from
wide triples in the field–LABEL:lastpage

##  1 Introduction

In recent years it had been shown that for ultra-wide stellar systems the host
galactic field is collisional (Kaib & Raymond, 2014; Antognini & Thompson,
2016; Michaely & Perets, 2016, 2019b, 2020; Michaely, 2020), that is one
cannot regard the system as isolated from its environment. Therefore, when
considering the evolution of wide stellar systems one needs to account for
random gravitational interactions with passing stars, including close flybys,
even in low density environments like the field of the host galaxy. In turn,
the interactions between flybys and wide systems change the wide system’s
orbital characteristics, both the semi-major axis (SMA) and its orbital
eccentricity. The change in eccentricity is usually the more pronounced
effect. (Lightman & Shapiro, 1977; Merritt, 2013).

Here we consider ultra wide systems to be either wide binaries or wide
hierarchical triples. Kaib & Raymond (2014) calculated the rate of stellar
collision in the Milky-Way (MW) Galaxy, caused by wide (SMA > $1000{\rm AU}$)
binaries interacting with random flyby stars in the field. They found that a
stellar collision between two main sequence (MS) stars happens every
$1000-7500{\rm yrs}$ in the MW. Expanding on that work Michaely & Perets
(2016) described the formation of low-mass X-ray binaries (LMXBs) from wide
binary systems composed of a stellar compact object, i.e. a black-hole (BH) or
a neutron star (NS) and a low mass companion. They found that if BHs are born
with little to no natal kicks then the formation rate of LMXBs is consistent
with the inferred rate from observations. Additionally, Michaely & Perets
(2019b) found that gravitational waves (GW) sources are also formed from wide
binary BHs interacting with flyby stars in their host galaxies and reported a
rate of $1-10{\rm Gpc^{-3}yr^{-1}}$.

Hierarchical triples are composed of an inner binary and an outer binary,
where the inner binary center of mass (COM) is consider one component of the
outer binary and the distant, third object is the second component. Michaely &
Perets (2020) calculated the GW merger rate for wide BH triples that become
unstable due to interactions with random field stars that excite their outer
eccentricity. As a result the triple becomes unstable and a GW merger can
happen either in the instability phase or at the endstate phase. Under their
assumptions they found an extremely high merger rate of $100-250{\rm
Gpc^{-3}yr^{-1}}$, which indicates that while the formation channel is robust
the underlying assumptions are very generous. Recently (Michaely, 2020,
hereafter: Paper I) calculated via similar dynamical processes the Type Ia
supernova (SN) rate of triple white-dwarf (WD) systems. The SN originate
either by a direct collision from double WDs (Raskin et al., 2009; Rosswog et
al., 2009; Thompson, 2011; Kushnir et al., 2013) or by a GW inspiral that
leads to double WDs coalescing in a Hubble time, a process called “the double-
degenerate (DD) channel" (Iben & Tutukov, 1984; Webbink, 1984).

Recently several studies were done on the dynamics of scattering events trying
to calculate, among others, the rates of DWD mergers or collisions as a source
for Type Ia SNe. Antognini & Thompson (2016) performed an extensive numerical
study on the nature of scattering events: binary-binary, triple-binary, and
triple-single (the focus of this manuscript). In Antognini & Thompson (2016)
only triples with outer SMA $<1000\rm{AU}$ were considered, unlike paper I and
this manuscript. They reported that the collision rates of DWD in the field is
$\sim 2\times 10^{-6}\rm yr^{-1}$. Population synthesis studies were carried
out in order to quantify the Type Ia SNe originating from triples, due to
secular evolution Hamers et al. (2013); Toonen et al. (2018). They showed that
the fine tuning of the mutual orbital inclination combined with stellar
evolution lower SN rates down to $0.1-1\%$ of the total rate. Additionally,
Hamers (2018) calculated the Type Ia SN originating from quadruple star
systems, due to mergers and collisions. It was found the combined rate for all
the possible channels that lead to a SN is 3 order of magnitude lower than the
observed rate. In Leigh et al. (2018) it was shown that gravitational
scattering is dominated by triple-single scattering events, similar to paper I
and this manuscript, rather than binary-binary or triple-binary scattering.

In this study we follow-up the work of paper I and relax the assumption that
all three components of the triples are WDs. Instead we focus on triples
wherein two of the components are WDs and the third is a low mass MS star.
Following a similar dynamical scenario as explored in paper I, we focus both
on the intermediate phase, with direct collisions between the components, and
the endstate where we calculate the rates where collisions are possible and
the inspiral of double WD (DWD) binaries due to GW emission.

The outcomes of a collision between a WD and a low mass MS star have been
explored in the literature (Shara & Shaviv, 1977, 1978; Shara & Regev, 1986;
Regev & Shara, 1987; Ruffert, 1992; Shara, 1999). Other stellar collisions
have been studied as well (Soker & Tylenda, 2006; Katz & Dong, 2012; Aznar-
Siguán et al., 2013). Here we consider the energetics, timescales and the
potential detectability of the expected transient generated by a WD-MS
collision and reserve a detailed study for future research.

