# Calorie Aware Automatic Meal Kit Generation from an Image

Ahmad Babaeian Jelodar, and Yu Sun A. B. Jelodar, and Y. Sun are with the
Department of Computer Science and Engineering, University of South Florida,
FL, 33620 e-mail: (ajelodar,yusun@usf.edu).

###### Abstract

Calorie and nutrition research has attained increased interest in recent
years. But, due to the complexity of the problem, literature in this area
focuses on a limited subset of ingredients or dish types and simple
convolutional neural networks or traditional machine learning. Simultaneously,
estimation of ingredient portions can help improve calorie estimation and meal
re-production from a given image. In this paper, given a single cooking image,
a pipeline for calorie estimation and meal re-production for different
servings of the meal is proposed. The pipeline contains two stages. In the
first stage, a set of ingredients associated with the meal in the given image
are predicted. In the second stage, given image features and ingredients,
portions of the ingredients and finally the total meal calorie are
simultaneously estimated using a deep transformer based model. Portion
estimation introduced in the model helps improve the calorie estimation and is
also beneficial for meal re-production in different serving sizes. To
demonstrate the benefits of the pipeline, the model can be used for meal kits
generation. To evaluate the pipeline, the large scale dataset Recipe1M is
used. Prior to experiments, the Recipe1M dataset is parsed and explicitly
annotated with portions of ingredients. Experiments show that using
ingredients and their portions significantly improves calorie estimation.
Also, a visual interface is created in which a user can interact with the
pipeline to reach accurate calorie estimations and generate a meal kit for
cooking purposes.

###### Index Terms:

Meal kit generation, Ingredient Prediction, Portion Estimation, Calorie
Estimation.

##  I Introduction

Cooking related applications have become a popular research area in recent
years spanning from tasks such as ingredient recognition [1], cooking motion
recognition [2, 3], cooking activity understanding [4], dish classification
[5, 6], recipe generation from a single image [7] to calorie estimation [8],
and recipe retrieval [9]. Food nutrition, and health are two important aspects
of our lives that require close monitoring and care and are strictly
associated with cooking. Specifically, the amount of calorie intake in a meal
is an important matter of health. Many research have addressed calorie
estimations from a single image, but they only use simple small sized datasets
with a few ingredients or dish types [8, 10, 11]. They also lack simultaneous
portion estimation of ingredients which can help improve calorie estimation,
and reproduce the meal in different serving sizes.

Figure 1: An example of meal kits generation from a single image.

Although ingredient recognition, and recipe generation from a single image
have become growing areas of research in recent years, ingredient portion
estimation is a neglected research area which includes very little literature
that only focus on a small set of ingredients. In this paper, we propose to
predict ingredients, and estimate their portions while estimating the calorie
of the given image. For example, rather than only predicting the containing
ingredients (e.g. carrot, cabbage, onion, etc) of a meal (e.g. cabbage
casserole with sausage) or generating recipes, we focus on predicting
ingredients and portions simultaneously (e.g. 1 carrot, 1.5 pounds cabbage, 1
onion) as shown in Figure 1. One direct application of this research is the
automatic generation of meal kits content.

Meal kit services, which offer and mail pre-portioned ingredients of specific
meals have become very popular in recent years. These services also offer
manually provided images and information about ingredients, and their
portions. Our proposed model can automatically extract knowledge about
ingredients and portions from a given image making this work directly
applicable to automatic meal kit generation. To our knowledge there is no
research that can automatically generate ingredients and portions at large
scale and be used for applications such as automatic meal kit generation.
Visual information extraction from cooking images can be useful for many other
applications such as calorie estimation, automatic recipe generation, and task
graph generation [12, 13] for automatic recipe preparation given a single
image.

In this paper we propose a two stage pipeline. In the first stage, using a
transformer based decoder [14], the main and optional ingredients of a meal
(illustrated in the given image) are generated sequentially. In the second
stage, all ingredients generated from the first stage are used for the task of
total calorie estimation using a deep model with multiple encoder modules.
Three encoders are deployed in the model to model per ingredient calories,
units, and portions and the total calorie intake of the image. We finally
introduce an application of this pipeline for meal kit generation. This work
has three main contributions:

  * •

Calorie estimation using ingredients and their portions information from a
single image.

  * •

Portion estimation for the purpose of meal reproduction.

  * •

Pipeline and interface for iterative meal kits generation.

Figure 2: The two stage pipeline for calorie estimation. Stage 1: Ingredient
set generation. Stage 2: Estimation of meal calorie using intermediate
estimates of ingredient portions, units, and calories.

The remainder of this paper is organized as follows: In Section II we have an
overview of the related work. In Section III and IV we discuss the first and
second stages of the model respectively. Finally, in Section V and VI we
analyze the results and conclude the paper.

##  II Related Work

###  II-A Dish Classification

Dish (or food) classification can be considered as an application of image
classification. Some work such as [5, 15] provide experimental studies on
small scale datasets to recognize food (or dish) types from a single given
image. Many applications of dish classification incorporate non-visual context
such as geo-location to increase dish classification accuracy [16, 17]. In
[6], to address the dynamic and changing nature of food and dish
classification, Horiguchi et al. proposed a personalization based model for
dish classification. The commonality between the proposed work in this area is
a small scale dataset and a deep learning model to address that. We in this
paper create our own set of dish types and use a state of the art deep
learning network to perform dish classification.

###  II-B Ingredient Recognition

Research in the area of ingredient recognition can be classified into two main
categories: retrieval based, prediction. In retrieval based applications, a
list of ingredients or the whole recipe is retrieved based on creating an
embedding and retrieving the appropriate image match from the dataset [9, 18,
19, 7]. This body of work requires the predicted combination of ingredients to
be a fixed set as seen in one of the datasets. To handle this issue,
ingredient prediction approaches inspired by multi-class modeling [20, 21,
22], recurrent image captioning [23, 24, 25], and auto-regressive list
prediction methods emerged [14, 1]. Ingredient state recognition is also
another field of study that has been under-studied. Introducing new
ingredients states datasets [26, 27], or addressing the states problem as
image classification or multi-class labeling (i.e. ingredient-state tuple)
problem [26, 28] are instances of research in this area.

One recent work that we use as our baseline for ingredient prediction is the
inverse cooking research [1]. In this work, both ingredients and recipe are
generated in auto-regressive manner using the transformer model [1].

###  II-C Portions Estimation

Research on ingredient portions has mainly been conducted in the context of
calorie estimation. In most of the work [29, 30, 31, 32] portions of
ingredients (e.g. apple) are identified after image segmentation [29] and size
computation [33] using various approaches (e.g. geometry, 3d modeling) for
calorie estimation of very simple cooking images in small scale datasets [31].
Also, approaching portions from a visual recognition and segmentation view may
not be feasible in meals where the ingredients is not visually discerned (e.g.
chicken in soup). Therefore we approach the portions problem in a self-
attention query based manner using a large scale dataset (i.e. Recipe1M).

###  II-D Calorie Estimation

Calorie estimation from image has gained attention in food and image
processing research. Some methods propose multi-stage pipelines to predict
food categories/ingredients, identify portions/sizes and estimate the calorie
intake of the food [29, 30, 31, 32]. Some of these methods take two input
images to define depth and segmentation of food in the image [29, 31]. These
algorithms use model based or deep learning based methods for the recognition
stage and standard nutritional fact tables for calorie estimation [32]. Some
literature directly provide estimates of calorie from a food image [34, 35] by
predicting the food category itself and directly mapping it to a calorie
intake for that meal with [35] or without a reference object. The drawback to
these methods is that they do not take into account the variety of ingredients
different versions of a meal can have. Some work propose a CNN-based direct
image method that take into account multiple food in one image but still do
not consider the containing ingredients of meals for calorie estimation [36,
37, 38]. Also, in [39] authors propose a food-estimation Bayesian framework
for food-balance estimation which considers a limited number of food
categories with a limited number of classes each with limited discrete values.

Most of the work done in the area of calorie estimation assumes that images
are of food with clear segmented boundaries [29, 31, 11] and do not consider
addressing more complex food such as mixed or cooked meals where the
containing ingredients are not clear. Another issue with these work is that
the dataset used is very small and low diversity and the images are captured
in a well-controlled setting.

On the other hand, there are a few literature that exploit ingredients for
image-based calorie estimation [40]. In [40] a deep learning based method is
proposed for simultaneous learning of calories, categories, ingredients and
cooking directions. Datasets such as Japanese calorie-annotated food photo
dataset and the American calorie-annotated food photo dataset [40] are
datasets with calorie annotations.

##  III Ingredient Generation

We propose a two-stage pipeline for calorie estimation of a given input image
as shown in Figure 2. In the first stage (i.e. ingredient generation), given
an image, the dish type, main ingredients and optional ingredients are
predicted sequentially. In the second stage, using the image and the generated
ingredients, an estimate of portions and calories of each of the ingredients
is provided. Hierarchically the entire calorie intake of the given image is
also estimated. In this section, we will discuss the first stage and in the
next section the second stage (i.e. portion and calorie estimation) is
elaborated.

