# Automated Detection of Satellite Trails in Ground-Based Observations Using
U-Net and Hough Transform

F. Stoppa1,2, P.J. Groot1,3,4, R. Stuik5, P. Vreeswijk1, S. Bloemen1, D.L.A.
Pieterse1, P.A. Woudt3 1 Department of Astrophysics/IMAPP, Radboud University,
PO Box 9010, 6500 GL Nijmegen, The Netherlands 2 Department of
Mathematics/IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The
Netherlands 3 Department of Astronomy and Inter-University Institute for Data
Intensive Astronomy, University of Cape Town,
Private Bag X3, Rondebosch, 7701, South Africa 4 South African Astronomical
Observatory, P.O. Box 9, Observatory, 7935, South Africa 5 Leiden Observatory,
Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands

(August 28, 2024
)

###### Abstract

Aims. The expansion of satellite constellations poses a significant challenge
to optical ground-based astronomical observations, as satellite trails degrade
observational data and compromise research quality. Addressing these
challenges requires developing robust detection methods to enhance data
processing pipelines, creating a reliable approach for detecting and analyzing
satellite trails that can be easily reproduced and applied by other
observatories and data processing groups.

Methods. Our method, called ASTA (Automated Satellite Tracking for Astronomy),
combines deep learning and computer vision techniques for effective satellite
trail detection. It employs a U-Net based deep learning network to initially
detect trails, followed by a Probabilistic Hough Transform to refine the
output. ASTA’s U-Net model was trained on a dataset with manually labelled
full-field MeerLICHT images prepared using the LABKIT annotation tool,
ensuring high-quality and precise annotations. This annotation process was
crucial for the model to learn and generalize the characteristics of satellite
trails effectively. Furthermore, the user-friendly LABKIT tool facilitated
quick and efficient data refinements, streamlining the overall model
development process.

Results. ASTA’s performance was evaluated on a test set of 20,000 image
patches, both with and without satellite trails, to rigorously assess its
precision and recall. Additionally, ASTA was applied to approximately 200,000
full-field MeerLICHT images, demonstrating its effectiveness in identifying
and characterizing satellite trails. The software’s results were validated by
cross-referencing detected trails with known public satellite catalogs,
confirming its reliability and showcasing its ability to uncover previously
untracked objects.

##  I Introduction

The rise of large satellite constellations has greatly enhanced global
communication and connectivity. However, this development introduces
significant challenges for ground-based telescopes, as satellite trails appear
as streaks of light in astronomical images, degrading data quality by
introducing noise and covering celestial objects (McDowell, 2020).
Historically, astronomical observations have faced various sources of
interference, but the rapid increase in satellite launches, especially with
mega-constellations such as Starlink and OneWeb, has exacerbated the issue
(Tyson  _et al._ , 2020, Hainaut and Williams, 2020, Mallama, 2022, Bassa  _et
al._ , 2022, Gallozzi  _et al._ , 2020, Groot, 2022). Even space-based
observatories, such as the Hubble Space Telescope, are not immune to these
effects, highlighting the pervasive nature of satellite interference (Kruk
_et al._ , 2023).

The anticipated surge in satellite deployments could significantly increase
the frequency and severity of interference, underscoring the need for
proactive measures and innovative solutions to ensure the continued
effectiveness of ground-based astronomy (Walker  _et al._ , 2020). Despite
their advanced capabilities, wide-field optical ground-based telescopes, such
as ATLAS (Tonry  _et al._ , 2018), GOTO (Steeghs  _et al._ , 2022), BlackGEM
(Groot  _et al._ , 2024), ZTF (Bellm  _et al._ , 2019), and Pan-STARRS
(Chambers  _et al._ , 2016), face significant challenges due to the increasing
presence of satellite trails in their observations. Upcoming large aperture
and wide field-of-view facilities like the Vera Rubin Observatory (Ivezić and
Kahn, 2019) will be even more affected by these bright satellite trails. These
streaks introduce noise and can obscure celestial objects, complicating the
detection and analysis of transient events (see e.g., Groot 2022).