In section 2 we describe the interaction of ultra-wide triples in the field of
a host galaxy. In section 3 we discuss the dynamics in the unstable phase of
the triple as multiple binary-single encounters. In section 4 we calculate the
expected rates of collisions and DD mergers in large spiral and elliptical
galaxies. We discuss the energetics and detectability of WD-MS collisions in
section 5. We estimate the delay-time distribution of WD-MS collisions in
section 6. We summarize our results in section 7.

##  2 Wide triples in the field

In what follows we briefly describe the gravitational interaction between a
wide hierarchical system, illustrated in Fig. 1, and a flyby star in the field
of the host galaxy. A complete mathematical description can be found in paper
I and references therein.

For simplicity we assume throughout this paper that all three components in
the triple have the same mass $m_{1}=m_{2}=m_{3}=0.6M_{\odot}.$ The inner
binary (masses $m_{1}$ and $m_{2}$) of the triple is characterized by a
circular orbit with a SMA $a_{1}$. The outer binary, composed of the third
object $m_{3}$ and the center of mass (COM) of the inner binary is
characterized by a SMA $a_{2}>1000{\rm AU}$ and an eccentricity $e_{2}$. The
nature of the interaction between the triple and a random flyby star is
determined by the local environment of the host galaxy. In particular, the
encounter is determined by the relative speed between the flyby star and the
triple, $v_{{\rm enc}}$ and the interaction rate, $f$. We identify the
relative speed to be the local velocity dispersion, $v_{{\rm enc}}=\sigma$.
The interaction rate is related to the local number stellar density $n_{*}$ by
$f=n_{*}\sigma_{{\rm CS}}v_{{\rm enc}}$ where $\sigma_{{\rm CS}}$ is the
geometrical cross-section of the flyby interaction.

In paper I it was shown that for an ensemble of wide triple systems with a
thermal distribution, $f\left(e_{2}\right)=2e_{2}$ of outer eccentricities,
there is a non-negligible probability that the third component’s pericenter
passage, $q=a_{2}\left(1-e_{2}\right)$, is sufficiently close to the inner
binary COM that the systems becomes unstable. Specifically, the instability
condition is $q\leq a_{1}$.

The instability of the triple can be described as follows. The triple
undergoes multiple binary-single encounters, during which a temporary binary
is formed from a random pair of the three system components, with a SMA
$a_{{\rm IMS}}$ and eccentricity $e_{{\rm IMS}}$, where the subindex IMS
stands for “intermediate state". The values of $a_{{\rm IMS}}$ are drawn
uniformly between $\left(a_{1},2a_{1}\right)$ and the $e_{{\rm IMS}}$ are
drawn from a thermal distribution (for a complete derivation see eq. 20-24 in
paper I). The third object is bound to the temporary binary on a wide
Keplerian orbit with $a_{2{\rm IMS}}$ determined by conservation of orbital
energy. The relatively wide outer orbit allows the inner binary to revolve
around its COM multiple times before the next pericenter passage, namely
before the next binary-single interaction where this process repeats.

For point-like particles the average number of binary-single encounters is
$\left\langle N\right\rangle=20$ (Michaely & Perets, 2020). The end state is
the formation of a compact binary with SMA $a_{{\rm ES}}<a_{1}$, and
eccentricity $e_{{\rm ES}}$ which follows a thermal distribution. The end
state binary is composed of two objects randomly chosen out of the triple. The
third object is ejected to infinity.

In order to determine the rate at which wide triple systems become unstable we
focus on the outer binary of the wide triple, with $a_{2}$ and $e_{2}$. This
wide binary interacts with flyby stars, of mass $m_{{\rm p}}$, which perturb
the outer binary. Lightman & Shapiro (1977) and Merritt & Poon (2004) showed
that the perturbation primarily torques the outer binary, changing its
eccentricity. We use the “loss-cone" formalism to determine the loss rate of
wide stable triple star systems.

We consider an ensemble of wide systems with a thermal distribution of the
outer eccentricity. The loss cone is the fraction of systems from the ensemble
that becomes unstable, namely the outer pericenter distance of the outer
binary is within the inner binary SMA, $q\leq a_{1}$. From paper I we get:

$F_{q}=\frac{2a_{1}}{a_{2}}$ (1)

which for hierarchical triples $F_{q}\ll 1$. We compare this value to the
smear-cone, $F_{s}$ (paper I) which essentially represents the change in
eccentricity due to the impulse interaction with the perturber.