(a) Predicting main ingredients iteratively (Main Ingredient Module).

(b) Predicting optional ingredients given main ingredients (optional
Ingredient Module).

Figure 3: Prediction of ingredients given image features. A list of main
ingredients are predicted in the first stage and afterwards a list of optional
ingredients are predicted given the list of main ingredients. The list of main
ingredients is predicted iteratively through iterations of correction.

###  III-A Dish Classification

The first stage of the pipeline starts with an estimation of the dish type
$d$, given the input image. A convolutional neural network (with a Resnet
base) is trained to classify the image into one of $N_{d}$ classes of dishes.
We name the predicted dish as $D$. In the next steps, the predicted dish type
is given along side the image as input to the ingredient generation model. The
dish vocabulary is also combined with the ingredient vocabulary to provide
token embeddings for dish names alongside ingredient embeddings.

###  III-B Ingredient Generation

The ingredient generation module utilizes a transformer decoder as in [14, 1]
and performs ingredient generation in two steps. In the first step, the
transformer generates a set of main ingredients, $I^{main}$, one step at a
time as shown in the equation below and Figure 3.a. The model provides
confidences for each ingredient at each step that can be useful for correcting
wrongful selections.

$I^{main}=p(I^{main}_{j+1}/I^{main}_{j},F,D)$ (1)

$I^{main}_{j}$ and $I^{main}_{j+1}$ are the j-th and (j+1)-th generated
ingredients, $D$ is the predicted dish type from previous stage and $F$ is the
image features extracted from the last convolutional layer of a pre-trained
Resnet mapped to $(n\times n)\times e_{size}$ where $e$ is the embedding size.
All ingredient and dish tokens are projected to embeddings of size $e_{size}$.
The $p$ is a transformer decoder that takes as input previous ingredients and
image features and generates ingredients one step at a time as shown in Figure
3.a. In the second step, the transformer module generates a set of optional
ingredients one step at a time given the main ingredients.

$I^{opt}=p(I^{opt}_{k+1}/I^{opt}_{k},I^{main},F,D)$ (2)

All generated ingredients are merged and used in next stages of the pipeline
$I=I^{opt}+I^{main}$.

####  III-B1 Formulation

The ingredient generation module takes image features and the predicted dish
class, $D$, as input. Image features are extracted from the final
convolutional layer, $V\in\mathbb{R}^{M\times n\times n}$. A $1\times 1$
convolution layer and a reshape layer are applied to make the image features
$F\in\mathbb{R}^{s_{1}\times e_{size}}$.

$F={reshape(conv_{1\times 1}(V))}$ (3)

The module also takes as input a one-hot token matrix comprising of a dish
token and ingredients. Therefore, the vocabulary includes all ingredients in
the dataset, $N_{i}$, combined with all dish vocabulary, $N_{d}$. Therefore,
the dish classes would be considered in the input vocabulary but not in the
output vocabulary making the total vocabulary size $N=N_{d}+N_{i}$. The input
matrix of tokens is of size $I\in\mathbb{R}^{s_{2}\times N}$ where $N$ is the
size of vocabulary and $s_{2}$ is the number of maximum ingredients the model
accepts as input. The first token in $I$ is always the predicted dish class
$D$ from the dish classification stage and the next tokens are generated
ingredients from the previous step of the transformer. The token matrix is
projected to an embedding matrix, $E\in\mathbb{R}^{s_{2}\times e_{size}}$,
through an embedding layer. The output of the transformer are generated
ingredients at each step, $O_{ingr}\in\mathbb{R}^{s_{2}\times N}$.

##  IV Portion and Calorie Estimation

In the second stage, a two stream network is proposed to estimate the total
calorie intake of a recipe (depicted in Figure 4) given the generated
ingredients and image features. The first stream of the model is the calorie
module that provides per ingredient calorie estimations. The second stream
includes a unit module, a portion module and an alignment module. The unit
module generates per ingredient unit predictions (e.g. teaspoon), the portion
module uses the estimated units and the input ingredients to generate per
ingredient portion estimations and the alignment module creates alignment
between the generated units and portions using per ingredient calorie
estimations. The model is trained end-to-end.

###  IV-A Inputs and Intermediate Outputs

The proposed model takes as input all generated ingredients $I$ and image
features $V$ and contains two streams with four modules and three intermediate
outputs. All encoders have two inputs; image features and ingredient
embeddings. Image features are extracted from the last convolutional layer,
$V\in\mathbb{R}^{M\times n\times n}$, and are projected and reshaped with a
($1\times 1$) conv layer to $F_{c}\in\mathbb{R}^{s_{1}\times e_{size}}$,
$F_{u}\in\mathbb{R}^{s_{1}\times\frac{e_{size}}{2}}$,
$F_{p}\in\mathbb{R}^{s_{1}\times e_{size}}$, and
$F_{a}\in\mathbb{R}^{s_{1}\times e_{size}}$ for the calorie encoder, unit
encoder, portion encoder, and alignment encoder respectively.

All encoders create intermediate embeddings. The calorie encoder, unit
encoder, portion encoder, and alignment encoder create intermediate embeddings
$E_{c}\in\mathbb{R}^{s_{2}\times e_{size}}$,
$E_{u}\in\mathbb{R}^{s_{2}\times\frac{e_{size}}{2}}$,
$E_{p}\in\mathbb{R}^{s_{2}\times e_{size}}$, and
$E_{a}\in\mathbb{R}^{s_{2}\times e_{size}}$ respectively. The details of how
these intermediate embeddings are generated are explained in Section IV-B.

Besides image features, each encoder has another set of inputs which is either
ingredient embeddings or a combination of ingredient embeddings and
intermediate embeddings from other encoders. The generated ingredients from
stage one are converted to ingredient embeddings through an embedding layer
and are fed to the calorie encoder, $I\in\mathbb{R}^{s_{2}\times e_{size}}$.
The original ingredient embeddings, $I$, are projected to smaller sized
embeddings $I_{u}\in\mathbb{R}^{s_{2}\times\frac{e_{size}}{2}}$ for the unit
encoder. The input for the portion encoder, $I_{p}\in\mathbb{R}^{s_{2}\times
e_{size}}$, and alignment encoder, $I_{a}\in\mathbb{R}^{s_{2}\times{size}}$,
are created by concatenating the smaller sized ingredient embeddings, $I_{u}$,
and the intermediate unit embeddings, $E_{u}$.

###  IV-B Encoder Structure

Each of the four modules include an encoder which follows the exact
architecture of a transformer decoder [14]. The transformer decoder has
multiple identical transformer decoder layers and takes as input three sets of
matrices; queries, keys and values. Each of the transformer decoder layers in
the encoder include two layers of multi-head attention layers with residual
connections. The general formulation of each of the multi-head attention
layers is shown in Equation 7.

$\begin{array}[]{l}MultiHead(Q,K,V)=Concat(head_{1},...,head_{h})W^{o}\\\
where:head_{i}=Attention(QW_{i}^{Q},KW_{i}^{K},VW_{i}^{V})\end{array}$ (4)

where the projections are parameter matrices
$W^{o}\in\mathbb{R}^{e_{size}\times s_{2}}$,
$W_{i}^{Q}\in\mathbb{R}^{\frac{e_{size}}{h}\times s_{2}}$,
$W_{i}^{K}\in\mathbb{R}^{\frac{e_{size}}{h}\times s_{2}}$,
$W_{i}^{V}\in\mathbb{R}^{\frac{e_{size}}{h}\times s_{1}}$ where $h$ is the
number of heads, and $Attention$ is the scaled dot product attention mechanism
from [14] and shown in Equation 5.

$Attention(Q,K,V)=softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V$ (5)

where $Q$ and $K\in\mathbb{R}^{\frac{e_{size}}{h}\times s_{2}}$, and
$V\in\mathbb{R}^{\frac{e_{size}}{h}\times s_{1}}$ are projected matrices of
queries, keys and values. In our model both queries and keys are a set of
ingredient embeddings and values are image features. The last component of a
transformer decoder layer is a position-wise feed forward network that is
applied after the two multi-head attention layers [14]. A stack of transformer
decoder layers makes a transformer decoder. The output of the transformer
decoder is a matrix of embeddings $E\in\mathbb{R}^{s_{2}\times e_{size}}$.
Each of the intermediate embeddings ($E_{c},E_{u},E_{p},E_{a}$) discussed in
IV-A are the output of a transformer decoder. For more details on transformer
decoders readers are referred to [14].