Currently, several methods are employed to detect and mitigate satellite
trails in astronomical images. These methods can be broadly categorized into
simple source detection, template fitting to line shapes, computer vision
techniques, and machine learning algorithms. Simple source detection, such as
using SExtractor (Bertin and Arnouts, 1996) and focusing on elongated shapes,
has been employed to detect streaks. However, this approach is less effective
at low signal-to-noise ratios (S/N) and tends to result in a high false-alarm
rate (Waszczak  _et al._ , 2017). In contrast, template fitting to line shapes
involves aligning a predefined streak shape to the image data and calculating
a weighted sum of the pixels along the line to assess the match quality. This
matched-filter approach (Turin, 1960), used by Dawson  _et al._ 2016, employs
the maximum likelihood method to detect streaks, assuming uncorrelated noise
and a constant known Point Spread Function (PSF). While accurate, it can be
computationally intensive and may require multiple templates or faster
computational techniques.

Computer vision techniques provide another set of tools for streak detection,
using methods developed for natural image processing. These techniques are
attractive due to their versatility and effectiveness in various imaging
contexts. Two popular methods are the Hough Transform (Duda and Hart, 1972)
and the Radon Transform (Radon, 1986). The Hough Transform is applied to
binary images following edge detection, while the Radon Transform is used on
grayscale images. Despite their different applications, they are
mathematically equivalent and have both been successful in finding streaks in
crowded fields and images with diffuse light sources (Cheselka, 1999, Virtanen
_et al._ , 2014, Bektešević and Vinković, 2017). Both the Hough Transform and
the Radon Transform have seen improvements in computational efficiency with
the development of the Probabilistic Hough Transform and the Fast Radon
Transform (FRT), respectively. These advanced methods, which often leverage
GPUs, provide significant improvements in speed and computational performance
(Zimmer  _et al._ , 2013, Andersson  _et al._ , 2015, Nir  _et al._ , 2018,
Borncamp and Lian Lim, 2019).

Despite recent progress, there remains a need for methods that can handle the
increasing complexity and volume of astronomical data. Deep learning, in
particular, offers new opportunities for improving streak detection. By
training convolutional neural networks (CNNs, LeCun  _et al._ , 1999) on large
datasets of labelled images, these methods can learn complex patterns and
features directly from data, improving detection rates and reducing false
positives (Paillassa  _et al._ , 2020, Elhakiem  _et al._ , 2023, Chatterjee
_et al._ , 2024). Building on these advancements, our study introduces ASTA
(Automated Satellite Tracking for Astronomy), a novel tool that uniquely
combines the strengths of deep learning and computer vision techniques for
detecting satellite trails in ground-based observations. We use a U-Net
architecture (Ronneberger  _et al._ , 2015) for initial satellite streak
detection, followed by a Probabilistic Hough Transform to refine the output
and extract satellite information.

A common criticism of machine learning methods is the difficulty in obtaining
high-quality and well-labelled training sets, which are crucial for achieving
accuracy and reliability. This issue often hinders the widespread adoption and
reproducibility of such methods. However, we have addressed this challenge by
using the LABKIT tool (Arzt  _et al._ , 2022) to meticulously annotate images
from the MeerLICHT telescope (Bloemen  _et al._ , 2016), ensuring the high-
quality training data necessary for effective model training. This approach
not only produces a reliable dataset for our application but also simplifies
the creation process, making it accessible to other researchers and
observatories. By providing a detailed account of our methodology and dataset
preparation, we aim to facilitate the widespread adoption of ASTA, enhancing
the ability of observatories worldwide to mitigate the impact of satellite
trails on astronomical data quality.

The paper is organized as follows: Section II describes the data and the
process of image selection and preparation. Section III outlines the machine
learning techniques and the Probabilistic Hough Transform used for refining
the detections, along with their validation. Section IV demonstrates the
application of our tool, ASTA, by searching for Geostationary and
Geosynchronous satellites in approximately 200,000 full-field MeerLICHT
images. This analysis focuses on identifying satellite trails and cross-
referencing the results with known public satellite catalogs. Finally, Section
V summarizes our key findings and discusses the significance of developing
sustainable solutions to ensure the advancement of astronomical research in
the era of satellite constellations.

ASTA is directly accessible on GitHub 111https://github.com/FiorenSt/ASTA. All
the images and masks used in this paper for training, test, and validation are
available on Zenodo 222https://zenodo.org/records/11642424 (Stoppa, 2024).

##  II Data

This study uses data from the MeerLICHT telescope, located in South Africa.
MeerLICHT, a prototype for the recently operational BlackGEM telescope array
(Groot  _et al._ , 2024), plays a crucial role in detecting and analyzing
transient astronomical phenomena. Both MeerLICHT and BlackGEM are designed to
capture high-quality astronomical data, but their ground-based nature makes
them susceptible to satellite trails, which can degrade the quality of the
data and impact the accurate analysis and interpretation of transient events.