$F_{s}=\frac{27}{4}\left(\frac{m_{p}}{M}\right)^{2}\left(\frac{GM}{a_{2}v_{{\rm
enc}}^{2}}\right)\left(\frac{a_{2}}{b}\right)^{4}$ (2)

where $M=m_{1}+m_{2}+m_{3}$ is the total mass of the triple and $b$ is the
closest approach of the perturber to the triple COM. These quantities
naturally create two regimes, the full loss cone and the empty loss cone
regimes. In the full loss cone regime, where $F_{s}\geq F_{q}$ the loss cone
is continuously full, namely the interactions are frequent and strong enough
to keep the loss cone full. In the empty loss cone regime, $F_{s}<F_{q}$ the
loss cone is primarily empty until a weak interaction with a flyby star occurs
which kicks a system into the loss cone.

Additionally, we account for binary ionization processes in the field. Because
the field is collisional for wide binaries we need to account for the loss of
wide binaries due to the disruption of wide binaries from random gravitational
processes. We use the standard half-life calculation from (Bahcall et al.,
1985)

$t_{1/2}=0.00233\frac{v_{{\rm enc}}}{Gm_{{\rm p}}n_{*}a_{2}}.$ (3)

The loss probabilities, the probability of a system with inner SMA $a_{1}$,
outer SMA $a_{s}$, located in a stellar environment with number density
$n_{*}$ at time $t$ to become unstable, are given by the following equations:
for the empty loss cone regime

$L\left(a_{1},a_{2},n_{*}\right)_{{\rm
empty}}=\tau\frac{2a_{1}}{a_{2}}n_{*}\pi\sqrt{\frac{27}{8}\left(\frac{m_{{\rm
p}}}{M}\right)^{2}\frac{GMa_{2}^{4}}{a_{1}}}\left(1-e^{-t/\tau}\right)$ (4)

where $\tau=t_{1/2}/\ln 2$. For the full loss cone regime we get

$L\left(a_{1},a_{2},n_{*}\right)_{{\rm
full}}=\tau\frac{2a_{1}}{a_{2}}\left(\frac{GM}{4\pi^{2}a_{2}^{3}}\right)^{1/2}\left(1-e^{-t/\tau}\right).$
(5)

Figure 1: Illustration of a triple system in hierarchical configuration,
$a_{1}\ll a_{2}$. The inner binary is circular with SMA, $a_{1}$. The outer
binary is defined by the outer SMA, $a_{2}$ and outer eccentricity, $e_{2}$.

##  3 Binary-single encounters

In this section we focus on the dynamics of the triple system when it becomes
unstable and enters the multiple binary-single interaction phase. In
subsection 3.1 we focus on the intermediate phase while in subsection 3.2 we
explore the end state of the intermediate phase, specifically the compact
binary that forms as a result of the intermediate phase.

The case where one of the components is a low mass star, in its MS phase, is
fundamentally different from the case presented in paper I and in Michaely &
Perets (2020). The reason is that the radius of a low mass MS star is roughly
$50$ times larger than the radius of a WD. Hence a direct collision between a
WD and the MS star becomes probable.

###  3.1 The intermediate phase

In order to calculate the probability of this scenario, a collision between a
MS star and a WD from a triple with two WDs and one MS, star we perform a
numerical calculation. We sample 20 values of the inner SMA $a_{1}$ from
$\left(10^{-2}{\rm AU},10^{2}{\rm AU}\right)$ equally distributed in log
space. For each value of $a_{1}$ we perform $N_{{\rm tot}}=10^{5}$ “scattering
experiments". In this context a single “scattering experiment" is $N_{{\rm
IMS}}=20$ binary-single interactions, where for each one a temporary binary is
formed with $a_{{\rm IMS}}$ and $e_{{\rm IMS}}$. $a_{{\rm IMS}}$ is drawn
uniformly from $\left(a^{\prime}_{{\rm L}},a^{\prime}_{{\rm U}}\right)$ and
$e_{{\rm IMS}}$ is drawn from a thermal distribution. The boundary values
$\left(a^{\prime}_{{\rm L}},a^{\prime}_{{\rm U}}\right)$ are determined by the
individual masses of the triple (see eq. 18-24 in paper I). Moreover, each
temporary binary is composed of two random components, such that either a
double WD is formed or a WD-MS binary is formed. The temporary pericenter is
$q^{\prime}=a_{{\rm IMS}}\left(1-e_{{\rm IMS}}\right)$ and we compare it to
the sum of the radii of the temporary binary, $R=r_{1}+r_{2}$. We calculate
the radius of the WD using Hamada & Salpeter (1961):

$R_{{\rm WD}}=1.3\times 10^{-2}R_{\odot}\text{$\left(\frac{M_{{\rm
WD}}}{0.6M_{\odot}}\right)$}^{-1/3},$ (6)

and for the low mass convective MS star we use

$R_{*}=0.6R_{\odot}\left(\frac{m_{*}}{0.6M_{\odot}}\right)$ (7)

taken from Mann et al. (2015).

where $m_{*}$ is the mass of the MS star. In the case where $q^{\prime}\leq R$
we flag the scattering experiment as a collision, check the nature of the
components that collide and terminate this experiment. In the case where
$q^{\prime}>R$ we calculate the outer binary orbital period, which is the time
of the next interaction and randomize the binary-single interaction again
until we reach $N_{{\rm IMS}}$ times.