###  IV-C The Final Network

The first stream of the network contains the calorie module and the second
stream contains three modules (unit, portion and alignment encoders). The
modules contain encoders that create intermediate embeddings. Calorie,
portion, and alignment intermediate embeddings, $E_{c}$, $E_{p}$, $E_{a}$,
$E_{u}$ are projected to intermediate outputs through (per ingredient)
identical fully connected layers, $f_{c}$, $f_{u}$, $f_{p}$, $f_{a}$.

$\begin{array}[]{l}o_{c}=f_{c}(E_{c})\\\ O_{u}=f_{u}(E_{u})\\\
o_{p}=f_{p}(E_{p})\\\ o_{a}=f_{a}(E_{a})\end{array}$ (6)

where $o_{c}\in\mathbb{R}^{s_{2}\times 1}$, $O_{u}\in\mathbb{R}^{s_{2}\times
N_{units}}$, $o_{p}\in\mathbb{R}^{s_{2}\times 1}$, and
$o_{a}\in\mathbb{R}^{s_{2}\times 1}$ are intermediate calorie, unit, portion,
and alignment outputs for $s_{2}$ ingredient inputs.

The combination of intermediate unit and portion outputs provides a
representation of the amount of each ingredient. The purpose of adding an
alignment module with the ingredient calories loss incorporated is to align
unit and portions with the calorie values.

Attention reduction layers are applied to the calorie embeddings, $E_{c}$,
from the first stream and alignment embeddings $E_{a}$ from the second stream
to create a reduced calorie vector $r_{c}\in\mathbb{R}^{1\times e_{size}}$ and
a reduced alignment vector $r_{a}\in\mathbb{R}^{1\times e_{size}}$
respectively. Projections $P_{cal}$ and $P_{align}$ are applied to $r_{c}$ and
$r_{a}$ respectively and their outputs are concatenated to produce the output
of the model (total calorie estimate). This two stream (four modules) model is
named $T_{upc}$ henceforth.

Figure 4: The calorie estimation model with individual ingredient units,
portions, and calories incorporation.

###  IV-D Losses

Two types of loss are used in the entire model in five different locations.
One MSE loss is used for ingredient calorie estimation in the first stream
($L_{1}$). The weighted cross-entropy loss is used for ingredient unit
classification ($L_{2}$). Units with less frequency in the training data are
assigned a larger weight for loss computation. Three MSE losses are used for
portion estimation ($L_{3}$), calorie estimations in the alignment module
($L_{4}$) and total calorie estimation ($L_{5}$). The final loss is computed
as below with $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$,
$\lambda_{5}$ being hyperparameters.

$\begin{array}[]{l}L=\lambda_{1}L_{1}+\lambda_{2}L_{2}+\lambda_{3}L_{3}+\lambda_{4}L_{4}+\lambda_{5}L_{5}\end{array}$
(7)

##  V Experiments and Results

This section comprises of an overview of the dataset used in experiments, some
of the implementation details, and all the experiments to evaluate the
proposed model.

###  V-A Dataset

We train and evaluate our models on the Recipe1M dataset [7], composed of
1,029,720 recipes scraped from cooking websites. The dataset contains 720,639
training, 155,036 validation and 154,045 test recipes, with a title, a list of
ingredients, a list of cooking instructions and/or an image. In our
experiments, we use only the recipes containing images, and remove recipes
with less than 2 ingredients resulting in 252,547 training, 54,255 validation
and 54,506 test samples [1].

Because the data was extracted by scraping cooking websites they are
unstructured and include redundant ingredients. We follow all the operations
in [1] (e.g. cluster 400 different cheese categories into one) and therefore
reduce the number of ingredients from 16,823 to 1488. Some of the ingredients
in [1] were clustered or split inaccurately. We performed a semi-automatic
correction of some of the inaccurate clusters in [1]. For example we separated
tomato from tomato sauce which were originally merged, or we merged sausages
that were classified as separate classes with their brand names into one
category. We only maintain high frequency ingredients (top 95%) which results
in 202 ingredients. After this stage we maintain 132,442 train and validation
recipes and 23,602 test recipes. We further automatically cluster recipes into
32 classes (i.e. dish names) using their ingredients list and recipe titles
and remove outliers to keep 67,359 train and validation recipes and 11,743
test recipes. We extract ingredients (e.g. tomato paste) and their portions
(e.g. 1 spoon) from the provided text for each ingredient in the dataset [7]
and remove the recipes that contain extreme portion outliers or include
ingredients with missing portions. The final dataset contains 42,455 train and
validation recipes and 7,575 test recipes, with a title, a list of
(ingredients, portions, units) tuples, and images.

###  V-B Implementation Details

Images were resized to 256 pixels in their shortest side and random crops of
$224\times 224$ were taken for training. For evaluation the central $224\times
224$ pixels were select. For the transformer encoders, we use a transformer
with 2 blocks and 8 multi-head attentions, each one with dimensionality 64.
For the last layer of the transformer, reduced embeddings of sizes 512, and
1024 were used. To obtain image embeddings we use the last convolutional layer
of ResNet-50 model which would be of size $2048\times 7\times 7$ (i.e.
$M=2048,n=7$ in Section III). All the word embeddings and all transformer
decoder input vectors were set to 1024. A maximum of 10 ingredients is used
for each recipe. The models are trained with the Adam optimizer [41] for 60
epochs. Loss hyperparameters are all set to 1 with the exception of
$\lambda_{4}$ being set to 0.1. All parts of the model are implemented with
PyTorch. A GUI is implemented for user correction using Python and the Flask
API.

###  V-C Results

We perform ingredient generation in the first step of the pipeline. State of
the art ingredient generation models [1] require much more improvements to be
applicable to real world problems. Therefore, we proposed a multi-level model
for ingredient generation. We also include semi-automatic user correction at
each level to provide accurate ingredients for the second stage. For this, we
created an interface using Flask and obtain user feedback on our results at
each level of the ingredient generation stage 111http://www.rpal-
eve.cee.usf.edu/.

####  V-C1 Ingredient Generation

In the first step of the experiments we implemented a Resnet based CNN for
dish classification and achieved near to 50% accuracy in classification of 32
dish types (e.g. omelette, cake, pizza, salad, etc). Ingredient generation is
performed as suggested in Section III. The ingredient vocabulary set comprises
of 202 ingredients (e.g. butter, chicken, strawberry, flour, etc). The models
for the ingredient generation include the dish types in their input vocabulary
(i.e. 234 input tokens). Table I shows intersection over union (IoU) results
for three different experiments with or without dish types given as input.
Results for a) main ingredients generation: estimation of up to five main
ingredients, b) optional ingredients generation: estimation of up to 10
optional ingredients given an image and main ingredients and c) all
ingredients generation is shown in the table. The results clearly show that
when the model contains verbal knowledge about the dish type it estimates a
more accurate ingredient list.

TABLE I: Intersection over union (IoU) for a) main ingredients, b) optional ingredients, and c) all ingredients generation. | Given dish | No dish given
---|---|---
Main Ingredients | 32.2% | 27.3%
Optional Ingredients | 49.2% | 47.7%
All Ingredients | 34.1% | 31.5%

The ingredient estimates from the model for main ingredient generation is
revised using user feedback. Table II shows results for estimating one
ingredient at a time given the previous ingredients are revised by the user.
As it can be observed in this table, the accuracy of generating main
ingredients is much higher when revision happens. To evaluate revision
accuracy, the revised ingredient is fed back to the model (and not the actual
ground-truth ingredient at that time step). Therefore, the number of
ingredients available at each time step in the test set is different for when
the dish name is given in comparison to when it is not given as input.
Furthermore, having the dish name as input for the model improves performance.

TABLE II: Main Ingredient Prediction (Given N main ingredients) Predicting | Dish name
---|---
Given | Not given
1st Ingredient | 67.0% | 56.8%
2nd Ingredient | 55.3% | 49.1%
3rd Ingredient | 52.4% | 44.4%
4th Ingredient | 45.3% | 37.1%
5th Ingredient | 9.2% | 21.9%

####  V-C2 Portions and Units Estimation (meal kits)

The portion encoder creates estimates of portions of the listed ingredients by
providing a value and a unit. The portions and units of an ingredient can be
used to generate automatic meal kits procedures for a given image. This stage
of the model is evaluated based on the mean absolute error distance between
the target and predicted portion and the accuracy of classification of units.
In our experiments six different units as shown in Table III is used. In Table
III, MAEs for predicted portions for each individual unit and the prior MAE
for each unit is shown. It can be observed that the calorie estimations are
much better than the prior but the portions are not as good. The reasoning is
that to have an accurate measurement of the amount of the ingredient an
accurate combination of portion and unit is needed. The unit estimation
accuracy of the model is 72.3%. In Figure 5, a few visual results of generated
ingredients and their portions and calorie intakes for meal kits generation
purposes is depicted.

