# Deep machine learning for meteor monitoring: advances with transfer learning
and gradient-weighted class activation mapping

Eloy Peña Asensio  Josep M. Trigo-Rodríguez  Pau Grèbol-Tomàs  David
Regordosa-Avellana  Albert Rimola

###### Abstract

In recent decades, the use of optical detection systems for meteor studies has
increased dramatically, resulting in huge amounts of data being analyzed.
Automated meteor detection tools are essential for studying the continuous
meteoroid incoming flux, recovering fresh meteorites, and achieving a better
understanding of our Solar System. Concerning meteor detection, distinguishing
false positives between meteor and non-meteor images has traditionally been
performed by hand, which is significantly time-consuming. To address this
issue, we developed a fully automated pipeline that uses Convolutional Neural
Networks (CNNs) to classify candidate meteor detections. Our new method is
able to detect meteors even in images that contain static elements such as
clouds, the Moon, and buildings. To accurately locate the meteor within each
frame, we employ the Gradient-weighted Class Activation Mapping (Grad-CAM)
technique. This method facilitates the identification of the region of
interest by multiplying the activations from the last convolutional layer with
the average of the gradients across the feature map of that layer. By
combining these findings with the activation map derived from the first
convolutional layer, we effectively pinpoint the most probable pixel location
of the meteor. We trained and evaluated our model on a large dataset collected
by the Spanish Meteor Network (SPMN) and achieved a precision of 98%. Our new
methodology presented here has the potential to reduce the workload of meteor
scientists and station operators and improve the accuracy of meteor tracking
and classification.

###### keywords:

meteorites, meteors, meteoroids , machine learning , convolutional neural
networks , transfer learning

††journal: Planetary and Space Science

[inst1]organization=Departament de Química, Universitat Autònoma de Barcelona,
addressline=Carrer dels Til·lers, city=Bellaterra, postcode=08193,
state=Catalonia, country=Spain

[inst2]organization=Institut de Ciències de l’Espai (ICE, CSIC),
addressline=Campus UAB, C/ de Can Magrans S/N, city=Cerdanyola del Vallès,
postcode=08193, state=Catalonia, country=Spain

[inst3]organization=Institut d’Estudis Espacials de Catalunya (IEEC),
addressline=Carrer del Gran Capità, 2, city=Barcelona, postcode=08034,
state=Catalonia, country=Spain

[inst4]organization=Spanish Meteor Network (SPMN, CSIC) , addressline=Campus
UAB, C/ de Can Magrans S/N, city=Cerdanyola del Vallès, postcode=08193,
state=Catalonia, country=Spain

##  1 Introduction

Meteors, popularly known as shooting stars, particularly the most luminous
ones called fireballs or bolides, are spectacular physical processes that have
fascinated mankind for centuries (Trigo-Rodríguez, 2022). These dazzling
streaks of light occur when a meteoroid enters the Earth’s atmosphere at
hypersonic velocity, causing intense heating through repeated collisions with
air molecules (Ceplecha et al., 1998; Silber et al., 2018; Trigo-Rodríguez,
2019). Meteors are formed due to the extreme heat produced by the interaction
with the gaseous environment and the rising atmospheric pressure which causes
the meteoroid to undergo rapid vaporization, a process known as ablation. This
ablation leads to the formation of a luminous trail composed of ionized gas
and fragmented debris, which can be observed and recorded from the ground
using optical devices. Current digital video imagery provides a sequential
recording useful to obtain complete light curves, high temporal and spatial
resolution measurements, and spectra (Hughes, 1978; Koschny et al., 2017;
Subasinghe et al., 2017; Drolshagen et al., 2021).

Traditionally, two classes of meteors are considered: those that are expected
to occur in a specific period of the year because they are associated with
meteoroid streams, and those that are sporadic and have no discernible
periodic pattern (Wiegert and Brown, 2004; Jopek and Williams, 2013; Dumitru
et al., 2017; Jenniskens, 2017; Vaubaillon et al., 2019; Peña-Asensio et al.,
2022, 2023). Although showers exhibit regular activity, the sporadic meteors
require constant sky monitoring to quantify the meteoroid flux and properties
of the different sources (Trigo-Rodríguez and Blum, 2022).

Meteors can provide valuable information about the composition, dynamics, and
origin of our Solar System (Koschny et al., 2019). By analyzing the physical
and chemical properties of meteorites, which are pieces of meteoroids that
survive their journey through the Earth’s atmosphere and land on the surface,
scientists can gain insight into the formation and evolution of comets and
asteroids, as they provide information about the age of the Solar System, the
composition of the early solar nebula, and the processes that led to the
formation of the planets (Bottke et al., 2002; Lauretta and McSween, 2006).

The growing interest in meteoritics has driven an increase in the number of
video meteor detection networks around the world (Ceplecha, 1987; Ceplecha et
al., 1998; Koten et al., 2019; Colas et al., 2020). Comprised of strategically
placed stations equipped with cameras and other sensors, these networks are
designed to monitor atmospheric volumes with clear views of the sky, aiming to
maximize the number of recorded meteors within common observing fields. A
noteworthy trend in recent years has been the rise of pro-am collaborations in
this field, involving professional scientists, amateur astronomers, and
citizen scientists working together to collect valuable data. These
collaborations have significantly expanded the reach of meteor networks,
enabling the recording of events from diverse locations and perspectives.

However, with the increasing number of detection stations, the accumulation of
video and image data has also surged. Consequently, this data influx has
created a bottleneck in processing and analysis, as traditional manual methods
prove to be excessively time-consuming and resource-intensive. To deal with
these issues, many networks are embracing automation as a means to efficiently
handle the significant volumes of generated data (Molau, 2001; Spurný et al.,
2007; Gural and Šegon, 2009; Brown et al., 2010; Gural, 2011; Weryk et al.,
2013; Howie et al., 2017; Suk and Šimberová, 2017; Nikolic, 2019; Peña-Asensio
et al., 2021a, b; Vida et al., 2021). These automated approaches allow meteor
scientists to analyze and interpret meteor data faster and more efficiently
than ever before, helping to uncover new insights into meteor behavior and
properties.

The detection of luminous sources moving in the sky is relatively easy to
solve as the camera control software only needs to store and overwrite the
last few minutes of recording, and in the event of a sudden increase in
illumination, permanently save this data. However, the trigger threshold must
be carefully calibrated to avoid missing any meteors while minimizing the
number of false positives. Defining this cut-off is complex as it can vary
depending on a number of factors, including general lighting conditions, which
need to be updated periodically to consider specific dusk and dawn
illumination conditions or the presence of the Moon.

The pipelines that attempt to automate the detection and tracking of meteors
face a difficult task because meteors are virtually random phenomena and can
occur in a variety of ways due to impact geometry, variable velocity, size,
shape, composition, viewing angle, sky conditions (clouds or illumination),
etc. In addition, meteors must be distinguished from false positives caused by
satellites, airplanes, helicopters, drones, birds, lightning, or artificial
light sources. The combination of possible characteristics that meteors can
exhibit makes it difficult to define fixed parameters that work in all cases.
As a result, many networks still rely on human experts to manually review the
footage and identify/classify meteors. However, human operators can
occasionally make errors, particularly when artificial events cause confusion
or ambiguity. However, there are also networks that use fully automated
approaches based on traditional computer vision techniques, such as image
processing algorithms with fixed instructions, e.g. CAMS (Jenniskens et al.,
2011), SonotaCo (SonotaCo, 2016), or EDMOND (Kornoš et al., 2014). Some of the
detection pipelines currently in use are MetRec, MeteorScan, and UFOCapture;
an overview of their capabilities is given in Molau and Gural (2005). These
automated approaches show a high percentage of events with suspicious
calculated results due to their reliance on fixed parameters that may not be
appropriate for all scenarios (Hajdukova et al., 2020).