In order to calculate the fraction of systems that undergo a collision between
a WD and a MS star in the intermediate phase, $f_{{\rm WD-
MS}}\left(a_{1}\right)$ we divided the number of mergers by $N_{{\rm tot}}$
for each $a_{1}$. The numerical results are presented in Fig. 2. We fit the
numerical results with two functions as a function of the inner SMA $a_{1}$.

$\log f_{{\rm WD-MS}}\left(a_{1}\right)=\begin{cases}-0.961\cdot\log
a_{1}-1.212&a_{1}>0.1{\rm AU}\\\ -0.542\cdot\exp\left(-\frac{\left(\log
a_{1}+0.49\right)}{0.682}\right)^{2}&a_{1}<0.1{\rm AU}\end{cases}.$ (8)

We note here that we ignore the rate of double WD collisions in the
intermediate stage, represented in Fig. 2 with red circles. One can expect the
rate of double WD collision scales as the ratio of the radii of WD and the MS
star, for $M_{{\rm WD}}=m_{*}=0.6M_{\odot}$

$\frac{R_{{\rm WD}}}{R_{*}}\approx 0.02.$ (9)

Figure 2: The fraction of systems that undergo a WD-MS collision in the
intermediate state. $f_{{\rm WD-MS}}$ as a function of the initial SMA,
$a_{1}$. For every initial value of $a_{1}$ we simulated $10^{5}$ scattering
experiments. In each experiment we run $N_{{\rm IMS}}=20$ binary-single
scattering events. For every event we randomize the temporary binary orbital,
SMA and eccentricity. We check if the temporary IMS leads to a WD-MS collision
or WD-WD collision. Black dots are the calculated fraction of WD-MS collisions
from our numerical experiment. Red circles are the fraction of WD-WD
collisions. The blue solid line is a broken fit to the numerical results.

###  3.2 End state

In the case where no collision occurs during the $N_{{\rm IMS}}$ times we
simulate the endstate configuration. The emerging compact binary has a
different SMA distribution than the distribution in the intermediate phase.
Specifically, the energy of the final binary is distributed according to the
following relation (Stone & Leigh, 2019)

$E_{{\rm ES}}\propto\left|E_{1}\right|^{-4}.$ (10)

where $E_{{\rm ES}}$ is the energy of the endstate binary and
$E_{1}=-Gm_{1}m_{2}/(2a_{1})$ is the initial binary energy. Similar to the
intermediate phase case the eccentricity of the compact binary $e_{{\rm ES}}$
is drawn from a thermal distribution (Stone & Leigh, 2019). Hence, a compact
binary is formed for every system that did not collide during the IMS with a
SMA $a_{{\rm ES}}\left(E_{{\rm ES}}\right)$ and eccentricity $e_{{\rm ES}}$.
This endstate binary is composed either of a double WD (DWD) or a MS star with
a WD companion. As a result the possible outcomes are either a direct
collision between the components or the ensuing evolution of a binary star.
Depending on the binary components, the evolution may lead to a merger through
gravitational waves in the case of a DWD, or any standard outcome allowed by
binary evolution theory, including common envelope evolution (CEE), a
cataclysmic variable (CV) etc. In the left plot of Fig. 3 we present the
fraction of systems that experienced a WD-MS collision in the post-resonance
phase as a function of the initial SMA. The numerical fit presented in the
figure is composed of two functions:

$\log f_{{\rm WD-MS}}^{{\rm ES}}\left(a_{1}\right)=\begin{cases}-0.94\cdot\log
a_{1}-2.18&a_{1}>0.1{\rm AU}\\\ -0.006\cdot\exp\left(-2.98\log
a_{1}\right)&a_{1}<0.1{\rm AU}\\\ -2.113\cdot\exp\left(0.51\log
a_{1}\right)\par\end{cases}.$ (11)

Next, we calculate the fraction of systems that undergo a double WD merger
through GW emission similar to the classic DD scenario. In order to do so we
calculate the GW merger timescale for eccentric binaries which is (Peters,
1964)

$t_{{\rm
merger}}\approx\frac{768}{425}T_{c}\text{$\left(a\right)$}\left(1-e^{2}\right)^{7/2}$
(12)

where $T_{c}=a^{4}/\beta$ is the merger timescale for a circular orbit and
$\beta=64G^{3}m_{i}m_{j}\left(m_{i}+m_{j}\right)/\left(5c^{2}\right)$. The
indices $i,j$ are the indices of the two WDs that ended up as the surviving
compact binary and $c$ is the speed of light. If $t_{{\rm merger}}<10^{10}{\rm
yr}$ we flag this systems as a DD inspiral, and a possible source for a Type
Ia SN. The numerical fit is

$\log f_{{\rm DD}}\left(a_{1}\right)=\begin{cases}-1.04\cdot\log
a_{1}-2.647&a_{1}>0.1{\rm AU}\\\ -0.009\cdot\exp\left(-2.79\log
a_{1}\right)&a_{1}<0.1{\rm AU}\\\ -2.77\cdot\exp\left(0.56\log
a_{1}\right).\end{cases}$ (13)

We emphasize here that the outcome of a WD-MS binary that does not collide is
still of interest. These binaries are the progenitors of CEE or CV. Once the
MS star companion in the binary evolves to the giant phase it may fill its
Roche-lobe and begin mass transfer. If the mass transfer is stable an
accretion disk forms around the WD and a CV is formed. In the case where the
mass transfer is unstable the system may inspiral to a CEE. The study of the
evolution of these binaries has been explored elsewhere (for a recent overview
see Beccari & Boffin (2019)).