TABLE III: MAE portion estimation of different units Metric | Calorie MAE
---|---
Prior | Estimated
pound | 323.7 | 162.8
ounce | 238.7 | 155.1
cup | 201.6 | 92.9
count | 144.6 | 52.5
tblsp | 99.4 | 68.9
tsp | 7.6 | 5.3
total | 144.2 | 78.5
Figure 5: Examples of results from different parts of the end-to-end pipeline
which includes predicted dish name, generated ingredients, portions and
calorie estimates. The total predicted calorie intake and its ground-truth
value is shown in the last column (gt: ground-truth).

####  V-C3 Calorie Estimation

For calorie estimation, we evaluated the proposed model using mean absolute
error ($MAE$) and percentage mean absolute error ($MAE_{\%}$) on unseen test
set. We compare the final model ($T_{upc}$) with a few variations of
transformer based models. In one of the models we remove the ingredient based
calorie training and only train the model for total recipe calories
($T_{calorie}$). In another version, we maintain the same base but we only
individually generate calorie values for each ingredient and a final calorie
of the entire recipe ($T_{calories}$).

We also compare the final model with a simple neural network for generating a
calorie for each ingredient individually and adding them up to generate the
final calorie ($NN_{calories}$) and a single ingredient based neural network
based on portions, units and calories of each ingredient and adding them up to
generate the final calorie ($NN_{upc}$).

We compare the model with a trained CNN on recipe calories ($CNN$), recipe
calories estimated using prior calorie of each ingredient ($P_{imean}$), and
recipe calories estimated estimated using prior dish names ($P_{dish}$).

TABLE IV: Calorie Estimation Model | Total Calorie MAE
---|---
$MAE$ | $MAE_{\%}$
$T_{upc}$ | 279.4 | 37.5%
$T_{calorie}$ | 394.5 | 49.9%
$T_{calories}$ | 283.5 | 38.1%
$NN_{upc}$ | 306.7 | 39.7%
$NN_{calories}$ | 310 | 40.9%
$CNN$ | 380 | 49.8%
$P_{imean}$ | 323.3 | 44.7%
$P_{dish}$ | 407 | 52.3%

We can observe from Table IV that the transformer model which uses
intermediate ingredient portion and calorie estimates performs the best in
both predicting ingredient calorie estimation and recipe calorie estimation.
Using ingredient based neural nets performs good but because the self
attention between different ingredients is not modeled it performs worse than
transformer based models. The CNN based model reaches to 49.8% $MAE_{\%}$ for
recipe calorie estimation showing the need for ingredient incorporation in
this application. Just using the dish name ($P_{dish}$) and only knowing the
ingredients ($P_{imean}$) perform relatively worse than the proposed model in
estimating meal and ingredient calorie.

###  V-D Discussion

Most state-of-the art models focus on ingredient retrieval given an image [9,
18]. The models that generate ingredients (non-retrieval) from a given image
have very low performance [1]. We incorporate a state-of-the-art method as
base with a slightly modified (i.e. corrected) dataset and add semi-automatic
correction into the model to enhance the ingredient generation model for meal
kits generation.

Our model also includes a stage (in the overall end-to-end pipeline) where
portions and units (e.g. 1 spoon of oil) are generated for each ingredient
using both attention between ingredients themselves and their underlying image
features using transformer properties. An element in the portions generation
stage that is unique to our model and beneficial is the containment of six
unit types in the output which can automatically be mapped to meal kit content
generation.

Another potential property of the model is the use of per ingredient unit and
portion estimation to backtrack and provide re-computed ingredient amounts
given a new serving amount of the meal.

To our knowledge, this work is the first work on large scale calorie
estimation on images with intermediate ingredient and portions estimation
where the ingredients may or may not be (e.g. salt) visually seen in the
image. Also, to prepare data for portions and calorie estimation we removed
instances that lacked enough data from the Recipe1M dataset and therefore the
dataset used in the experiments is uniquely tailored for this application
making it impossible to compare the results with any baseline methods.

##  VI Conclusion and Future Work

The main objective of this paper was calorie estimation and identifying the
underlying ingredients and their portions for meal kits generation. We
proposed a pipeline using deep encoders to extract image features and generate
ingredients (and their portions) from a given image. The ingredients are
extracted using an auto-regressive encoder which captures both relative
association between ingredients through self attention and ingredients and the
image features through the transformer and Resnet features. The portions and
furthermore calories of ingredients are also extracted with the assumption
that knowing all ingredients contained in a recipe image can contribute to the
portion knowledge base of a recipe (i.e. self-attention between portions of
ingredients). The total calorie is estimated using all generated mid-level
knowledge in an end-to-end manner. The current pipeline can semi-automatically
generate ingredients with portions, and calories and a final calorie estimate
of the entire recipe. In future work, we plan to extend experiments in a wider
range of ingredients and recipes where more accurate portion estimates is
available. We also plan to integrate this pipeline with robotic manipulation
and use the predicted ingredients and their portions, state changes to infer
manipulation tools based on the relationships between ingredients and tools
[42, 43]. Eventually the complete system will be able to infer a robot
manipulation task graph for cooking the meal in an image from its meal kit
[44, 45].

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# New Dynamical Channel: Wide Binaries in the Galactic Center as a Source of
Binary Interactions

Erez Michaely  Department of Physics and Astronomy, University of California,
Log Angeles, CA 90095, USA  Mani L. Bhaumik Institute for Theoretical Physics,
University of California, Log Angeles, CA 90095, USA  Smadar Naoz  Department
of Physics and Astronomy, University of California, Log Angeles, CA 90095, USA
Mani L. Bhaumik Institute for Theoretical Physics, University of California,
Log Angeles, CA 90095, USA

###### Abstract

The inner $500\rm pc$ in the galactic center is dense with stars and a
dynamically hot environment. Here, we focus on wide binaries as a source of
tidally or collisional interactions. These binaries were previously ignored as
sources of binary interaction because they are too wide to have a close
pericenter passage, or they will quickly become unbound (ionized) due to
gravitational interactions with passing neighbors. However, we show that wide
binaries tend to interact more frequently with neighboring stars due to their
larger cross-section for gravitational impulse interactions. These
interactions mainly torque the wide system, causing it to change its
eccentricity. As a result, the eccentricity might be excited to sufficiently
high values, causing the binary to interact at the pericenter. As a proof of
concept, we present four channels of such interactions: binary main-sequence
(MS), white-dwarf (WD) - MS, black hole - MS, and lastly, WD-WD. During
Galaxy’s lifetime, we predict tens of thousands of MS-MS interacting binaries
that may form G2-like objects later appear younger than their environment.
X-ray signatures and, perhaps, supernovae may result from thousands of WD-MS
and WD-WD interacting binaries from this channel. Lastly, we predict a few
hundred BH-MS interacting binaries at the inner $500$ pc.

stars: kinematics and dynamics — keyword2 — keyword3

##  1 Introduction

It is well known that almost every galaxy, our own Milky Way (MW) included,
host a supermassive black hole (SMBH) at it’s center, with masses between
$10^{6}M_{\odot}-10^{9}M_{\odot}$ (Ghez et al., 1998, 2000, 2008; Gillessen et
al., 2009; Genzel et al., 2003, 2010; Ferrarese & Ford, 2005; Kormendy & Ho,
2013). The SMBHs are surrounded by dense regions of stars and compact objects
(e.g. Ghez et al., 2003, 2005; Alexander, 2005; Hopman, 2009; Chu et al.,
2018, and others). In the context of the stellar dynamics in the center of a
galaxy, we consider here two distinct regions. One is the inner region which
is completely dominated by the gravitational potential of the SMBH, and the
other is an extended region all the way toward the Galactic field.

The inner region, called ”the sphere of influence,” extends to a radius where
each object is dominated by the SMBH gravitational potential. In most galaxies
within the sphere of influence reside a very dense star cluster named nuclear
star cluster (Ghez et al., 2003, 2005; Alexander, 2005; Hopman, 2009; Chu et
al., 2018; Neumayer et al., 2020), though it can extend to larger radii. These
NSCs are the densest star clusters in the Universe, hence host to very
interesting dynamical scenarios (e.g., Magorrian & Tremaine, 1999; Merritt &
Poon, 2004; Hopman & Alexander, 2005; Antonini et al., 2010, 2011; Prodan et
al., 2015; Aharon & Perets, 2016; Sari & Fragione, 2019; Linial & Sari, 2022;
Naoz et al., 2022; Rom et al., 2023, and others). The mass within the sphere
of influence is approximately two times the mass of the central SMBH. The size
of the sphere of influence is given by $r_{h}\approx GM_{\rm
SMBH}/\sigma_{0}^{2}$, where $M_{\rm SMBH}$ is the SMBH mass and $\sigma_{0}$
is the velocity dispersion. In the MW $\sigma_{0}\approx 75\rm{kms^{-1}}$
(Gebhardt et al., 1996, 2000a, 2000b) this corresponds to an estimated radius
of $\sim 3.1\rm{pc}$ enclosing millions of stars.