Consequently, addressing the challenges of meteor monitoring requires the
adoption of artificial intelligence techniques. In this paper, we delve into
the utilization of new methodologies for meteor classification and processing
tasks. Specifically, we investigate how Machine Learning (ML) approaches can
effectively enhance the accuracy and efficiency of automated pipelines, given
the massive volume of data generated by meteor networks. We present a fully
automated pipeline leveraging Convolutional Neural Networks (CNNs) for meteor
detection and tracking using transfer learning and the Gradient-weighted Class
Activation Mapping (Grad-CAM) technique.

##  2 Artificial intelligence for meteor detection

Advances in computer technology and hardware performance have fueled the
remarkable progress of ML, particularly artificial neural networks with
multiple layers, which are classified as deep learning. Neural networks have
become increasingly popular in various domains due to their exceptional
performance in image classification and recognition. Among them, CNNs have
gained popularity for their fault tolerance and self-learning capabilities
through multi-layer feedforward networks with a convoluted structure (Gu et
al., 2018). They can handle complex environments and unclear background
problems with significantly better generalization ability compared to other
methods. A typical CNN architecture includes an input layer, multiple
convolutional layers, pooling layers, a fully connected layer, and an output
layer. CNNs can be used for both supervised and unsupervised learning, and are
applied in diverse fields such as computer vision, natural language
processing, and others (Hastie et al., 2001).

In the context of meteor monitoring, current research efforts using ML focus
on two main objectives. First, some studies concentrate on determining the
presence of a meteor in a given event, with the goal of distinguishing
meteoric events from non-meteoric phenomena. Alternatively, other works rely
on ML for meteor tracking, facilitating the accurate localization and
monitoring of meteoroids throughout their bright atmospheric trajectory. The
primary challenge is to effectively distinguish false positives caused by non-
meteor objects such as airplanes, birds, and insects, or atmospheric
conditions (e.g. clouds). These innovative approaches are being increasingly
employed within meteor networks worldwide, including the Global Meteor Network
(GMN) (Gural, 2019), AllSky7 Fireball Network Germany
(FNG111https://allsky7.net/), Meteorite Orbits Reconstruction by Optical
Imaging (MOROI) (Nedelcu et al., 2018), Canadian Automated Meteor Observatory
(CAMO) (Weryk et al., 2013), Cameras for All-sky Meteor Surveillance (CAMS)
(Jenniskens et al., 2011), EXOSS meteor network222https://exoss.org/, and
Meteor Automatic Imager and Analyzer (MAIA) (Vítek et al., 2011).

In image classification, it is customary to use transfer learning techniques
with pre-trained models of CNNs (Sennlaub et al., 2022; Marsola and Lorena,
2019; Galindo and Lorena, 2018). These methods allow inheriting the ability to
detect objects from those pre-trained models, which need to be re-trained on
meteors. With this methodology, optimal results are achieved with a smaller
number of training data compared to starting from an uninitialized model. As
underlined in Galindo and Lorena (2018), uttermost results are achieved when
the proper pre-trained dataset is selected. They compared the performance of
ImageNET and Fashion-MNIST (Xiao et al., 2017) with fine-tuning, concluding
that the latter is the most optimal as it is already trained to work with
black and white images. They also checked whether the CNN could distinguish
meteors if the image was previously tweaked (slightly zoomed, rotated, or
flipped). The results showed that these transformations could produce
unrealistic apparent trajectories and worsen the classification. In order to
solve these types of problems, Ganju et al. (2023) use a windowing technique
to create new frames from existing ones. With this, all meteor detection would
have the same number of frames, easing the posterior analysis. In Cecil and
Campbell-Brown (2020) is shown a comparison of different sets of combinations
of different techniques used in image processing, such as convolutions and
max-pools.

Even though most CNN meteor detection algorithms have been trained to reach a
satisfying prediction percentage ($>99\%$), particularly when considering
large sample sizes of more than $\sim$10,000 events, some anomalies are still
misclassified as meteors for small datasets. The next step in meteor detection
algorithms is to consider the intrinsic properties of meteors on camera images
to discard these misleading anomalies. Additionally, the main drawback of the
current meteor tracking algorithms is the runtime required to analyze high-
definition 1080p video images. However, they perform well when dealing with
small, low-resolution video images.

Beyond CNNs, other ML techniques are often used. It is the case of Recurrent
Neural Networks (RNN), Gradient Boost (GB), or Random Forest (RF), which can
also be used as complementary analysis tools (Gural, 2019; Anghel et al.,
2022). Temporal resolution can be introduced in the analysis by using other
networks such as Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU),
Temporal Convolutional Network (TCN), Time Delay Neural Network (TDNN),
Support Vector Machine (SVM), or VGG16 (Siladi et al., 2015; Simonyan and
Zisserman, 2015; Sennlaub et al., 2022), supporting 16 layers. A particular
additional technique is the discrete pulse transform (DPT) (Rohwer and Laurie,
2006), in which image signals are decomposed in pulses. Vítek and Nasyrova
(2019) introduced DPT to characterize the number of pulses related to the
meteor compared to those of stars. Even though it is not specified if the
results are better than other works, they do underline that using DPT is
faster than other methods used in the MAIA data.

Sennlaub et al. (2022) also classified those false positives based on their
origin. They did not achieve solid statements but pointed out similarities
among false positive subgroups. As sketched in Le Lan and Dinh (2021) a future
algorithm to classify the false positives would include prior knowledge of
each subgroup. These could also be expanded to classify the re-entry of
artificial space debris. They could also include a cross-matching
identification between different stations as a method of improving the overall
accuracy (Anghel et al., 2023).

Table 1 provides a comprehensive summary of the outcomes achieved in these
works accompanied by the data source, the number of samples, the technique
used, and the results such as F1 score and accuracy. The F1 score is a
performance metric that combines precision and recall to measure the accuracy
of a classification model. Precision represents the proportion of true
positive predictions out of all positive predictions, while recall represents
the proportion of true positive predictions out of all actual positive
instances. The F1 score considers both precision and recall, making it useful
when the dataset is imbalanced or when false positives and false negatives
have different consequences. Accuracy is a common evaluation metric used to
measure the overall correctness of a classification model. It calculates the
proportion of correctly predicted instances (both true positives and true
negatives) out of the total number of instances in the dataset.

Table 1: Summary of the main previous works on ML applied to meteor detection.
For each contribution, we denote the meteor network source, the number of
samples used to train the models, their goal, the transfer learning method
used (if any), the different models compared, and the best F1 score and/or
accuracy achieved (when provided). F1 score and accuracy have been normalized
between 0 and 1.

Article | Meteor network | # samples | Goal | Transfer learning | Model | F1 score | Accuracy
---|---|---|---|---|---|---|---
Ganju et al. (2023) | CAMS | 19,152 | Detection | BiLSTM | - | 0.91 | 0.89
Anghel et al. (2023) | MOROI | 9,858 | Detection | - | CNN, GB and RF | 0.989 | 0.998
Sennlaub et al. (2022) | AllSky7 FNG | 20,000 | Detection | ImageNET | CNN: GRU and SVM | - | 0.991
Cecil and Campbell-Brown (2020) | CAMO | 50,745 | Tracking | No | CNN: multi-layer and max pool | 0.997 | 0.998
Gural (2019) | CAMS | 200,361 | Tracking | MeteorNet | RNN and LSTM | 0.974 | 0.981
Marsola and Lorena (2019) | EXOSS | 400 | Detection | ImageNET | VGG16 | - | 0.8435
Vítek and Nasyrova (2019) | MAIA | - | Tracking | No | DPT | - | -
Galindo and Lorena (2018) | EXOSS | 1,660 | Detection | Fashion-MNIST | - | 0.94 | 0.960

##  3 CNN, transfer learning, and Grad-CAM

Our work aims to use ML techniques to achieve two main goals: detecting the
presence of meteors in images and tracking the motion of meteors in the field
of view. To achieve this, and based on the review of the scientific
literature, we have decided to develop a CNN model that classifies images into
two groups, ”Meteors” and ”No-meteors”, using transfer learning. In addition,
we implemented a novel application of Grad-CAM to track the coordinates of the
meteor’s motion.