Figure 3: Left plot: Similar to Fig. 2, but the fraction of systems that
undergo a WD-MS collision in the endstate. $f_{{\rm WD-MS}}^{{\rm ES}}$ as a
function of the initial SMA, $a_{1}$. If the triple survived the intermediate
state, the final outcome is a compact binary with SMA drawn from eq. (10) and
the eccentricity is drawn from a thermal distribution. We check if a compact
binary leads to a WD-MS collision. Black dots are the calculated fraction of
WD-MS collisions from our numerical experiments. Blue solid line is a broken
fit to the numerical results. Right plot: Same as left plot but we calculate
the fraction of DWD systems that merge via GW emission within a Hubble time.

##  4 Galactic rates

In what follows we use the numerical fits found in the previous section in
order to compute rates in galaxies for WD-MS collisions in the intermediate
phase and in the endstate. Additionally, we compute the galactic rates of DD
inspiral.

###  4.1 Galaxy models

As described in section 2, the process that destabilizes wide triples depends
on the local stellar environment. Specifically these are the local stellar
density, $n_{*}$ and the encounter velocity, $v_{{\rm vec}}$ which is
determined by the local velocity dispersion. Hence, one needs to model these
properties of the host galaxy.

First, we model the MW Galaxy as a prototype of a large spiral galaxies in the
local universe, using the same model described in paper I which follows (Jurić
et al., 2008). The number stellar density is given by

$n_{{\rm*s}}\left(r\right)=n_{0}e^{-\left(r-r_{\odot}\right)/R_{l}}$ (14)

where $n_{{\rm*s}}$ represents the number stellar density for and
$n_{0}=0.1{\rm pc}^{-3}$ is the number stellar density in the solar
neighborhood, $R_{l}=2.6{\rm kpc}$ is the Galactic length scale and
$r_{\odot}=8{\rm kpc}$ is the distance of the Sun from the Galactic center.
The velocity dispersion we use is that of the flat rotation curve of the
Galaxy, i.e. $\sigma=50{\rm kms^{-1}}$ which in turn is set to be the
encounter velocity.

We define ${\rm dN_{s}\left(r\right)}$, the number of stars in a volume of the
disk at a distant $r$ from the center of the Galaxy, by

${\rm dN_{s}(r)}={\rm n_{*s}\left(r\right)\cdot 2\pi\cdot r\cdot h\cdot dr}$
(15)

where $h=1{\rm kpc}$ is the scale height of the disk.

Second, for simplicity we model an elliptical galaxy the same way we did in
paper I following Hernquist (1990). The density profile is given by

$n_{{\rm*e}}\left(r\right)=\frac{M_{{\rm galaxy}}}{2\pi
r}\frac{r_{*}}{\left(r+r_{*}\right)^{3}}$ (16)

where $n_{{\rm*e}}$ is the stellar density for elliptical galaxy and
$r_{*}=1{\rm kpc}$ is the scale length of the galaxy, $M_{{\rm
galaxy}}=10^{11}M_{\odot}$ is the total stellar mass of the galaxy. Hence,

${\rm dN_{{\rm e}}\left(r\right)}=\frac{n_{{\rm*e}}}{\left\langle
m\right\rangle}{\rm dV}$ (17)

is the number of stars within some local volume ${\rm dV}$ at a distance $r$
from the center. $\left\langle m\right\rangle=0.6M_{\odot}$ represents the
average stellar mass of the galaxy. The velocity dispersion for a typical
elliptical galaxy is $\sigma=160{\rm kms^{-1}}$ (Cappellari et al., 2013). In
both types of galaxies we set the mass of the perturber to be $0.6M_{\odot}$,
the average mass of a star in the galaxy.

###  4.2 WD-MS collision rate

In this section we calculate the WD-MS collision rate in the intermediate
stage and endstate in both model spiral and elliptical galaxies. To do this we
must estimate the fraction of triple system which host two WDs and a MS star.
There are two possibilities for such systems to exist. The first is a DWD as
the inner binary and the stellar companion is the tertiary; we term these
systems as WWM. The second, is a WD-MS system as an inner binary and
additional WD as the tertiary; we term these systems as WMW.