Beyond the radius of influence, the density of stars remains high and slowly
decreases as it converges with the galactic bulge (Ballero et al., 2007;
Athanassoula et al., 2017; Barbuy et al., 2018) and further away from the
Galactic field. At this regime, the gravitational potential is dominated by
both the SMBH and the stellar mass enclosed. In the MW this corresponds to a
maximal radius of $500-1000\rm{pc}$ with $10^{9}-10^{10}$ of stars. Aside from
stars and compact objects inside the galactic bulge, there is a presence of
unusually high concentration of molecular gas. These molecular gas clouds are
conducive to star formation and the birth of new stars in the bulge and this
central molecular zone (CMZ Mills, 2017; Langer et al., 2017; Sormani et al.,
2020, e.g.).

In the context of this work, we largely neglect the gas contribution to the
dynamics because the gas density is still low and will have a negligible
effect on the dynamics. On the other hand, the star formation that takes place
in the CMZ may give rise to young stars and binary stars. Particularly, in the
field, binaries and higher multiplicity systems are very common (Tokovinin,
1997; Raghavan et al., 2010; Sana et al., 2014; Moe & Di Stefano, 2016).
Further, binaries and high multiples white dwarfs have been shown to be
prevalent in the local neighborhood (e.g., Kraus & Hillenbrand, 2009; El-Badry
et al., 2021; Heintz et al., 2022; Shariat et al., 2023).

Additionally, the detection of a few binaries in the MW’s NSC (e.g., Ott et
al., 1999; Martins & Plez, 2006; Pfuhl et al., 2014) indicated a broader
binary population in the NSC. Furthermore, other observational studies point
out that the OB binary fraction in the NSC is comparable to the fraction found
in young massive stellar clusters (e.g., Ott et al., 1999; Rafelski et al.,
2007), and the fraction of Eclipsing young OB binaries in the galactic center
is consistent with the local fraction observed (Chu et al., 2018; Gautam et
al., 2019). The theoretical study (Stephan et al., 2016) showed that the
binary fraction is $\sim 70\%$ for the population from the most recent star
formation episode, 6 Myr ago, in the NSC (e.g., Lu et al., 2013). Lastly, the
peculiar configuration of the young stellar disk in the galactic center
suggests it hosts a high fraction of binaries (Naoz et al., 2018).

In recent years, it was well established that wide systems ($>1000\rm AU$), of
either binaries or triples, in the field of their host galaxy are a rich
source of binary interaction (Kaib & Raymond, 2014; Michaely & Perets, 2016,
2019a, 2020; Michaely, 2020; Michaely & Shara, 2021; Michaely, 2021; Raveh et
al., 2022; Michaely & Naoz, 2022; Grishin & Perets, 2022; Rozner & Perets,
2023). Notably, while the galactic field is generally considered collisionless
for many dynamical processes, it is not the case for wide binaries. Wide
binaries, due to their large geometrical cross-section, which is proportional
to their semi-major axis (SMA), interact frequently with flyby field stars.
These interactions are well modeled by the impulse approximation, and their
rate is given by the standard $\Gamma=n_{*}\sigma_{\rm{geo}}v_{\rm{env}}$.
Where $n_{*}$ is the local stellar density, $\sigma_{\rm{geo}}$ is the
geometric cross-section, and $v_{\rm{env}}$ is the relative velocity between
the binary and the flyby of the encounter, which we take to be the local
velocity dispersion.

In this letter we study the interaction between wide binaries in the galactic
center and flyby stars. Notably, these interactions can naturally result in
observational signatures. Particularly here, we explore four different
dynamical channels as proof of the importance of the proposed dynamical
process. These channels involve main-sequence stars, white dwarfs, and black
holes, see below.

This paper is organized as follows: In section 2 we present the dynamical
scenario qualitatively and quantitatively. In section 3 we describe the
population model used in our calculations, in section4 we present the four
channels the rates, and the distribution of the systems of interest. In
section 5 we discuss the results, and we summarize in section 6.

##  2 Physical Picture

As mentioned in Sec. 1 the galactic field is collisional for wide systems
(Kaib & Raymond, 2014; Michaely & Perets, 2016, 2019a, 2020; Michaely, 2020;
Michaely & Shara, 2021; Michaely, 2021; Raveh et al., 2022; Michaely & Naoz,
2022). Expanding on this body of work we describe the interaction of wide
binaries in the inner $500\rm{pc}$ of the galactic center. We note that in
this manuscript we focus on binaries, we leave the treatment for triples to
future study.

In what follows we describe, in a qualitative manner, the physical picture of
the dynamical scenario, namely the gravitational interaction between wide
systems and random flyby stars in the galactic center. The galactic center is
a collisional environment, in which the soft-hard binary (Heggie, 1975) is
determined by the average kinetic energy of the stars. We focus on the soft
binaries, hereafter wide binaries. Wide binaries interact randomly with flyby
stars, the encounter velocity is determined by the local velocity dispersion.
The gravitational interaction is impulsive by nature, e.g. the interaction
timescale $t_{{\rm int}}\equiv b/v_{{\rm enc}}$ is much shorter than the
orbital period of the soft binary, $P$. Where $b$ is the closest approach of
the flyby to the binary’s center of mass. It was shown in the literature that
these interactions change the binary orbit characteristics, namely the orbital
energy (decrease/increase the SMA), but more significantly, torque the system
and change the binary eccentricity $e$ (e.g., Lightman & Shapiro, 1977;
Merritt, 2013).

Such consecutive interactions change the orbital eccentricity and at times
might excite the eccentricity to critical values that lead to interaction at
pericenter. This process occurs at the same time with the ”ionization” process
of soft binaries. Due to random interactions of these soft binaries with flyby
stars they gradually are ”ionized”, decreasing the number of available
binaries to interact at pericenter due to the process described here. The
timescale for the ionization is given by the average halt-life time of the
binary is (Bahcall et al., 1985):

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

As a result of the random interaction, we expect to find an increasing
pericenter interaction probability as a function of time, due to the random
nature of the encounters. However, the ionization of the binaries leaves the
total binaries number a decreasing function of time, as the total number of
binaries available for interaction decreases. We note here that as the
ionization probability is a function of the SMA, we expect the contribution of
the widest systems only happens at early times since formation.

We Consider an ensemble of isotropic wide binaries in the galactic center,
each is comprised of two objects with masses $m_{1}$ and $m_{2}$, and the
total (reduced) mass is denoted by $M\left(\mu\right)$. All binaries have the
same SMA $a$, and a thermal distribution of orbital eccentricities,
$f(e)de=2ede$. We define a length scale called the interaction distance to be
$d$, this length scale is set by the type of interactions we are interested in
the pericenter.

As an illustrative example in (Michaely & Perets, 2016), we were interested in
calculating the formation of an X-ray source from an initially wide BH-MS
binary, hence we set $d$ to be the tidal radius for which the binary will lose
enough orbital energy and eventually become an X-ray source, specifically we
set $d=5R_{*}$.

Once one sets the interaction distance $d$, the critical eccentricity,
$e_{c}$, for which each pericenter distance, $q=a(\left(1-e\right)$ is equal
or smaller than $d$ is given by

$e_{{c}}=1-\frac{d}{a}.$ (2)

Next, we calculate the loss cone fraction, $F_{q}$, namely, the fraction of
systems, given thermal distribution of eccentricities, that satisfies $q\leq
d$, which is equivalent to $e\geq e_{c}$

$F_{q}=\int_{e_{c}}^{1}2ede=1-e_{c}^{2}=\frac{2d}{a}.$ (3)

In the last Equation, we used the approximation $d/a\ll 1$.

Now, we are ready to introduce the interaction with the environment,
specifically with flyby stars. The environment in which the ensemble of wide
systems resided is characterized by its local stellar density, which is a
function of distance from the center of the host galaxy,
$n_{*}\left(r\right)$, the local velocity dispersion also as a function of
distance, and the average mass of stars. We model the stellar density function
by the following (Genzel et al., 2010; Rose et al., 2020)

$n_{*}\left(r\right)=\frac{M_{\bullet}}{2\pi r_{h}^{3}m_{\rm
mean}}\left(\frac{r}{r_{h}}\right)^{-\alpha}\ ,$ (4)

where $r_{h}=GM_{\bullet}/\sigma^{2}_{0}$ is the radius of influence, and
$\sigma_{0}$ is the velocity dispersion in the radius of influence, set to be
$\sigma_{0}=75\rm{kms^{-1}}$. The SMBH mass is $M_{\bullet}$ and we set it to
be $M_{\bullet}=4\times 10^{4}M_{\odot}$. The velocity dispersion is a
decreasing function of $r$ from the radius of influence until the boundary of
the field; we adopt the following toy model

$\sigma\left(r\right)=\rm{Max}\left(\sigma_{0}\left(\frac{r_{h}}{r}\right)^{1/2},\sigma_{\rm
field}\right)\ .$ (5)

The flyby mass is taken to be $m_{p}=1M_{\odot}$, $\sigma_{\rm
field}=25\rm{kms^{-1}}$ and we use $\alpha=2$.