###  3.1 Detection

To build our model, we chose to use ResNet-34, which is a 34-layer pre-trained
CNN from the Residual Network family (He et al., 2015). This allowed us to
quickly specialize the model for our specific use case using transfer learning
techniques with a small dataset and rapidly inheriting all detection skills
already learned by the network. ResNet-34 mainly consists of an input layer,
convolutional layers, residual blocks, shortcut connections, downsampling,
global average pooling, and fully connected layers. One of the key elements of
this network is the residual building block, which is its infrastructure. As
shown in Figure 1, the residual building block consists of several
convolutional layers (Conv), batch normalizations (BN), a rectified linear
unit (ReLU) activation function, and a link. This block is used for all 34
layers of ResNet-34, as depicted in Figure 2. The output of the residual block
is given by the formula $y=F(x)+x$, where $F$ is the residual function and $x$
and $y$ are the input and output of the residual function, respectively. The
entire residual network is composed of the first convolutional layer and
multiple basic blocks, making it a highly effective and efficient deep
learning architecture for image recognition tasks.

Figure 1: A basic-block building block of ResNet-34.  Table 2: The structure of ResNet-34. Layer name | Output size | 34-layer
---|---|---
Conv1 |  112$\times$112 | $\begin{matrix}7\times 7,64,stride2\\\ 3\times 3\,\,max\,\,pool,stride2\end{matrix}$
Conv2_x |  56$\times$56 | $\begin{bmatrix}3\times 3,64\\\ 3\times 3,64\end{bmatrix}\times 3$
Conv3_x |  28$\times$28 | $\begin{bmatrix}3\times 3,128\\\ 3\times 3,128\end{bmatrix}\times 4$
Conv4_x |  14$\times$14 | $\begin{bmatrix}3\times 3,256\\\ 3\times 3,256\end{bmatrix}\times 6$
Conv5_x |  7$\times$7 | $\begin{bmatrix}3\times 3,512\\\ 3\times 3,512\end{bmatrix}\times 3$
|  1$\times$1 | average pool, 1000-d fc, softmax

For training and testing the model, we used a dataset of 982 images of meteors
detected by optical devices of the Spanish Meteor Network (SPMN) network
stations (Trigo-Rodríguez et al., 2006), along with 56,285 images without
meteor detection collected over the year 2021, particularly from the Pujalt
observatory. To balance the two groups, we generated a dataset of 982 images
with meteors and 1,050 without detections for training. To ensure reliable
model performance, a portion of these images, specifically 20%, is allocated
for validation, which is utilized to evaluate the training progress. From this
dataset, 300 images were specifically set aside for testing purposes, 150 from
the meteor class and 150 from the no-meteor. The test set serves as an
independent dataset to evaluate the final performance of the trained model
after the completion of the training process.

The dataset consisted of grayscale long exposure (30 seconds) images that were
pre-processed to enhance the meteor trail and remove static visual elements
from the background by subtracting consecutive images. This included
converting the images to black and white, resizing them to 400x400, and
subtracting successive images to remove the background. To facilitate the
generalization of the model and reduce overfitting, data augmentation
techniques were used during the transfer learning process. Specifically, each
batch of images received by the CNN during the 35 epochs of training was
modified with geometric transformations such as randomly flipping, cropping,
rotating, and translating the images, or applying lighting modifications.

###  3.2 Tracking

The final layer of our CNN exhibits activations corresponding to neurons that
are specifically triggered when a meteor is detected in an image. While such
activations provide initial utility, leveraging the subsequent classification
layer’s weights in conjunction with these activations further enhances their
significance. It is important to note that the classification layer possesses
a comprehensive understanding of the meteor classification task, enabling it
to assign appropriate weights to the activated neurons. Hence, certain
previously activated neurons may be deemed less influential in the final
classification decision. This is the foundation of the so-called Class
Activation Mapping (CAM) technique (Zhou et al., 2015).

The CAM technique is usually employed to generate a heatmap for a given class
(class ”meteor” in our case). This technique involves capturing the
activations from the last convolutional layer of the CNN and multiplying them
by the corresponding weights from the last fully connected layer responsible
for the classification task. By performing this operation, the CAM technique
effectively highlights the areas within the image that contribute the most to
the meteor classification, that is, the Region of Interest (ROI). Figure 2
illustrates the overall procedures of this method.

However, in order to further enhance the performance of our model, we opted to
incorporate an advanced variant of CAM known as Grad-CAM (Selvaraju et al.,
2016). Grad-CAM builds upon the CAM methodology by integrating gradient
information instead of the weights of the classification layer. This provides
a more fine-grained localization of important regions within an image. Grad-
CAM computes the gradients of the target class of a specific layer with
respect to the activations of the same layer. By multiplying the activations
from the last convolutional layer with the average of the gradients across the
feature map of that layer, Grad-CAM obtains the importance weights for the
activation maps. The weighted combination of the activation maps produces the
final heatmap, which visually highlights the critical regions within the input
image for the classification of the target class.

Figure 2: The general process of class activation mapping method. Adapted from
Jiang et al. (2021).

To capture finer details for properly tracking the meteor motion, we focus on
the activations of the initial convolutional layer, restricting our attention
to activations falling within the defined ROI delineated by the Grad-CAM.
Within the CNN, the activation of the deepest layers offers a more detailed
resolution map (56x56). However, these layers are more prone to noise and less
reliable in accurately identifying the ROI related to meteor detection. To
tackle these difficulties, we apply a noise reduction strategy by selectively
retaining only the cells that align with the cells from the higher precision
but lower resolution Grad-CAM (7x7) with a non-zero value. By doing so, we
filter out noisy activations and focus on the cells that have a meaningful
impact on the meteor classification. Subsequently, we extract the cells with
the maximum activation values from the refined high-resolution activation map.
By calculating the average position among these selected cells, we are able to
project a single point onto the original image. This refinement process
significantly enhances the accuracy of meteor detection by precisely
pinpointing the location of meteors within the frames generated by our model.
Figure 3 illustrates the described meteor detection and tracking process.

Figure 3: Illustrative diagram of the proposed process of meteor detection and
tracking using CNN and Grad-CAM. The image exemplifies a video detection made
by the SPMN station in Estepa, Seville operated by Antonio J. Robles.

Note that we do not multiply the activations of the initial layer with the
weights of the classification layer because the spatial and semantic gap
between the initial convolutional layer and the classification layer limits
the effectiveness of such an approach. Although Grad-CAM allows the analysis
of any layer in the network, we observed that focusing just on the activations
from the initial convolutional layer within the ROI calculated using Grad-CAM
on the last layer yields good results.

##  4 Training and results

Figure 4 shows the evolution during the training process of the loss function
as a function of the batches processed, which is a metric that measures the
deviation between predicted and actual values. The primary goal during
training is to minimize this function. The training set consists of the images
used to train the CNN model, while the validation set consists of a subset of
150 images randomly selected and reserved for evaluating the model’s
performance and generalization ability. By evaluating the loss function on
both sets, we monitor the progress of the model and detect any signs of
overfitting or underfitting.

A batch refers to a group of images that are processed together during each
iteration of the training algorithm. The total number of batches processed can
be calculated using the formula $Batches=N*(I/BS)$, where $N$ is the number of
epochs, which is the number of times the entire training data set is passed
through the network during training. $I$ denotes the total number of images in
the training dataset, including both meteor and non-meteor images. Finally,
$BS$ refers to the batch size, which represents the number of images fed to
the network in each training iteration. For this study, we used a batch size
of 32.

By substituting these values, we determined that a total of 1,515 batches were
processed during the training process. This corresponds to the maximum value
observed in Figure 4, indicating the completion of all batches. Understanding
the relationship between the loss function, the batches processed, and the
progress of the training and validation sets provides valuable insight into
the training dynamics and performance evaluation of the neural network model.
As every batch comprises 32 images and is subjected to various random
transformations, the overall training process, including data augmentation,
incorporates a varied array of 48,480 distinct images. Table 3 compiles the
evolution of different metrics during the training process.