In this section, we explain the data used to create ASTA and the dataset-
building process. This involves selecting, preparing, and manually labelling
images to ensure high-quality training data for our machine learning
algorithm.

###  II.1 MeerLICHT and BlackGEM Telescopes Array

Figure 1: Full-field, $10560\times 10560$ pixels, MeerLICHT image with a
60-second exposure, showcasing five satellite trails overlaying dozens of
sources.

The MeerLICHT telescope, with its 65 cm aperture and a high-resolution
$10.5k\times 10.5k$ pixel CCD, offers a wide field-of-view of 2.7 square
degrees, sampled at $0.56^{\prime\prime}$/pixel. MeerLICHT is equipped with
the Sloan-Gunn type $u,g,r,i,z$ filter set, along with an additional wide-band
$q$ filter (440-720 nm), enhancing its observational versatility. The
telescope uses 60s integrations by default and reaches a limiting point-source
magnitude of $q_{\text{AB}}>20.5$ under standard conditions. As a consequence,
any satellite in Low Earth Orbit (LEO) up to Geostationary orbit (GEO) is very
well detected and causes strong streaks in the observations. This causes
interference in the photometric measurements of astrophysical objects but, at
the same time, offers the opportunity to monitor the satellite’s presence in
Earth’s orbit.

Images captured by MeerLICHT are promptly processed at the IDIA/ilifu facility
using BlackBOX 333https://github.com/pmvreeswijk/BlackBOX image processing
software (Vreeswijk et al., in prep). The processing pipeline includes source
detection via SourceExtractor (Bertin and Arnouts, 1996), astrometric and
photometric calibration (Lang  _et al._ , 2010), PSF determination (Bertin,
2011), image subtraction, and transient detection (Zackay  _et al._ , 2016,
Hosenie  _et al._ , 2021). Furthermore, a new set of deep learning methods has
been developed specifically for the MeerLICHT/BlackGEM telescopes. These
methods are currently being tested and compared against traditional pixel-
based counterpart techniques as detailed in Stoppa  _et al._ , 2022, 2023a,
2023b.

Figure 1 illustrates a typical full-field MeerLICHT image, showcasing the
impact of multiple satellite trails.

###  II.2 Dataset preparation

Figure 2: Examples of satellite trail annotations in MeerLICHT images. The
figure shows two columns and four rows: (a) Original images with satellite
trails of varying intensities and types in the first column, (b) Corresponding
labelled ground truth masks in the second column.

Creating a high-quality training dataset is fundamental for developing an
effective machine learning model. In our study, we require an accurately
labelled dataset where pixels belonging to a satellite trail are assigned to
one class, and all other pixels—including those representing the sky
background, stars, and common linear artefacts seen in CCD images, such as
diffraction spikes, cosmic ray hits, and charge bleeding—are assigned to
another class. With such a dataset, the network can effectively learn the
relationship between the original image and the segmentation mask, enabling
accurate and reliable detection of satellite trails.

However, a significant challenge for researchers attempting to reproduce
machine learning methods for their specific applications is the lack of a
well-labelled dataset and uncertainty about how to create one from scratch.
This obstacle often discourages the adoption of machine learning techniques.
To address this issue, we provide a detailed explanation of how to build a
reliable dataset for satellite trail detection, offering a clear guide for
other researchers and observatories.

We started by collecting 178 full-field MeerLICHT images, all visually
inspected to identify the presence of satellite trails. These images included
trails of varying lengths and signal-to-noise ratios, ensuring a diverse and
comprehensive dataset. As can be seen in Fig. 2, MeerLICHT’s spatial
resolution is high enough that seeing- and tracking-variations during the
integration can be seen as a ‘wavy’ nature of the trail, making them
significantly deviate from just a straight line.

To facilitate the creation of the ground truth segmentation masks, we used
LABKIT (Arzt  _et al._ , 2022), a plugin for the Fiji image processing
package, which simplifies the annotation process with its pixel classification
algorithm for quick automatic segmentation.