We start by estimating the fraction of WWMs out of the stellar population,
$f_{{\rm WWM}}.$ We assume all stars with mass in the range of
$1M_{{\rm\odot}}-8M_{\odot}$ become WDs and we count only MS stars in the mass
rage of $0.4M_{\odot}-1M_{\odot}$. We do so to be consistent with our
assumption that the triple systems is composed of stars of equal masses. We
use the initial mass function (IMF) given by (Kroupa, 2001) and find that
$f_{{\rm primary}}\approx 0.1$ of all stars are in the range of
$1M_{{\rm\odot}}-8M_{\odot}$ which turn into WDs in $10{\rm Gyrs}$. This is an
upper limit because binary evolution can hasten WD faster via interactions
between the binary stars. The binary companion mass is calculated from a
uniform mass ratio distribution (Moe & Di Stefano, 2016), $Q_{{\rm
inner}}\in\left(0.1,1\right)$. For the tertiary we calculate the outer mass
ratio by the following expression, $Q_{{\rm
outer}}=m_{3}/\left(m_{1}+m_{2}\right)$ and its value is drawn from a power
law distribution (Moe & Di Stefano, 2016) $f_{Q_{{\rm outer}}}\propto Q_{{\rm
outer}}^{-2}$ where $Q_{{\rm outer}}\in\left(0.1,1\right)$. For WWM systems we
count only the systems in which the primary mass is within the range
$1M_{{\rm\odot}}-8M_{\odot}$, the secondary is in the range
$1M_{{\rm\odot}}-8M_{\odot}$ and the tertiary is in the range
$0.4M_{\odot}-1M_{\odot}$. Given these distributions we find that the fraction
of secondaries in the WD production range is $f_{{\rm secondary}}\approx
0.44$, while for the tertiaries it is $f_{{\rm tertiary}}\approx 0.44$. The
triple fraction is set to be $f_{{\rm triple}}=0.2$ (Duchêne & Kraus, 2013)
and the fraction of wide outer binaries greater than $1000{\rm AU}$ from a
log-uniform distribution, $f_{a_{2}}$, is $f_{{\rm wide}}=0.2.$ Combining
these estimates we get

$f_{{\rm WWM}}=f_{{\rm primary}}\times f_{{\rm secondary}}\times f_{{\rm
tertiary}}\times f_{{\rm triple}}\times f_{{\rm wide}}\approx 7.6\times
10^{-4}.$ (18)

A similar calculation can be made for the WMW case. Then the primary is a WD,
$f_{{\rm primary}}\approx 0.1$ the secondary is a MS star, $f_{{\rm
secondary}}\approx 0.4$ and the tertiary is a WD, $f_{{\rm tertiary}}\approx
0.14$.

$f_{{\rm WMW}}\approx 2.2\times 10^{-4}.$ (19)

We define

$f_{{\rm model}}=f_{{\rm WWM}}+f_{{\rm WMW}}.$ (20)

We note that this approach to estimate the fraction of triples is a
simplification of a very complex calculation. In order to calculate the wide
triple systems out of a certain stellar population more accurately one needs
to numerically evolve large numbers of systems and take into account both
single and binary stellar evolution. This, in turn, will change the initial
SMA and eccentricity distributions while also changing the masses. In
particular, common envelope evolution (Ivanova et al., 2013) modifies the
inner SMA and even the outer SMA due to mass loss from the inner binary
(Michaely & Perets, 2019a; Igoshev et al., 2020).

Next, we compute the total galactic rate for WD-MS collisions for both types
of galaxies. The rate, $\Gamma$ is given by integrating the loss cone (4) and
(5) for all outer SMAs, $a_{2}$ between $10^{3}-10^{5}{\rm AU}$, the local
stellar density in the galaxy $n_{*}$ from equations (14) and (16). In order
to integrate the inner binary SMA we use the following limits
$10^{-1}\left(10^{-2}\right)-10^{2}{\rm AU}$. We choose two minimal values of
the inner binary in order to roughly estimate the uncertainties caused by our
lack of knowledge of the real distribution:

$\Gamma=\int\int\int\frac{L_{{\rm
collision}}\left(a_{1},a_{2},n_{*}\right)}{10{\rm Gyr}}da_{1}da_{2}{\rm
dN\left(r\right)}$ (21)

where $L_{{\rm collision}}\equiv
L\left(a_{1},a_{2},n_{*}\right)f_{a_{1}}f_{a_{2}}f_{{\rm model}}f_{{\rm WD-
MS}}$ and we define

${\rm dL}\equiv\frac{L_{{\rm collision}}\left(a_{1},a_{2},n_{*}\right)}{10{\rm
Gyr}}da_{1}da_{2}{\rm dN\left(r\right)}.$ (22)

Now we plug in the function from eq. (8)

$\Gamma_{{\rm MW}}=\int_{{\rm 0.5kpc}}^{{\rm 15kpc}}\int_{{\rm
10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$ $\approx 1.9\
\left(3.69\right)\times 10^{-4}{\rm yr^{-1}}$ (23)

and for a typical elliptical

$\Gamma_{{\rm elliptical}}=\int_{{\rm 0.1kpc}}^{{\rm 30kpc}}\int_{{\rm
10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$ $\approx 3.44\
\left(6.34\right)\times 10^{-4}{\rm yr^{-1}.}$ (24)

These results are averaged over a $10{\rm Gyr}$ lifetime of the galaxies, In
subsection 6 we approximate the delay-time distribution of these collisions.