Figure 4: Training and validation loss during model training.  Table 3: Metrics evolution during the training process, including for each epoch the error rate, accuracy, precision, recall, and F1 score. Epoch | Error | Accuracy | Precision | F1
---|---|---|---|---
0 | 0.245665 | 0.754335 | 0.763323 | 0.753592
1 | 0.210983 | 0.789017 | 0.788704 | 0.788804
2 | 0.205202 | 0.794798 | 0.798389 | 0.794714
3 | 0.187861 | 0.812139 | 0.823898 | 0.811444
4 | 0.127168 | 0.872832 | 0.872594 | 0.872726
5 | 0.161850 | 0.838150 | 0.842479 | 0.838064
6 | 0.104046 | 0.895954 | 0.895722 | 0.895867
7 | 0.138728 | 0.861272 | 0.875206 | 0.860709
8 | 0.086705 | 0.913295 | 0.913068 | 0.913222
9 | 0.072254 | 0.927746 | 0.928404 | 0.927527
10 | 0.063584 | 0.936416 | 0.936314 | 0.936382
11 | 0.049133 | 0.950867 | 0.950896 | 0.950775
12 | 0.060694 | 0.939306 | 0.939599 | 0.939294
13 | 0.063584 | 0.936416 | 0.936896 | 0.936244
14 | 0.049133 | 0.950867 | 0.950896 | 0.950775
15 | 0.095376 | 0.904624 | 0.912476 | 0.904489
16 | 0.066474 | 0.933526 | 0.936153 | 0.933521
17 | 0.049133 | 0.950867 | 0.950693 | 0.950834
18 | 0.052023 | 0.947977 | 0.948137 | 0.947865
19 | 0.037572 | 0.962428 | 0.962805 | 0.962337
20 | 0.052023 | 0.947977 | 0.948137 | 0.947865
21 | 0.043353 | 0.956647 | 0.956495 | 0.956586
22 | 0.052023 | 0.947977 | 0.948109 | 0.947961
23 | 0.046243 | 0.953757 | 0.953543 | 0.953719
24 | 0.037572 | 0.962428 | 0.962282 | 0.962375
25 | 0.037572 | 0.962428 | 0.962805 | 0.962337
26 | 0.049133 | 0.950867 | 0.950896 | 0.950775
27 | 0.034682 | 0.965318 | 0.965915 | 0.965224
28 | 0.043353 | 0.956647 | 0.956688 | 0.956566
29 | 0.049133 | 0.950867 | 0.950693 | 0.950834
30 | 0.049133 | 0.950867 | 0.950693 | 0.950834
31 | 0.052023 | 0.947977 | 0.947761 | 0.947933
32 | 0.046243 | 0.953757 | 0.953551 | 0.953702
33 | 0.046243 | 0.953757 | 0.953551 | 0.953702
34 | 0.034682 | 0.965318 | 0.965525 | 0.965244

The next step involves defining the crucial hyperparameter for the training
process: the learning rate. By evaluating the loss function values across
various learning rates, we can identify the region of sustained and
substantial loss reduction, disregarding transient peaks and irregular drops
as they do not represent reliable trends. Once this region is identified, we
pinpoint the midpoint of the steepest descent line, which corresponds to the
most significant loss reduction. This specific learning rate is then selected
for the subsequent training procedure. For our specific study, we determined a
learning rate of 0.003 to be the optimal choice.

The training phase of our pipeline ends with an F1 score of 0.94, indicating
the high precision and recall achieved during the training process. We then
evaluated the performance of the trained model on the test dataset to assess
its generalization capabilities. The evaluation results show that our model
achieved an accuracy of 0.96 on the test dataset. This accuracy metric
indicates the model’s ability to correctly classify meteor and non-meteor
images with a high degree of accuracy. Furthermore, when specifically
considering the meteor class, our model achieved a precision of 0.98.

To further analyze the performance of the model, a confusion matrix was
constructed as it provides insight into the classification performance by
showing the distribution of the predicted labels against the true labels. The
confusion matrix for our two labels ”Meteor” and ”No-Meteor” is shown in
Figure 5.

Figure 5: Confusion matrix of the trained model with normalized values in
parentheses.

In the confusion matrix, the rows correspond to the true labels, while the
columns represent the predicted labels. The matrix values indicate the
proportion of images belonging to each category. The confusion matrix shows
that 47% of the images were correctly classified as meteors, while 3.3% of the
images were incorrectly classified as non-meteor images. Furthermore, 1% of
the images were incorrectly classified as meteor images, while 49% were
correctly identified as non-meteor images. The high accuracy achieved by the
pipeline demonstrates its robustness in accurately detecting and classifying
meteor images. The low misclassification rates for both meteor and non-meteor
classes indicate the effectiveness of the trained CNNs in distinguishing
between these classes, then minimizing the required human time for these time-
consuming tasks.

It is worth noting that the proposed model shows the ability to generalize and
accurately detect meteor trails even in color frames that have not undergone
the previous frame subtraction process. This is particularly noteworthy
because these frames may contain stationary elements such as clouds, the Moon,
buildings, or other obstructions that can cause interference and affect the
accuracy of meteor detection. Despite these challenges, the model can
effectively distinguish and identify meteor trails, providing robust and
reliable results. This further validates the model’s ability to operate under
real-world conditions, making it a valuable tool for meteor scientists and
enthusiasts alike.

However, this very capability in detecting meteor trails renders it
susceptible to satellite misidentifications, as they often exhibit a similar
trace when reflecting the sunlight. An instance of such incorrect
classification is illustrated in the top panel of Figure 6 with a transit of a
SpaceX Starlink satellite train, necessitating retraining the network to
encompass this new class. Conversely, the bottom panel shows an undetected
meteor, possibly due to its proximity to the full moon and dimming by a cloud-
covered field of view.

Figure 6: Top panel: False positive of a SpaceX Starlink satellite track as it
exhibits similar characteristics as a meteor trail. Recording obtained from
the Alpicat SPMN station under the operation of Marc Corretgé-Gilart. Bottom
panel: False negative of SPMN070523G superbolide recorded near the full moon
with a cloudy sky. Recording obtained from Bartolo-Castelló SPMN station under
the operation of Vicente Ibañez

.

We used data augmentation techniques during the training process, which is a
common strategy that helps prevent overfitting and improves the model’s
ability to generalize well to unseen data. However, it is worth noting that
Xiao et al. (2017) suggested data augmentation could potentially degrade model
performance in certain cases. Therefore, it is possible that our use of data
augmentation in the training process may have resulted in slightly lower
performance compared to some of the models reported in Table 1. Despite this
slight performance difference, our pipeline still demonstrates a high level of
accuracy and efficiency in meteor detection and classification. The inclusion
of data augmentation techniques is crucial to promote better generalization
and robustness in the model, even though it may have had a slight impact on
the overall results compared to other models in the comparison.

We compared our results with those obtained by Galindo and Lorena (2018) and
Marsola and Lorena (2019), who used datasets of similar size to ours. In this
comparison, our methodology equals or outperforms existing approaches,
demonstrating superior performance. Furthermore, our pipeline incorporates the
added complexity of meteor tracking, which presents a significant challenge
due to the smaller luminous trace in each frame. Meteor tracking enables the
computation of the velocity curve, a key factor in both discriminating between
artificial and natural objects and in determining the heliocentric orbit and
the potential meteorite strewn field. Automating this process facilitates the
extraction of orbital elements for hundreds or thousands of meteors detected
nightly, providing valuable insights into both the sporadic meteoroid
background flux and the characteristics of meteoroid streams.

In subsequent phases, we intend to enhance the pipeline by refining the
balance of false positives, encompassing a diverse spectrum of potential false
positive sources including satellites, planes, birds, bugs, and other light
sources in the training process.

##  5 Conclusions

The implementation of automated detection software has led to a massive
increase in the amount of data collected and reduced every year by meteor
networks. However, the need for human oversight to filter out false positives
and organize the records has created a bottleneck, and the traditional
computer vision techniques implemented have limited performance due to the
random and specific characteristics of each meteor event. We employed CNNs to
address these challenges.