In LABKIT, we first manually classified a really small subset of pixels in the
images ($\sim 0.001\%$) into two classes: satellite trails and background.
This initial step involved selecting pixels that represent the trails and
marking the sky background, sources, and other linear features as background.
LABKIT then used these labelled pixels and a set of features automatically
extracted from the images to train a random forest classifier. The feature
vector for each labelled pixel is obtained by applying a set of filters to the
image, such as Gaussian, a difference of Gaussians, or Laplacian filters.
These filters emphasize different features of the input image, and their
responses for each pixel are added to the feature vector. The final feature
vectors, paired with their respective ground-truth classes, constitute the
training set for the random forest, which consists of a hundred decision
trees. LABKIT then proceeds to predict the segmentation mask for the entire
image and provides a quick, although rough, approximation of all the trails.

After LABKIT’s initial automated classification, the software easily provides
tools to manually refine the annotations to ensure their accuracy and
reliability. Using LABKIT’s interface, we adjusted the pixels and specific
parts of trails or spurious detections that were assigned to the wrong class.
This comprehensive annotation process resulted in a full binary mask for each
full-field MeerLICHT image and was completed by a single person in three days.

After labelling, the full-field images and their associated ground truth masks
were divided into smaller patches of 528x528 pixels to facilitate efficient
training. Data augmentation techniques, including 90-degree rotations, flips,
and shifts, were applied exclusively to patches containing satellite trails.
This approach enhanced the model’s exposure to diverse trail characteristics,
thereby improving detection accuracy. This selective augmentation was
necessary to prevent the dataset from becoming unbalanced, as most patches
from a full-field image do not contain satellite trails. Without this, the
dataset would be dominated by patches with empty masks, making the model
training more difficult and less effective.

Figure 2 shows four examples of patches with satellite trails and their
corresponding ground truth masks.

##  III Method

This section outlines the methodology employed by ASTA for detecting and
analyzing satellite trails in astronomical images. ASTA leverages a U-Net
architecture for the initial segmentation of satellite trails, providing a
robust framework for distinguishing trails from other features. To enhance the
precision of trail delineation, the initial segmentation is refined using the
Probabilistic Hough Transform (Galamhos  _et al._ , 1999). This combination of
deep learning and classical image processing techniques ensures high accuracy
and reliability in identifying and characterizing satellite trails.

###  III.1 Detection of Satellite Trails Using U-Net

Figure 3: Comparison of ground truth segmentation mask and U-Net predicted
segmentation map. The top panel shows the ground truth segmentation mask with
satellite trails marked in black. The bottom panel shows the U-Net predicted
segmentation map, with pixel values ranging from 0 to 1, indicating the
likelihood of each pixel belonging to a satellite trail. The insets provide a
zoomed-in view to highlight the detailed accuracy of the predictions.

Astronomical images are often crowded with various linear and non-linear
features, including stars, galaxies, cosmic rays, and diffraction spikes from
bright stars. These complexities present a significant challenge for
accurately identifying satellite trails. The U-Net architecture (Ronneberger
_et al._ , 2015), originally developed for biomedical image segmentation,
excels at identifying complex patterns in images, making it well-suited for
detecting satellite trails against such diverse backgrounds.

U-Net’s architecture is designed to understand and reconstruct the context of
an image through two main pathways: a contracting path that compresses the
image to grasp its broader context, and an expansive path that reconstructs
the image’s details for precise localization of features. This structure
allows U-Net to process the image at multiple scales, capturing both the
overall patterns and the fine details. By integrating features from both
paths, U-Net maintains a balance between contextual understanding and detailed
segmentation.

Our network consists of convolutional layers with LeakyReLU activations that
adjust the spatial dimensions and number of filters, from 8 to 128. To prevent
overfitting, dropout layers are integrated within the network. The final layer
produces a predicted segmentation map, indicating the likelihood of each pixel
belonging to a satellite trail with values ranging from 0 to 1. Overall, the
model has approximately 485,000 trainable parameters, ensuring it is both
lightweight and efficient for large astronomical datasets.

To optimize the model during training, we used the Combo loss (Asgari
Taghanaki  _et al._ , 2018), a combination of binary cross-entropy (BCE) loss
(Mannor  _et al._ , 2005) and Dice loss (Sudre  _et al._ , 2017). This
approach balances pixel-wise accuracy with segmentation performance, improving
detection accuracy in class-imbalanced datasets. For a more detailed
description of this loss function, we refer readers to Stoppa  _et al._ 2022.

Figure 3 illustrates a comparison between a ground truth segmentation mask and
the corresponding prediction made by the U-Net model. The predicted
segmentation map effectively captures the trail, and the pixel values decrease
rapidly to zero beyond the edges of the trail, indicating the U-Net’s ability
to accurately identify and differentiate the trail from the background and
other artefacts.