Each interaction with the perturber changes the relative velocity between the
binaries components (e.g., see eq.(7)) in Michaely & Perets (2016). One can
calculate the smear cone, $F_{s}$, which is the fraction of phase space to
which the initial relative velocity could be kicked into after a single
passage of the perturber as a function of $b$

$F_{s}=\frac{\pi\theta^{2}}{4\pi}=\frac{27}{4}\left(\frac{m_{p}}{M}\right)^{2}\left(\frac{G\mu}{av_{{\rm
enc}}^{2}}\right)\left(\frac{a}{b}\right)^{4}\ .$ (6)

For the loss cone to be continuously filled, two conditions must be fulfilled.
First, the interaction between the perturber and the wide binary should be
sufficiently strong in order to fill the loss cone. This condition is achieved
when $F_{s}\geq F_{q}$. The second, the replenishment rate into the loss cone,
is equal to the rate of losing binaries from the loss cone, specifically the
binary dynamical timescale. The rate of replenishment is given by the rate of
interaction, which is

$f\equiv\frac{1}{t_{{\rm enc}}}=n_{*}\sigma\left(b\right)v_{{\rm enc}}\ .$ (7)

The condition $F_{s}\geq F_{q}$ is actually a condition of the closest
approach $b$,

$b_{\rm kick}^{2}\leq v_{\rm
enc}^{-1}\left(\frac{27}{8}\frac{m_{p}^{2}}{\mu}\frac{G\mu\pi}{d}a^{4}\right)^{1/2}\
.$ (8)

Now, we are set to find the critical SMA from which the loss cone is
continuously full, under the impulse approximation, as a function of the local
environment characteristics. Thus, for $F_{s}=F_{q}$ and $1/t_{{\rm
enc}}=n_{*}\sigma v_{{\rm enc}}$, we find:

$a_{{\rm crit}}=\left(\frac{2}{27\pi^{4}}\frac{\mu
M}{m_{p}^{2}}\frac{d}{n^{2}_{*}}\right)^{1/7}\ .$ (9)

Binaries with SMA equal to the $a_{{\rm crit}}$ have the highest probability
of interacting at the pericenter. Binaries with SMA smaller than $a_{{\rm
crit}}$ are harder to perturb. Therefore, the probability rate decreases.
Binaries with SMA larger than $a_{{\rm crit}}$ also decrease in probability
due to the smaller loss cone, additionally, these binaries are more likely to
become unbound (ionized) during their life-time. This interplay leads to a
sharp transition in the merger probability, as highlighted in panel a of
Figure 1.

In other words, the full loss cone regime is for all systems with $a>a_{{\rm
crit}}$. The loss rate of systems from the full loss-cone is governed by the
timescale where the binaries are lost from the cone. This timescale is the
orbital period, $P$. Hence, ($\dot{L}_{{\rm full}}$), is given by

$\dot{L}_{{\rm full}}=\frac{F_{q}}{P}.$ (10)

On the other hand, tighter binaries are less susceptible to change due to a
fly-by, this is evident from equation (6). Therefore for $a<a_{{\rm crit}}$ we
expect that the loss cone will not be full all the time, in this “empty loss
cone” regime the loss rate depends on the rate of orbits being kicked into the
loss cone is just the interaction rate, $f$:

$\dot{L}_{{\rm empty}}=F_{q}f=\frac{F_{q}}{t_{{\rm enc}}}$ (11)

From the critical SMA we can calculate the merger probability for each regime,
for $a<a_{{\rm crit}}$ (empty loss cone) and for $a>a_{{\rm crit}}$ (full loss
cone). We note here that by considering the critical SMA we implicitly include
the dependency on the impact parameter, $b$.

Next, we calculate the loss probability which is time and space-dependent for
the two regimes, the empty and full cones.

The loss cone $F_{q}$ is the fraction of systems that are lost from the
ensemble, hence $\left(1-F_{q}\right)$ represents the fraction of binaries
that survive after a single flyby with the relevant timescale. Therefore,
$\left(1-F_{q}\right)$ is a monotonically decreasing function of time. Hence,
the surviving fraction of binaries after time $t$ and the relevant timescale
for the empty loss cone $f$ is $\left(1-F_{q}\right)^{tf}$ (e.g., Kaib &
Raymond, 2014; Michaely & Perets, 2016, 2019a). Therefore the probability for
a wide binary interaction, which complements this expression to unity is

$L_{a<a_{{\rm crit}}}=1-\left(1-F_{q}\right)^{tf}$ (12)

where $t$ is the time since the birth of the binary. As one can expect the
probability only depends on the size of the loss cone and the rate of
interactions. For the limit of $tfF_{q}\ll 1$ we can expand Equation (12) and
take the leading term, to find the loss probability to be approximated by

$L_{a<a_{{\rm crit}}}=tf(b)F_{q}.$ (13)

For the full loss cone regime, the limiting factor is not the value of $f$,
but rather the orbital period $P$. Therefore, the full expression for the loss
probability for $a>a_{crit}$ is

$L_{a>a_{{\rm crit}}}=1-\left(1-F_{q}\right)^{t/P}.$ (14)

For the limit $F_{q}\cdot t/P\ll 1$ we can approximate the probability by

$L_{a>a_{{\rm crit}}}=tF_{q}~{}\frac{1}{P(a)}.$ (15)

Figure 1:  Left panel (panel a): The probability for BMS interaction as a
function of the binary SMA. The probability is calculated at $t=50\times
10^{6}\rm{yr}$ for two MS stars each with mass of $1M_{\odot}$. For two
locations in the galactic center. The blue solid line is at $r=50\rm{pc}$ and
the black dotted line and the red dashed line are at $r=500\rm{pc}$. Right
panel (panel b), the BMS case. The blue solid line is the critical SMA from
eq. (9). The red dashed line is $a_{\rm max}$ the upper cut-off of the BMS
binaries see eq. (19). For binaries inside $\lesssim 200\rm pc$ the $a_{\rm
crit}>a_{\rm max}$. Namely, only the empty loss cone contributes to the rates,
and for $r\gtrsim 200\rm pc$ an added contribution from the full loss cone.

The probabilities calculated up to here ignore the ionization of the binaries.
To account for this process, we use the half-life time approximate relation
given by (Bahcall et al., 1985), see eq. (1)

Correcting for this we get for Equation (13) and Equation (15) the following
approximations:

$L_{a<a_{{\rm crit}}}=\tau f(b)F_{q}\left(1-e^{-t/\tau}\right)$ (16)

and

$L_{a>a_{{\rm crit}}}=\tau\frac{F_{q}}{P(a)}\left(1-e^{-t/\tau}\right),$ (17)

where $\tau=t_{1/2}/\ln 2$ is the mean lifetime of the binary. For a more
detailed derivation see equations (11) and (12) in Michaely & Perets (2019a).

The formation rate in the galactic center at time $t$ is given by the
following integral:

$\Gamma=\int_{r_{h}}^{500\rm pc}\int_{a_{\rm min}}^{a_{\rm max}}Lf_{\rm
a}\left(a\right)dN\left(r\right)da,$ (18)

where the first integral is on the position in the galactic center, and the
second is on the SMA of the binaries. $L$ is given by equations (16) and (17).
$f_{\rm a}\left(a\right)$ is the fraction of wide systems from the underlining
distribution of the SMA. The total number of systems within the differential
part of the galactic center $dr$ is $dN\left(r\right)$ given be eq. (4).

In order to calculate this integral we need to define the physical
characteristics of the population we consider. We do that in the next section.

##  3 Binaries in the inner 500 pc

Here, we apply the general equations presented above and apply them for the
particular characteristics of the wide population of binaries at the inner
$500$ pc.

###  3.1 Upper SMA cut-off

As mentioned before the galactic center is dense to the point the half-life
time of binaries is shorter than Hubble time. As a result, the expected SMA
distribution for binaries changes as a function of time. In the galactic
center, we calculate the upper cut-off of the SMA from the evolution time and
the ionization time. For example, if we are interested in white dwarf binary
(see section 4.4), we set the evolution time, $t_{\rm evolv}$ to be the time
it take to form the white dwarf. This timescale corresponds to a SMA with a
similar ionization timescale. This is the upper cutoff of the population we
consider. Specifically,

$a_{\rm max}\left(t_{1/2}\right)=\frac{0.00233v_{\rm
enc}}{Gm_{p}n_{*}t_{1/2}}\ .$ (19)

###  3.2 Lower SMA cut-off

We restrict our treatment to the impulsive approximation. Hence, we introduce
a lower cutoff to the SMA. The typical kinetic energy per unit mass is higher
at the galactic center; therefore, systems are regarded wide for much more
compact binaries. We set the minimal value of the SMA to be such that the
Keplerian velocity of the binary will be equal to the local velocity
dispersion times a numerical factor, $\beta=5$. This gives the lower cut-off
of the SMA distribution, namely:

$a_{\rm min}=\frac{G\mu}{\beta\sigma}^{2}\ .$ (20)

###  3.3 Wide binary fraction

In what follows, we calculate the fraction of binaries that reside in the span
between $a_{\rm min}$ and $a_{\rm max}$. In order to compute this fraction at
the beginning of the integration time, we assume the log-uniform distribution
of the SMA, $a$ from the minimal value of $a_{\rm absmin}=2R_{\odot}$ that
represents the most compact binary in the ensemble to $a_{\rm max}$ from eq.
(19). The wide binary fraction is defined as the fraction of binaries the
reside between between $a_{\rm min}$ and $a_{\rm max}$, namely

$f_{\rm wide}=\int_{a_{\rm min}}^{a_{\rm max}}c\times\frac{1}{a}da\ ,$ (21)

where $c$ is the normalization constant to the log-uniform distribution of the
SMA.