In our study, we used a dataset of 982 meteor images along with 1,050 images
without meteors detected by SPMN stations in 2021 to train a CNN model. A
transfer learning technique was applied, and Grad-CAM was used for accurate
tracking. Our main results are as follows:

1) Our approach utilized ResNet-34, a deep learning architecture consisting of
34 pre-trained layers. By using pre-trained layers, we capitalized on the
knowledge and representations gained from a large dataset during the initial
training phase, resulting in improved model performance. In addition, data
augmentation techniques were employed to facilitate the model’s ability to
accurately generalize and mitigate overfitting. The results achieved
demonstrate a precision of 98% for meteor classification.

2) Grad-CAM was used to track the coordinates of the meteor within each image.
This technique involved analyzing deeper layers of the neural network, which
have higher accuracy but lower resolution. ROI information was extracted using
the gradients of the last convolutional layer and then combined with
activation information from the initial layer, which is characterized by
higher resolution but lower accuracy. This fusion of information allowed the
identification of the most critical pixel corresponding to the position of the
meteoroid. This technique allows for the precise localization of meteor
positions within frames.

3) The high performance achieved by our pipeline underscores its robustness in
precisely detecting and classifying meteor images. The success rate, even with
a relatively small dataset, highlights the potential of our method to
significantly reduce the workload of meteor scientists and station operators
involved in meteor data analysis. This potential is further enhanced by one of
the most notable advancements in our methodology: the novel use of Grad-CAM
for meteor tracking in combination with the initial activation map.

In summary, our study highlights the significant potential of applying ML
techniques to meteor monitoring. It illustrates the effectiveness of CNNs and
transfer learning in reducing false positives and correctly identifying
meteors in images. By automating the meteor monitoring process, our pipeline
increases the efficiency of meteor detection and tracking using the Grad-CAM
technique. This, in turn, facilitates the study of meteoroid fluxes, aids in
population characterization, and improves our ability to distinguish between
meteorite-dropping events, thereby increasing fresh extraterrestrial material
recovery rates.

## Acknowledgments

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 865657) for the project “Quantum Chemistry on
Interstellar Grains” (QUANTUMGRAIN). JMT-R, EP-A, and PG-T acknowledge
financial support from the project PID2021-128062NB-I00 funded by
MCIN/AEI/10.13039/501100011033. AR acknowledges financial support from the
FEDER/Ministerio de Ciencia e Innovación – Agencia Estatal de Investigación
(PID2021-126427NB-I00, PI: AR). This work was also partially supported by the
program Unidad de Excelencia María de Maeztu CEX2020-001058-M. We thank all
SPMN station operators and photographers whose continuous dedication has
allowed to record incessantly the meteors of the Iberian Peninsula, the
Balearic Islands, and the Canary Islands. We express our gratitude to David
Puiggròs-Figueras for his assistance in the CAM and Grad-CAM methodology. The
following Python packages were extensively used in the study: Numpy, OpenCV,
Matplotlib, PyTorch, and fast.ai.

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# Self-similar solutions of viscous and resistive ADAFs with thermal
conduction

Kazem Faghei

###### Abstract

We have studied the effects of thermal conduction on the structure of viscous
and resistive advection-dominated accretion flows (ADAFs). The importance of
thermal conduction on hot accretion flow is confirmed by observations of hot
gas that surrounds Sgr A∗ and a few other nearby galactic nuclei. In this
research, thermal conduction is studied by a saturated form of it, as is
appropriated for weakly-collisional systems. It is assumed the viscosity and
the magnetic diffusivity are due to turbulence and dissipation in the flow.
The viscosity also is due to angular momentum transport. Here, the magnetic
diffusivity and the kinematic viscosity are not constant and vary by position
and $\alpha$-prescription is used for them. The govern equations on system
have been solved by the steady self-similar method. The solutions show the
radial velocity is highly subsonic and the rotational velocity behaves sub-
Keplerian. The rotational velocity for a specific value of the thermal
conduction coefficient becomes zero. This amount of conductivity strongly
depends on magnetic pressure fraction, magnetic Prandtl number, and viscosity
parameter. Comparison of energy transport by thermal conduction with the other
energy mechanisms implies that thermal conduction can be a significant energy
mechanism in resistive and magnetized ADAFs. This property is confirmed by
non-ideal magnetohydrodynamics (MHD) simulations.

00footnotetext: School of Physics, Damghan University, Damghan, Iran
e-mail<EMAIL_ADDRESS>

Received: 14 October 2011 / Accepted: 30 November 2011

Keywords accretion, accretion discs – conduction – magnetohydrodynamics: MHD

##  1 Introduction

The observational features in active galactic nuclei (AGN) and X-ray binaries
can be successfully explained by the standard geometrically thin, optically
thick accretion disc model (Shakura & Sunyaev 1973). The motion of the matter
in the standard thin model of the accretion disc is approximately Keplerian,
and the energy released in the accreting gas is radiated away locally. In the
past two decades, another type of accretion flow has been considered that the
energy released due to heating processes in the flow may be trapped within
accreting gas. As, only the small fraction of the energy released in the
accretion flow is radiated away due to inefficient cooling, and most of the
energy is stored in the accretion flow and advected to the central object.
This type of accretion flow is called as advection-dominated accretion flow
(ADAF). The models of ADAF have been studied by a number of researchers (e.g.
Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994; Abramowicz et al. 1995;
Blandford & Begelman 1999; Ogilvie 1999).

The observations of black holes confirm existence of hot accretion flow that
contrasted with classical cold and thin accretion disc model (Shakura &
Sunyaev 1973). Hot accretion flows can be seen in supermassive black holes of
galactic nuclei and during quiescent of accretion onto stellar-mass black
holes in X-ray transients (e.g., Lasota et al. 1996; Esin et al. 1997, 2001;
Narayan et al. 1998a; Menou et al. 1999; Di Matteo et al. 2000; see Narayan et
al. 1998b; Melia & Falcke 2001; Narayan 2002; Narayan & Quataert 2005 for
reviews). Chandra observations provide constraints on the density and
temperature of gas at or near the the Bondi capture radius in Sgr A∗ and
several nearby galactic nuclei (Loewenstein et al. 2001; Baganoff et al. 2003;
Di Matteo et al. 2003; Ho et al. 2003). Tanaka & Menou (2006) exploited these
constraints to calculate mean free path of the observed gas. They suggested
accretion in such systems are under weakly collisional condition. Moreover,
they suggested thermal conduction as a possible mechanism by which the
sufficient extra heating is provided in hot accretion flows. In following,
Johnson & Quataert (2007) studied the effects of electron thermal conduction
on the properties of a spherical hot accretion flows. Their model is
applicable for Sgr A∗ in the Galactic centre. Because, electron heat
conduction is important for low accretion rate systems. They also found a
supervirial temperature in the presence of thermal conduction. Similar to
Tanaka & Menou (2006), they assumed a steady state model, but they solved
their equations numerically. Abbassi et al. (2008) presented a set of self-
similar solutions for ADAFs with a toroidal magnetic field in which the
saturated thermal conduction has a important role in the energy transport in
the radial direction.

Since the observational evidences and magnetohydrodynamics (MHD) simulations
have expressed the toroidal magnetic field and the magnetic diffusivity are
important in accretion flows (see Faghei 2011a and references therein), Faghei
(2011a) examined the self-similar solutions of viscous and resistive ADAFs in
the presence of a toroidal magnetic field. But, he did not consider the
effects of thermal conduction in his model. While, the recent studies of
resistive accretion flows have represented that thermal conduction can play an
importance role in such systems (e.g. Sharma et al. 2008; Ghanbari et al.
2009). Sharma et al. (2008) studied the effects of thermal conduction on
magnetized spherical accretion flows using global axisymmetric MHD
simulations. In their model, when the magnetic energy density becomes
comparable to the gravitational potential energy density, the plasma due to
resistivity is heated to roughly the virial temperature, the mean inflow
becomes highly subsonic, and most of the energy released by accretion is
transported to large radii by thermal conduction and the accretion rate became
much smaller than Bondi accretion rate. Moreover, they found that for a larger
values of conductive parameter, energy transport through thermal conduction
becomes the dominant energy transport mechanism at small radii. Ghanbari et
al. (2009) studied a two-dimensional advective accretion disc bathed in the
poloidal magnetic field of a central accretor in the presence of thermal
conduction. They did not consider toroidal component of magnetic field and for
simplicity assumed the resistivity as a constant. They studied induction
equation of magnetic field in a steady state that is not according to anti-
dynamo theorem (e.g. Cowling 1981) and is useful only in particular systems
where the magnetic dissipation time is much longer than the age of the system.
On the other hand, this assumption implies that the flow is in balance between
escape and creation of the magnetic field.