In the case where a direct collision did not occur during the IMS phase the
triple is disrupted and a compact binary is formed. We calculate the collision
rate by using, $\log f_{{\rm WD-MS}}^{{\rm ES}}$ from (11) to get for the MW-
like Galaxy

$\Gamma_{{\rm MW}}^{{\rm ES}}=\int_{{\rm 0.5kpc}}^{{\rm 15kpc}}\int_{{\rm
10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$ $\approx 1.56\
\left(1.14\right)\times 10^{-5}{\rm yr^{-1}}$ (25)

and for the elliptical galaxy a rate of

$\Gamma_{{\rm elliptical}}^{{\rm ES}}=\int_{{\rm 0.1kpc}}^{{\rm
30kpc}}\int_{{\rm 10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$ $\approx 2.81\
\left(2.00\right)\times 10^{-5}{\rm yr^{-1}.}$ (26)

###  4.3 Double degenerate inspiral rate

Similar to the previous subsection here we calculate the galactic rate for an
endstate binary WD inspiral the may lead to Type Ia SN. In this case we use
eq. (13) and insert it in eq. (22) to get

$\Gamma_{{\rm MW,DD}}^{{\rm ES}}=\int_{{\rm 0.5kpc}}^{{\rm 15kpc}}\int_{{\rm
10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$ $\approx 7.56\
\left(5.38\right)\times 10^{-6}{\rm yr^{-1}}$ (27)

and for a typical elliptical galaxy

$\Gamma_{{\rm elliptical,DD}}^{{\rm ES}}=\int_{{\rm 0.5kpc}}^{{\rm
15kpc}}\int_{{\rm 10^{3}AU}}^{10^{5}{\rm AU}}\int_{{\rm
10^{-1}\left(10^{-2}\right)AU}}^{10^{2}{\rm AU}}{\rm dL}$

$\approx 1.34\ \left(0.96\right)\times 10^{-5}{\rm yr^{-1}}.$ (28)

These rates are of order one percent of the total Type Ia rate observed in
large galaxies:$\sim$ $10^{-3}{\rm yr^{-1}}$ (Maoz et al., 2014).

##  5 WD-MS collision

In what follows we discuss the physics and detectability of a WD-MS collision
event. Shara & Shaviv (1977) studied the timescales and energetics of WD-MS
collisions, while Shara & Shaviv (1978); Shara & Regev (1986); Regev & Shara
(1987) carried out 2D hydrodynamics simulations of head-on collisions,
including a simple power law prescription to allow for nuclear energy release.
In their study a direct collision between a MS star with mass $M_{*}$ and
radius $R_{*}$ and a WD with mass $M_{{\rm WD}}$ and radius $R_{{\rm WD}}$ was
considered. The relative velocity, at the collision, was set $v_{{\rm
coll}}\sim v_{*{\rm esc}}$ to the escape velocity of the MS star in Shara &
Regev (1986), and to 2000 or 6000 $kms^{-1}$ in Regev & Shara (1987).

As the WD approaches the MS star tidal forces act on the MS and stretch its
outer envelope. Energy conversion (from kinetic to thermal) and nuclear energy
lgeneration in the collision event begins as the mass from the outer edges of
the MS star impacts the surface of the WD, decelerating and being deflected.
The collision time is approximated by

$\tau_{{\rm col}}\approx\frac{R_{*}}{v_{{\rm coll}}}$ (29)

For $M_{*}=M_{{\rm WD}}=0.6M_{\odot}$ and $R_{*}=0.6R_{\odot}$ this yields,
$\tau_{{\rm col}}\approx 900{\rm sec}.$ During this time, during which the WD
moves supersonically through the envelope of the MS star, a roughly spherical
shock is formed when the collision velocity is lower than the WD’s escape
velocity, $v_{{\rm esc,WD}}\approx 7\times 10^{3}{\rm kms^{-1}}.$ The shock
compresses the envelope by a factor of $\sim 4-10$ to average values of
$\rho\approx 10^{3}{\rm gr\cdot cm^{-3}}$ and heats it to $\sim 3-5\times
10^{8}K$. At this point radiation pressure becomes dominant over gas pressure
in the MS star’s envelope. Under these conditions proton capture and the hot
CNO cycle are the dominant energy sources in the star, and He burning may also
be initiated. The nuclear burning consumes a few percent of the available
hydrogen and releases $E\sim 2\times 10^{48}{\rm ergs}$, during $\tau_{{\rm
col}}$ and deposits it in the escaping and optically thick stellar envelope.
The binding energy of a $0.6M_{\odot}$ star is $E_{G}\approx 10^{48}{\rm
erg}$; this implies that the MS is totally disrupted by the collision. For a
10 $M_{\odot}$ MS star Regev & Shara (1987) the energy release approaches s
$\approx 10^{50}{\rm erg}$.