The last ingredient of the model is the formation time of the binary. The
values depend on the specific binary we are studying. For example, for a
binary MS, the formation time is the pre-MS lifetime, and for the binary WD,
the formation time is the MS lifetime.

##  4 Binary channels and predictions

This section presents four channels for interaction in the galactic center.
For each channel, we describe the underlying population and present the
outstanding results.

###  4.1 MS-MS and G2-like objects

As the MS binary interacts at the pericenter, it loses energy and angular
momentum and forms a compact binary. Several outcomes are plausible from this
scenario; as a proof of concept in what follows, we only present them
generally and reserve a more comprehensive study for future work. As mentioned
above if the pericenter distance is sufficiently small the resulting outcome
is may be either a merger or even a collision between the two MS stars. A
merger BMS can lead to a merged object more massive and luminous than was is
expected from its environment. These objects are known as blue-stragglers
(Leonard, 1989; Shara, 1999; Carney et al., 2001; Genzel et al., 2003; Carney
et al., 2005; Perets & Fabrycky, 2009; Kaib & Raymond, 2014).

Another possible outcome from a merger of two MS stars is a transient event
called a luminous red nova which is characterized by a prominent red color and
a slow decaying light curve.(e.g. Kulkarni et al., 2007; Soker & Kashi, 2012;
Pejcha et al., 2016a, b; Pastorello et al., 2019).

Another possibility for an outcome is a common envelope evolution, a binary
evolutionary stage where the two stars cross each other Roche limit and share
an envelope (e.g. Webbink, 1984; Ivanova et al., 2013; Michaely & Perets,
2019b; Igoshev et al., 2020; Röpke & De Marco, 2023; Di Stefano et al., 2023)

We note, that this scenario is consistent with the idea that the G2-like
objects host a stellar component (Gillessen et al., 2012; Witzel et al., 2014;
Prodan et al., 2015; Stephan et al., 2016; Witzel et al., 2017; Stephan et
al., 2019; Gillessen et al., 2019; Ciurlo et al., 2020). Collisions between
two main sequence single stars have been proposed to be consistent with the G2
population closest to the SMBH (Rose et al., 2023). Here, we predict a
population of G2-like objects at much larger distances from the SMBH.

For this channel, we set the following parameters. The masses
$m_{1}=1M_{\odot}$ and $m_{2}=1M_{\odot}$, represent the mass of a MS star.
The interaction radius, $d=5R_{\odot}$, represents the distance in which tidal
interaction is sufficient to dissipate orbital energy. From Kroupa IMF
(Kroupa, 2001), we calculate the fraction of the primary to be an MS during
the integration time (Moe & Di Stefano, 2016). In order to do so, we limited
the mass range to be between $0.5M_{\odot}-3M_{\odot}$. The fraction of the
primary to be in that mass span is $f_{m_{1}}=0.25$. Assuming a uniform
distribution of the mass ratio $q\in\left(0.3,1\right)$, the fraction of the
companion to be in the same mass range is $f_{m_{2}}=0.54$. The binary
fraction for this mass scale is $f_{\rm binary}=0.5$. The product of these
fractions gives the fraction of MS-MS systems from a given population,
$f_{a}=f_{m_{1}}\times f_{m_{2}}\times f_{\rm binary}\times f_{\rm wide}$ from
the We emphasis here for simplicity we represent all MS stars with a single
mass of $1M_{\odot}$.

In Figure 2, we present the number interacting MS-MS (BMS) as a function of
separation from the SMBH after Hubble time, $t=10^{10}\rm yr$, blue solid
line. In Figure 3, we present the total number of systems of interest from BMS
(in blue solid line) as a function of time (see labels in both Figures).

Figure 2: Left Y axis: Total number of systems of interest as a function of
distance from the SMBH for all 4 channels after Hubble time,$10^{10}\rm yr$
for the simplistic case of a single starburst at $t=0$. Blue solid line,
denotes the BMS channel. The Red dashed line denotes the BH-MS channel. The
purple dotted line represents the WD-MS channel. Black dashed-dot line is the
DWD channel. All channels show similar structure, albeit with different total
systems due to the different initial systems for each channel. The break at
$r\approx 30\rm pc$ is due to the change in the velocity dispersion. The right
Y axis shows the velocity dispersion, toy model (see Eq. (5), as a function of
distance from the SMBH, from the radius of influence, $r_{h}$ up to $500\rm
pc$. The peak all channel exhibit around $r\approx 200\rm pc$ is explained in
section 5 and in Figure 4 Figure 3: The total number of systems of interest
for all channels. The color code is as Figure 2. The total number of system is
integrated over the galactic center. Specifically from the radius of
influence, $r_{h}$ to the edge of the field, $r=500\rm pc$. Each channel has a
different starting point due to the different time it takes the initial binary
to evolve.

###  4.2 BH-MS and stellar transients

The outcome of BH-MS interaction at the pericenter could lead to an X-ray
source, (e.g., Michaely & Perets, 2016; Naoz, 2016; Hoang et al., 2022; Rose
et al., 2023). Furthermore, in some cases these systems may lead to tidally
disrupting the star (e.g., Perets et al., 2016; Fragione et al., 2019; Kremer
et al., 2022, 2023; Xin et al., 2023). For a high eccentric encounter, these
tidal disruption events may leave short optical or x-ray transient.

In what follows we adopt $m_{1}=10M_{\odot}$ for the BH mass and
$m_{2}=1M_{\odot}$ for the main sequence star. Using Kroupa IMF we estimate
the fraction of these BHs from the population up to $500$ pc. We limited the
mass range to be bigger than $20M_{\odot}$ at the zero-age MS. Thus we find
that the fraction of the primary is $f_{m_{1}}=2\cdot 10^{-3}$. Assuming power
law distribution of the mass ratio $q\propto m^{-2}$ from $\left(0.1,1\right)$
(Moe & Di Stefano, 2016), the fraction of the companion to be an MS star is
$f_{m_{2}}=0.1$. The binary fraction for this mass scale is $f_{\rm
binary}=1$. We emphasize here for simplicity; we represent all BH and MS stars
with a single mass of $10M_{\odot}$ for the BH and $1M_{\odot}$ for the MS
star. Finally, the interaction radius, $d=5R_{\odot}$, represents the distance
in which tidal interaction is sufficient to dissipate orbital energy. In
Figure 2, we present the number of systems of interest from the BH-MS channel
as a function of the distance from the SMBH after Hubble time, $t=10^{10}\rm
yr$ in red dashed line. In Figure 3 we present the total number of systems of
interest in the red dashed line as a function of time.

###  4.3 WD-MS and CV/ novea

WD-MD interactions can lead to a myriad of observational signatures, such as
cataclysmic variable (e.g., Muno et al., 2006, 2009; Knigge et al., 2011),
novae and other transients (e.g., Shara & Shaviv, 1977, 1978; Shara & Regev,
1986; Webbink et al., 1987; Chomiuk et al., 2021), Type Ia supernova
progenitor (for the single degenerate scenario, e.g., Raskin et al. (2009,
2010); Rosswog et al. (2009); Hawley (2012); Soker (2019), but see Lorén-
Aguilar et al. (2010)). for the purposes of this work, we are interested in
the general electromagnetic signature and leave the analysis of the specific
electromagnetic signature to future work.