In this paper, we adopt the presented solutions by Tanaka & Menou (2006) and
Faghei (2011a). Thus, we will investigate the influences of thermal conduction
on a viscous and resistive ADAF in the presence of a toroidal magnetic field.
Moreover, it is assumed that magnetic diffusivity in the present model is not
constant, and escaping and creating of magnetic field are unbalanced. From
some aspects will be shown that the present model is in according with the
observations and the resistive MHD simulations. The paper is organized as
follows. In section 2, the basic equations of constructing a model for quasi-
spherical magnetized advection dominated accretion flow with thermal
conduction will be defined. In section 3, self- similar method for solving
equations which govern the behaviour of the accreting gas was utilized. The
summary of the model will appear in section 4.

##  2 Basic Equations

We suppose a rotating and accreting gas around a Schwarzschild black hole of
mass $M$. The flow is assumed to be in advection dominated stage, where
viscous and resistive heating are balanced by the advection cooling and
thermal conduction. We use a spherical coordinate ($r$, $\theta$, $\phi$)
centred on the accreting object. Furthermore, the flow is assumed to be steady
and axisymmetric ($\partial_{t}=\partial_{\phi}=0$), and the equations will be
considered in the equatorial plane, $\theta=\pi/2$. Thus, all flow variables
are a function of only $r$. For the sake of simplicity, the general
relativistic effects are ignored and Newtonian gravity is used. The magnetic
field in the present model has only a toroidal component. Under assumptions,
the model is described by the following equations:

The continuity equation with mass loss is

$\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\rho v_{r})=\dot{\rho},$ (1)

where $\rho$ is density, $v_{r}$ is the radial infall velocity, and
$\dot{\rho}$ the mass-loss rate per unit volume.

The radial equation of momentum is

$v_{r}\frac{dv_{r}}{dr}=r(\Omega^{2}-\Omega^{2}_{K})-\frac{1}{\rho}\frac{d}{dr}(\rho
c^{2}_{s})-\frac{c^{2}_{A}}{r}-\frac{1}{2\rho}\frac{d}{dr}(\rho c^{2}_{A}),$
(2)

where $\Omega$ is the angular velocity of the flow,
$\Omega_{K}=\sqrt{GM/r^{3}}$ is the Keplerian angular velocity, $c_{s}$ the
isothermal sound speed, which is defined as $c_{s}^{2}=p_{gas}/\rho$,
$p_{gas}$ being the gas pressure, and $c_{A}$ is Alfven speed, which is
defined as $c_{A}^{2}=B_{\varphi}^{2}/4\pi\rho=2p_{mag}/\rho$, $p_{mag}$ being
the magnetic pressure. The angular momentum transfer equation is

$\rho v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r^{2}}\frac{d}{dr}\left[\nu\rho
r^{4}\frac{d\Omega}{\partial r}\right],$ (3)

where the right-hand side of above equation describes the effects of viscous
torques due to shear ($\nu$, here, is kinematic coefficient of viscosity). As
noted in the introduction, we assume both of the kinematic coefficient of
viscosity and the magnetic diffusivity due to turbulence in the accretion
flow. Thus, it is reasonable to use these parameters in analogy to the
$\alpha$-prescription of Shakura & Sunyaev (1973) for the turbulent,

$\nu=P_{m}\eta=\alpha\frac{c_{s}^{2}}{\Omega_{K}},$ (4)

where $P_{m}$ is the magnetic Prandtl number, which is assumed a constant of
order of unity, $\eta$ is the magnetic diffusivity, and $\alpha$ is a free
parameter less than unity.

The energy equation becomes

$\displaystyle\frac{v_{r}}{\gamma-1}\frac{d}{dr}(\rho
c^{2}_{s})+\frac{\gamma}{\gamma-1}\frac{\rho
c^{2}_{s}}{r^{2}}\frac{d}{dr}\left(r^{2}v_{r}\right)=$ $\displaystyle
Q_{diss}-Q_{rad}+Q_{cond},$ (5)

where $Q_{diss}=Q_{vis}+Q_{resis}$ is the dissipation rate by viscosity
$Q_{vis}$ and resistivity $Q_{resis}$, $Q_{rad}$ represents the energy loss
through radiative cooling, and $Q_{cond}$ is the energy transported by thermal
conduction. For the right-hand side of the energy equation, we can write

$Q_{adv}=Q_{diss}-Q_{rad}+Q_{cond}$ (6)

where $Q_{adv}$ is the advective transport of energy. We exploit the advection
factor, $f=1-Q_{rad}/Q_{diss}$, that describes the fraction of the dissipation
energy which is stored in the accretion flow and advected into the central
object rather than being radiated away. In general, the advection factor
depends on the details of the heating and radiative cooling mechanisms and
will vary by position (e.g. Watari 2006, 2007; Sinha et al. 2009). Here, we
assume a constant $f$ for simplicity. Clearly, the flow in the case of $f=1$
is in the extreme limit of no radiative cooling and in the limit of efficient
radiative cooling, we have $f=0$.

As mentioned, the inner regions of hot accretion flows are collisionless and
the electron mean free path due to Coulomb collision is larger than the
radius. This property is described as saturation (Cowie & McKee 1977). The
traditional equation of heat flux due to thermal conduction,
$F_{cond}=-\kappa\nabla T$ which $\kappa$ being the thermal conduction
coefficient, is not valid for such systems. Because this equation is suitable
for the collisional plasma, in which mean free path of electron energy
exchange is smaller than temperature scale height. Cowie & McKee (1977)
derived the heat flux for the collisionless plasma as

$F_{sat}=5\phi_{s}\rho c_{s}^{3}=5\phi_{s}p\sqrt{\frac{p}{\rho}},$ (7)

where $\phi_{s}$ is a factor is less than unity and is called as saturation
constant. Now, the viscous and resistive heating rates and the energy
transport by thermal conduction are expressed as

$Q_{vis}=\nu\rho r^{2}\left(\frac{\partial\Omega}{\partial r}\right)^{2}$ (8)
$Q_{resis}=\frac{\eta}{4\pi}{\bf J}^{2}$ (9)
$Q_{cond}=-\left|\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}F_{sat}\right)\right|$ (10)

where ${\mathbf{J}}=\nabla\times{\mathbf{B}}$ is the current density,
${\mathbf{B}}$ being the magnetic field. We used a minus sign in equation
(10). Because we want thermal conduction as energy transport mechanism
outward. On the other hand, heat conduction flux will be behaved like a
cooling mechanism in accretion flow. Finally, since we consider only the
toroidal field, the induction equation with field escape can be written as

$\frac{1}{r}\frac{d}{dr}\left[rv_{r}B_{\varphi}-\eta\frac{d}{dr}(rB_{\varphi})\right]=\dot{B}_{\varphi}.$
(11)

where $B_{\varphi}$ is the toroidal component of magnetic field and
$\dot{B}_{\varphi}$ is the field escaping/creating rate due to a magnetic
instability or dynamo effect. This induction equation is rewritten as

$\displaystyle\frac{1}{r}\frac{d}{dr}\left[\sqrt{4\pi\rho
c^{2}_{A}}\left(rv_{r}-\frac{\alpha}{4\beta
P_{m}}\frac{1}{r\rho\Omega_{K}}\frac{d}{dr}(r^{2}\rho
c^{2}_{A})\right)\right]=$ $\displaystyle\dot{B}_{\varphi},$ (12)

where $\beta$ is the degree of magnetic pressure to the gas pressure and can
be defined by

$\beta=\frac{p_{mag}}{p_{gas}}=\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2}.$
(13)

In this paper, we apply the steady self-similar methods to solve the system
equations. Thus, this parameter will be constant throughout the disc. While,
Khesali & Faghei (2008, 2009) showed that it varies by position. In hot
accretion flows, typical value of $\beta$ is in the range $0.01$-$1$ (e.g. De
Villiers et al. 2003; Beckwith et al. 2008). Here, we will also consider the
magnetically dominated case ($\beta>1$). Because, when thermal instability
happens in an ADAF, the MHD numerical simulations imply that the thermal
pressure rapidly decreases while the magnetic pressure increases due to the
conservation of magnetic flux (Machida et al. 2006). This will result in large
$\beta$ and forms a magnetically dominated accretion flow (Bu et al. 2009).