The envelope escapes with a velocity of order a few x$10^{3}{\rm kms^{-1}}$,
assuming adiabatic expansion during the time it takes the optically thick
envelope to become optically thin, namely $\rho\approx 10^{-11}{\rm gr\cdot
cm^{-3}}$ is about $t\approx 10^{6}-10^{8}{\rm sec}$. Combining the time to
achieve an optically thick expanding shell, $t$ and the total energy released
from the nuclear burning, $E$ one can estimate the bolometric luminosity to be

$L\approx\frac{E}{t}\approx\frac{10^{48}{\rm erg}}{10^{6}-10^{8}{\rm
sec}}=10^{7}-10^{9}L_{\odot}.$ (30)

This luminosity is similar to that of a kilonova, but intermediate and heavy
elements are unlikely to be produced during such events unless helium burning
is achieved.

We note that all of the quoted 2D simulations were based on head-on collisions
with simplistic nuclear energy release models. 3D hydrodynamics, including a
realistic nuclear reactions network and off-center collisions are essential to
better understand the outcomes of WD-MS star collisions. We reserve such
studies to future efforts, noting that a head-on collision is far from the
most representative case.

##  6 Delay time distribution (DTD)

In this subsection we calculate the expected delay-time distribution (DTD) of
the transients described in section 5. The DTD is the hypothetical rate of
transients that follow a brief star formation episode. The main channel of WD-
MS collisions originates during the multiple binary-single encounter when the
triple becomes unstable. The collision occurs on a dynamical timescale which
is extremely short compared to the stellar evolution time that was needed to
produce two WDs. Therefore, we only calculate the time since the star
formation event which produced the WD and RD stars.

The DTD is determined by the numbers of available triples that become unstable
as a function of time. This depends on the initial mass function and the
stellar evolution time for each mass. As already assumed, the rate of flyby
interaction is constant in time if one disregards binary ionization via other
processes (such as mass loss during binary evolution). Therefore we can write
the following dependency

$\frac{d{\scriptscriptstyle N}}{dt}\propto\frac{d{\scriptscriptstyle
N}}{dm}\frac{dm}{dt}.$ (31)

The first term is the initial mass function which is similar to those of
Kroupa and Salpeter (Kroupa, 2001; Salpeter, 1955) $d{\scriptscriptstyle
N}/dm=m^{-2.3}$. The second term is just the MS life time, $t_{{\rm MS}}$,
i.e. the time it takes a MS star to evolve into a WD, $dm/dt=t^{-4/3}$. For
these simplifying assumptions we get

$\frac{d{\scriptscriptstyle N}}{dt}\propto t^{2.3/3}t^{-4/3}=t^{-0.56}\approx
t^{-3/5}.$ (32)

The DTD distribution given here is the predicted DTD of the transients
discussed in subsection 5. Moreover, due to the functional form of the DTD, we
expect to find these events not just in star forming spiral galaxies but also
in ellipticals.

##  7 Discussion and summary

###  7.1 Summary

In this study we described the interactions of triple systems composed of two
equal mass WDs and a low mass MS star with field stars of the host galaxy. We
found that a significant fraction of these systems become unstable due to
interactions with random flyby stars. Those flybys can excite the outer
eccentricity of the triple sufficiently to destabilize the triple system. The
instability manifests itself via multiple binary-single encounters during
which a collision between the MS star and a WD is plausible. In the case where
the triple survives the chaotic evolution, i.e. there is no collision between
any two components, the systems disrupts to the final endstate of a compact
binary, formed of any two object of the triple, and an “escaper” that is
ejected to infinity. The newly formed compact binary can, in turn cause a WD-
MS collision, or a DWD inspiral through GW emission, similar to the classic DD
scenario that leads to a Type Ia SN.

We found that the collision rates are very sensitive to the initial triple
population and its characteristics. For a set of plausible assumptions we
expect a collision rate of $\sim 1$ event every $10^{4}{\rm yr}$ in the MW or
any similar spiral galaxy, and $\sim 1$ event every $5000{\rm yr}$ in an
elliptical galaxy with total mass of $10^{11}M_{\odot}$.

Furthermore, we expect these events to be luminous with $L\approx
10^{7}-10^{9}L_{\odot}$ for weeks to years. We speculate that some sub-
luminous SNe or other luminous transients might result. Additionally, we
calculated the predicted DTD to be $t^{-3/5}$, and note that we expect to find
these events in both star forming galaxies and ellipticals. Finally, we
estimate that the total rate of these events is of order $1\%$ of the SNIa
rate in galaxies.

## Acknowledgments

E.M. thanks the University of Maryland CTC prize-fellowship for supporting
this research, and Nathan Leigh for improving this manuscript.

Data availability The data underlying this article will be shared on
reasonable request to the corresponding author.

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