We adopt $m_{1}=0.6M_{\odot}$ and $m_{2}=1M_{\odot}$ for the primary WD and
the secondary MS star, respectively. In this case, the primary is the WD, even
though it is the less massive component because the primary/secondary
distinction is defined in the MS lifetime. $m_{1}$ represents the more massive
MS star that evolved to be the WD sooner than its companion. From Kroupa IMF
we find the fraction of the primary to be a WD. We limited the mass range to
be between $3M_{\odot}-8M_{\odot}$. In this mass range, we assume the primary
evolves to a WD in $10^{8}\rm yr$, resulting in a fraction of the primary
$f_{m_{1}}=0.02$. Assuming a uniform distribution of the mass ratio
$q\in\left(0.3,1\right)$, the fraction of the companion to be an MS star is
$f_{m_{2}}=0.56$. The binary fraction for this mass scale is $f_{\rm
binary}=0.5$. We emphasize here for simplicity we represent all WD and MS
stars with a single mass of $0.6M_{\odot}$ for the MS and $1M_{\odot}$ for the
MS star. The interaction radius, $d=1R_{\odot}$, represents the distance in
which tidal interaction between the WD and the MS. In 2 we present the number
of systems of interest from WD-MS channel as a function of radius from the
SMBH after Hubble time, $t=10^{10}\rm yr$ in purple dotted line. In Figure 3
we present the total number of systems of interest in the purple dotted line
as a function of time.

###  4.4 WD-WD and SN/ LISA sources

In the last channel, we present here the following parameters that represent
the binary population of DWD. The masses are set to be $m_{1}=0.6M_{\odot}$
and $m_{2}=0.6M_{\odot}$. The interaction radius, $d=1R_{\odot}$, represents
the distance in which tidal interaction between the binary WD. Similar to the
previous channel we limit the MS mass range to be $3M_{\odot}-8M_{\odot}$,
hence from Kroupa IMF we find $f_{m_{1}}=0.02$, and under the assumption of
uniform distribution of the mass ratio $q\in\left(0.3,1\right)$ the fraction
of the companion to be a MS star is $f_{m_{2}}=0.5$. Similar to the previous
channel the binary fraction for this mass scale is $f_{\rm binary}=0.42$.

The possible outcomes from a compact DWD binary are: Type Ia supernova
progenitor, as the two WD inspiral toward each other while omitting GWs and
eventually merge and exceeding the Chandrasekhar limit which in turn leads to
detonation of the merged product (if the double degenerate scenario works).
Another way to achieve detonation in the binary systems is by a head-on
collisions between the two WDs (e.g., Katz & Dong, 2012; Kushnir et al.,
2013).

As mentioned above inspiraling DWD are a source for millehertz GW signals.
These signals, if close enough, i.e. in our galaxy, might be detectable by the
future LISA mission (Marsh, 2011; Amaro-Seoane et al., 2012). In Figure 2 we
present the number of systems of interest from DWD channel as a function of
radius from the SMBH after Hubble time, $t=10^{10}\rm yr$ in purple dotted
line. In Figure 3 we present the total number of systems of interest in the
purple dotted line as a function of time.

##  5 Discussion

Given the galactic center’s extensive observational campaign (e.g., Ghez et
al., 2000; Gillessen et al., 2009; Støstad et al., 2015; Chen et al., 2023),
the spatial distribution of the interacting binaries can be used to constrain
the star formation process in the inner $500$ pc. As depicted in Figure 2, the
probability of interaction at the binary’s pericenter is a function of time
and position in the galactic center. We consider a single starburst model;
thus, initially, the interacting binaries are localized closer to the SMBH
($\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}20$ pc). As time goes by (above $\sim$Gyr), the interacting
binaries are localized at the outer parts ($lsim$few tens of parsec.
Therefore, it is straightforward to understand the continuous or episodic star
formation rate at this vicinity (e.g., Lu et al., 2009; Chen et al., 2023),
from future observations of interacting binaries.

The localization features and the peaks of the rate as a function of distance
from the SMBH are straightforward to explain. Particularly, ignoring
ionization, the probability of interaction is an increasing function of time.
However, the ionization, which decreases the number of available binaries in
this scenario, is extremely efficient in a high-density and high-velocity
dispersion environment. As a result, we expect decreasing efficiency as a
function of time, starting from the radius of influence and working its way to
the edge of the Galactic field (heuristically referred to as $\sim 500$ pc).
Simultaneously, the probability of interactions in the empty loss cone
increases with higher velocity dispersion. Hence, we expect to see a high
interaction rate at the radius of influence at early times, which decreases in
space as $\sigma$ decreases. Once the velocity dispersion reaches its constant
field value, the merger probability change comes only as a function of the
local density via the critical SMA, $a_{\rm critical}$. These behaviors are
depicted in Figure 4.

As shown in Figures 2 and 4, the number of systems and the derivative have a
maximum at $\sim 200\rm pc$. Particularly, the total number of systems
interacting at the pericenter is the result of integrating the probability,
(Eq. (18)), twice, once over the binary’s SMA and once over the binary
position from the center. Thus, the boundaries of the interaction of the SMA
are set by the lower cut-off, from impulse approximating considerations, and
the upper cut-off, set by the ionization timescale (see Section 2). In the
denser regions of the galactic center, the upper cut-off is smaller than the
critical SMA, $a_{\rm critical}$. As the highest probability for interaction
occurs for binaries with SMA equal to $a_{\rm critical}$. Thus, the total
number of binaries increases as we approach the location in the galactic
center that is adding the contribution of the full loss cone, i.e., binaries
with $a>a_{\rm critical}$, this crossover occurs in $r\approx 200\rm pc$, see
Figure 1 panel b.

Figure 4: Left panel, the differential contribution to the interacting
binaries as a function of position for different times since formation. As the
blue solid line is the closet to star formation it exhibits the highest
contribution in the more denser regions with highest velocity dispersion. As
time progress we see less and less contribution in the more dense regions and
more contributions in the less dense region. The grin solid line presents the
contribution of the last time step, there is no contribution from the inner
region all ”new” interactions happen closer to the edge of the field. Right
panel, the delay time distribution for all channels considered. The color code
is as Figure 2. All channels exhibit the same peak of formation rate after
$10^{9}\rm yr$.

The delay time distribution (DTD), $dN/dt$, is the rate of forming the systems
of interest as a function of time since star formation. Specifically is the
time derivative of Equation (18) and the time derivative of Figure 3. In the
previous subsection, we discussed the DTD as a function of position in the
galactic center. In what follows, we present the DTD after integration on the
distance from the SMBH so we are only considering the total number of
interacting binaries as a function of time. We present the DTD for all channel
studies in the letter in Figure 4. The interaction probability peaks at
$t=10^{9}\rm yr$ after star formation.

##  6 Summary

Close binaries are often considered as a main source of exotic and transient
phenomena Typically, studies focus on binaries in the field of galaxies (e.g.,
Iben & Tutukov, 1984; Kashi & Soker, 2010; Soker & Kashi, 2012; Ilkov & Soker,
2012; Belczynski et al., 2002; Andronov et al., 2006, and many others), and
even very close to an SMBH (e.g., Antonini & Perets, 2012; Hoang et al., 2018;
Stephan et al., 2019; Hoang et al., 2020; Wang et al., 2020; Rose et al.,
2023, and others). However, the intermediate regime between the SMBH’s radius
of influence and the field. This unique regime has a dense environment such
that soft binaries are collisional. Unlike the environment in the field, the
stellar density is high. Furthermore, despite the high stellar density, these
binaries remain soft over long timescales. Thus allowing them to interact with
flyby stars, which can excite the binaries’ eccentricities to high values
(Michaely & Perets, 2016, 2019a, 2020; Michaely, 2020; Michaely & Shara, 2021;
Raveh et al., 2022; Michaely & Naoz, 2022).

The mentioned unique environment yields a smaller binary separation, compared
to the field, that maximizes the interaction at the pericenter due to flyby
(see Figure 1 panel a). This separation depends on the location of the binary
in the galactic center (as also depicted in Figure 1 panel b). Interestingly,
inward to $\sim 100$ pc the binaries separation is limited by the ionization
process. Thus, the binary “wideness,” depends on their location and varies
from roughly $10-10,000$ au, for $\sim 4-500$ pc, respectively.

Here, we presented a novel dynamical channel that produces binary interaction
from wide binaries in the galactic center. Wide binaries were thought not to
play a role in binary stellar evolution due to their susceptibility to being
ionized and because they were too wide to interact. However, we show that due
to their larger cross-section, wide binaries tend to interact with local flyby
stars. These interactions change the eccentricity and might gradually excite
it to the point of interaction at the pericenter.

As a proof of concept, we consider four interesting interacting binaries
formed via this channel. Particularly we consider: MS-MS, BH-MS, WD-MS, and
WD-WD. We choose these binaries due to the rich possible outcomes from their
interactions. We show that the DTD peak at $t=10^{9}\rm yr$ and the location
of the interaction gradually changes from the radius of influence, close to
star formation, to the edge of the field as time progresses. We find that
$10^{5}$ BMS in the galactic center interact due this process during the life-
time of the MW galaxy. DWD and WD-MS interact $10^{3}$ of times and BH-MS
$10^{2}$ of times.

## Acknowledgments

EM and SN thank the Bhaumik Institute for Theoretical Physics and Howard and
Astrid Preston for their generous support. We also acknowledge the partial
support from NASA ATP 80NSSC20K0505 and from NSF-AST 2206428

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