##  3 Self-Similar Solutions

###  3.1 Analysis

The self-similar method is useful to understand of the physics of accretion
flows. This method is familiar due to its wide range of applications in many
research fields of astrophysics. Self-similar solution, although constituting
only a limited part of problem, is often useful to understand the basic
behaviour of the system. Thus, in order to seek similarity solutions for the
above equations, we seek solutions in the following form:

$v_{r}(r)=-c_{1}\alpha\sqrt{\frac{GM_{*}}{r}}$ (14)

$\Omega(r)=c_{2}\sqrt{\frac{GM_{*}}{r^{3}}}$ (15)

$c^{2}_{s}(r)=c_{3}\frac{GM_{*}}{r}$ (16)

$c^{2}_{A}(r)=\frac{B^{2}_{\varphi}}{4\pi\rho}=2\beta c_{3}\frac{GM_{*}}{r}$
(17)

where coefficients $c_{1}$, $c_{2}$, and $c_{3}$ are determined later. We
assume a power-law relation for density

$\rho(r)=\rho_{0}r^{s}$ (18)

where $\rho_{0}$ and $s$ are constant. By using above self-similar quantities,
the mass-loss rate and the field escaping/creating rate must have the
following form:

$\dot{\rho}(r)=\dot{\rho}_{0}r^{s-3/2}$ (19)

$\dot{B}_{\varphi}(r)=\dot{B}_{0}r^{\frac{s-4}{2}}$ (20)

where $\dot{\rho}_{0}$ and $\dot{B}_{0}$ are constant.

Substituting the above solutions in the continuity, momentum, angular
momentum, energy, and induction equations [(1)-(3), (5), and (11)], we can
obtain the following relations:

$\dot{\rho}_{0}=-\left(s+\frac{3}{2}\right)\alpha\rho_{0}c_{1}\sqrt{GM_{*}},$
(21)

$-\frac{1}{2}c^{2}_{1}\alpha^{2}=c^{2}_{2}-1-c_{3}\left[s-1+\beta(1+s)\right],$
(22)

$c_{1}=3(s+2)c_{3},$ (23) $\displaystyle-\alpha
c_{1}\left[\frac{2(s-1)+3\gamma}{\gamma-1}\right]=\alpha
f\left[\frac{9}{2}c^{2}_{2}+\frac{\beta}{P_{m}}c_{3}(1+s)^{2}\right]$
$\displaystyle-10\phi_{s}c_{3}^{1/2}\left|s+\frac{1}{2}\right|,$ (24)

$\dot{B}_{0}=-\frac{\alpha s}{2}GM_{*}\sqrt{2\pi\rho_{0}\beta
c_{3}}\left[2c_{1}+\frac{c_{3}}{P_{m}}(1+s)\right].$ (25)

Above equations express for $s=-3/2$, there is no mass loss, while for
$s>-3/2$ mass loss (wind) exists. After algebraic manipulations, we obtain an
algebraic equation for $c_{3}$:

$\displaystyle{\frac{81}{8}}\,{\alpha}^{3}(s+2)^{2}f{c_{{3}}}^{2}+c_{3}\Bigg{[}\frac{3\,\alpha(s+2)}{2}\times\frac{2(s-1)+3\gamma}{\gamma-1}$
$\displaystyle-\frac{9\alpha
f}{4}\,\Bigg{(}\frac{2\beta}{9P_{m}}(s+1)^{2}+(s+1)\beta+s-1\Bigg{)}\Bigg{]}$
$\displaystyle-5\phi_{s}\left|s+\frac{1}{2}\right|\sqrt{c_{{3}}}-\frac{9}{4}\,f\alpha=0,$
(26)

and the rest of the physical variables are

$\dot{\rho}_{0}=-3\alpha\rho_{0}\sqrt{GM_{*}}(s+2)\left(s+\frac{3}{2}\right)c_{3},$
(27) $c_{1}=3c_{3}(s+2),$ (28)
$c_{2}^{2}=1-\frac{9\alpha^{2}}{2}(s+2)^{2}c_{3}^{2}+c_{3}\left[(s+1)\beta+s-1\right],$
(29) $\displaystyle\dot{B}_{0}=-3\alpha
sGM_{*}(s+2)^{3/2}c_{3}^{3/2}\sqrt{6\pi\rho_{0}\beta}\times$
$\displaystyle\left[1+\frac{s+1}{6P_{m}(s+2)}\right].$ (30)

We can solve algebraic equation (26) numerically and clearly only real roots
which correspond to positive $c_{1}$ are physically acceptable. Without
thermal conduction, i.e. $\phi_{s}=0$, (26) and similarity solutions reduce to
Faghei (2011a). But our main algebraic equation includes thermal conduction.

Fig. 1 : Physical quantities of the flow as a function of saturation constant
for $\gamma=1.3$, $\alpha=0.5$, $f=1$, and $P_{m}=100$. Solid, dotted, dashed,
and dot-dashed lines represent $\beta=0.1,0.3,0.7$, and $1.5$.

###  3.2 Numerical Results

Now, we consider the behaviour of solutions in the presence of thermal
conduction. But in this paper, only case of no wind ($s=-3/2$) is considered
that $\dot{\rho}_{0}=0$ and $\dot{B}_{\varphi}\propto r^{-11/4}$. In addition
to introduced coefficients, we also define a new parameter of $c_{4}$ that is
the right-hand side of equation (24)

$c_{4}=\alpha
f\left[\frac{9}{2}c^{2}_{2}+\frac{\beta}{P_{m}}c_{3}(1+s)^{2}\right]-10\phi_{s}c_{3}^{1/2}\left|s+\frac{1}{2}\right|.$
(31)

Equations (5), (24), and (31) imply that the parameter $c_{4}$ is the
advection transport of energy. The behaviour of coefficients $c_{i}$ as a
function of $\phi_{s}$ are shown in Figures 1-3. Moreover, Figure 1 represents
the profiles of physical quantities for several values of the magnetic
pressure fraction, i. e. $\beta=0.1,0.3,0.7$, and $1.5$. The value of $\beta$
measures the strength of magnetic field, and a larger $\beta$ denotes a
stronger magnetic field. Figure 2 represents the profiles of physical
quantities for several values of magnetic Prandtl number, i. e.
$P_{m}=\infty,1,2/3$, and $1/2$. The smaller values of $P_{m}$ denotes a
stronger magnetic diffusivity, $\eta$. Figure 3 shows the profiles of physical
quantities for several values of adiabatic index, i. e. $\gamma=1.2,1.25,1.3$,
and $1.35$.

Fig. 2 : Same as Fig. 1, but $\beta=1.0$, and solid, dotted, dashed, and dot-
dashed lines represent $P_{m}=\infty,1,2/3$, and $1/2$.

Fig. 3 :  Same as Fig. 1, but $\beta=1.0$, and solid, dotted, dashed, and dot-
dashed lines represent $\gamma=1.2,1.25,1.3$, and $1.35$.

The solutions in Figures 1-3 imply that the radial infall velocity, $c_{1}$,
and the sound speed, $c_{3}$, both decrease with the magnitude of conduction,
while the squared angular velocity, $c_{2}^{2}$, increases. These properties
are qualitatively consistent with dynamical analysis of Faghei (2011b). One
and two dimensional simulations of hot accretion flows have also shown that
thermal conduction reduces the flow temperature (Sharmal et al. 2008; Wu et
al. 2010). The profiles of advection transport of energy, $c_{4}$, in Figures
1-3 imply that thermal conduction behaves as a cooling mechanism, resulting in
a local decrease of the gas temperature relative to the original ADAF
solution. At the same time, the gas adjust its angular velocity (which
increases the level of viscous dissipation) and reduces its inflow speed.

Figure 1 shows the radial velocity, sound speed, and advection transport of
energy decrease by adding the magnetic pressure fraction, $\beta$. These
properties are qualitatively consistent with results of Bu et al. (2009) and
Faghei (2011a). Moreover, the angular velocity decreases with the magnitude of
magnetic field that is in according with results of Khesali & Faghei (2009)
and Faghei (2011a).

Figure 2 shows the magnetic diffusivity has the opposite effects of thermal
conduction on the physical variables. As, by adding the magnetic diffusivity,
$P_{m}^{-1}$, the radial velocity, sound speed, and advection transport of
energy increase, while the rotational velocity decrease. These results are
similar to resistive ADAF models without thermal conduction (e. g. Faghei
2011a).

Figure 3 represents the gas adiabatic index similar to magnetic diffusivity
has the opposite effects of thermal conduction on the physical quantities. As,
with the magnitude of $\gamma$, the inflow and sound speed increase, while the
rotational velocity decreases. This is in accord with dynamical study of hot
accretion flow (Faghei 2011b). Moreover, Figure 3 shows that the gas adiabatic
index contributes with thermal conduction to reduce advection transport of
energy. It can be due to decrease of rotational velocity by adding $\gamma$,
which reduces the level of viscous dissipation.

The studies of hot accretion flows (e. g. Shadmehri 2008) imply that the
solution for a given set of the input parameters reaches to a non-rotating
limit at a specific of $\phi_{s}$ which we denote it by $\phi_{s}^{c}$. With
zero insertion of $c_{2}$ in equations (22)-(24), $\phi_{s}^{c}$ can be
written as

$\displaystyle\phi_{s}^{c}=\frac{1}{60}\left[\frac{\beta
f}{P_{m}}-\frac{5/3-\gamma}{\gamma-1}\right]\times$
$\displaystyle\sqrt{2(\beta+5)\left[-1+\sqrt{1+\frac{18\alpha^{2}}{(\beta+5)^{2}}}\,\right]}.$
(32)

We can not extend the studies beyond $\phi_{s}^{c}$, because the right-hand
side of equation (29) becomes negative and a negative $c_{2}^{2}$ is clearly
unphysical. As, the rotational velocity profiles in Figures 1-3 show, we have
selected the input parameters that $c_{2}^{2}$ is positive. The behaviour of
critical saturation constant, $\phi_{s}^{c}$, as a function of $\beta$ for
several values of magnetic Prandtl number is shown in Figures 4. In left panel
of Figure 4, the viscosity parameter value is $\alpha=0.1$, and in right panel
is $\alpha=0.2$. The profiles of Figure 4 show the critical saturation
constant highly depends on magnetic pressure fraction, $\beta$, magnetic
Prandtl number, $P_{m}$, and viscosity parameter, $\alpha$. As, higher values
of $\beta$ corresponds to larger $\phi_{s}^{c}$. Critical saturation constant
also increases with higher value of $\alpha$. Since, magnetic diffusivity is
proportional to $P_{m}^{-1}$, Figure 4 shows that the magnetic diffusivity
similar to magnetic field increases $\phi_{s}^{c}$ value.

As mentioned in the introduction, Sharma et al. (2008) by resistive MHD
simulation studied a spherical accretion with thermal conduction. They found
for even modest thermal conductivities, conduction is the significant
mechanism of energy. Here, to compare conduction mechanism to others, we study
the ratio of energy transport by thermal conduction, $Q_{cond}$, to the gas
heating rate by viscosity $Q_{vis}$ and resistivity $Q_{resis}$. Such
solutions are shown in Figure 5 for two cases of non-resistive (left-panel)
and resistive (right-panel) flows. The solutions imply that thermal conduction
is the significant energy mechanism in the flow. This result confirms
simulation of Sharma et al. (2008). Moreover, the ratio of $Q_{cond}$ to
$Q_{diss}$ increases slightly by adding the magnetic field and does not change
for different values of magnetic Prandtl number. Because, a large fraction of
$Q_{diss}$ is generated by viscous dissipation.

##  4 Summary and Discussion

The collision timescale between ions and electrons in hot accretion flows is
longer than the inflow timescale. Thus, the inflow plasma is collisionless,
and transfer of energy by thermal conduction can be dynamically important. The
low collisional rate of the gas is confirmed by direct observation,
particularly in the case of the Galactic centre (Quataert 2004; Tanaka & Menou
2006) and in the intracluster medium of galaxy clusters (Sarazin 1986).

In this paper, the structure of a magnetized ADAF in the presence of
resistivity and thermal conduction is investigated. We assumed the magnetic
field has a purely toroidal component. We adopted the presented solutions by
Tanaka & Menou (2006) and Faghei (2011a). Thus, we assumed that angular
momentum transport is due to viscous turbulence and the $\alpha$\-
prescription is used for the kinematic coefficient of viscosity. We also
assumed the flow does not have a good cooling efficiency and so a fraction of
energy accretes along with matter on to the central object. In order to solve
the equations that govern the structure behaviour of magnetized ADAF with
thermal conduction, we have used steady self-similar solution.

Fig. 4 :  The critical saturation constant as a function of the ratio of
magnetic pressure to gas pressure. Solid, dotted, dashed, and dot-dashed lines
represent $P_{m}=10,5,1$, and $0.5$. The input parameters are set to
$\gamma=5/3$, $f=1$, $s=-3/2$, and the viscous parameter $\alpha$ in left-
panel is $0.1$ and in right-panel is $0.2$.

The solutions showed the radial infall velocity and sound speed in the
presence of thermal conduction both decrease, while angular velocity increase.
These properties are consistent with dynamical study of hot accretion flow (e.
g. Faghei 2011b) and from some aspects also are in accord with simulations of
Sharma et al. (2008) and Wu et al. (2010). Moreover, the solutions represent
the magnetic diffusivity and thermal conduction have the opposite effects on
physical quantities. For a moderate thermal conduction, the solutions imply
that thermal conduction can play an important role in energy mechanism of the
system. This property is qualitatively consistent with non-ideal simulations
of Sharma et al. (2008).

In the present model, accretion flow is studied in one-dimensional approach
and ignored from latitudinal dependence of physical quantities. There are some
researches in two-dimensional approach that express the importance of such
studies (Tanaka & Menou 2006; Ghanbari et al. 2009; Wu et al. 2010 ). Thus,
latitudinal study of present model can be investigated in other research.
Here, we used a saturated heat conduction flux. While, there are some studies
with unsaturated heat flux (e. g. Shcherbakkov & baganoff 2010) that show a
good agreement with observations. Thus, the study of the present model in a
unsaturated case will be interesting.

Fig. 5 :  The ratio of energy transport by thermal conduction, $Q_{cond}$, to
the gas heating rate by viscosity $Q_{vis}$ and resistivity $Q_{resis}$ as a
function of saturation constant. Solid, dotted, dashed, and dot-dashed lines
represent $\beta=0.0,1.0,3.0$, and $5.0$. The input parameters are set to
$\gamma=4/3$, $f=1$, $s=-3/2$, $\alpha=0.5$, and the magnetic Prandtl number
in left-panel is $\infty$ and in right-panel is $0.5$.

## Acknowledgements

I wish to thank the anonymous referee for very useful comments that helped me
to improve the initial version of the paper. I would also like to thank Roman
V. Shcherbakov for his helpful comments.

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