After the U-Net processes the images, we apply a threshold to its output to
create binary segmentation masks. This step is crucial as we need to identify
which pixels belong to satellite trails and which do not. Pixels with values
above the threshold are classified as satellite trails (value of 1), while
those below are classified as background, including the sky and astronomical
sources (value of 0).

To evaluate the effectiveness of the U-Net and determine the optimal
threshold, we tested the model on 20,000 patches, both with and without
satellite trails. We used several metrics, including precision, recall,
F1-score, and Intersection over Union (IoU), to assess the model’s performance
at various threshold levels. Figure 4 shows these metrics across different
thresholds.

Figure 4: Performance metrics for U-Net across different threshold levels:
IoU, Precision, Recall, and F1-score. A threshold of 0.58 provides a balanced
result in terms of all metrics tested.

Precision measures the accuracy of the model’s positive predictions, while
recall assesses its ability to identify all relevant instances. The F1-score
balances precision and recall, providing a single metric that accounts for
both false positives and false negatives. Intersection over Union (IoU)
measures the overlap between the predicted segmentation and the ground truth,
offering a comprehensive view of segmentation quality. Independently of the
threshold, all metrics indicate that U-Net performs well in predicting trails
and distinguishing them from other linear artefacts. A threshold of 0.58
provides a balanced result and is therefore used as the default value for the
successive analyses in this paper.

###  III.2 Refinement with Probabilistic Hough Transform

Figure 5: Sequential steps of satellite trail detection and refinement: (a)
Original patch image, (b) Ground truth segmentation mask, (c) U-Net predicted
segmentation map, (d) Final result after applying the Probabilistic Hough
Transform. This workflow demonstrates the process from initial detection to
refined trail delineation, ensuring precise identification and continuity of
satellite trails.

While U-Net predictions effectively identify satellite trails, they may
exhibit gaps due to factors such as tumbling of the satellites, bright stars,
detector defects, or unfortunate locations in the $528\times 528$ pixel
patches. To address these gaps in the recombined full-field binary masks, we
applied the Probabilistic Hough Transform (Galamhos  _et al._ , 1999), which
is highly effective for detecting linear patterns in images. The primary
function of the Hough Transform in this context is to fill in splits or gaps
in the predicted trails, ensuring continuous and accurate representation. This
refinement step maintains the integrity of trail detection, particularly in
areas with discontinuities, and ensures more accurate statistics about
satellites. This is crucial for further steps, such as estimating the total
number of satellites and matching them with known satellite catalogues.

The Hough Transform works by translating spatial relationships within an image
into a parameter space, making it effective for detecting linear patterns like
satellite trails. In this space, any line in the image can be represented as a
point defined by the equation $r=x\cos(\theta)+y\sin(\theta)$, where $r$ is
the perpendicular distance from the origin to the line, and $\theta$ is the
angle of this perpendicular line with the horizontal axis. For each pixel that
might belong to a satellite trail, the Hough Transform evaluates every
possible line through that pixel, represented by various ($r$, $\theta$)
combinations, resulting in a sinusoidal curve in the parameter space for each
pixel. The intersection of these curves from different points indicates a
consensus on the presence of a line in the image space, with accumulations in
an array highlighting the most significant lines. A threshold is applied to
distinguish meaningful lines from noise, ensuring that only genuine satellite
trails are identified.

The Probabilistic Hough Transform, a variant of the standard Hough Transform,
improves computational efficiency by considering a subset of points in the
image to detect lines. Optimizing the resolution of the parameter space,
particularly the granularity of the ($r$, $\theta$) bins, is essential for
accurately detecting satellite trails, balancing sensitivity and computational
efficiency. This adjustment allows for precise identification of satellite
trails, enhancing the reliability of our analysis. Figure 5 illustrates two
cases where the U-Net prediction is improved by the Hough Transform,
demonstrating the process from initial detection to refined trail delineation
and ensuring precise identification and continuity of satellite trails.

###  III.3 Contour Analysis and Feature Extraction

Following the refinement by the Hough Transform, we obtain a binary mask
consisting solely of satellite trails. The final step involves extracting
features of the detected trails, such as length, location of start and end
points, inclination, and brightness.

To achieve this, we identify the contours of the trails in the refined binary
mask using the cv2.findContours function from the OpenCV package. If each
satellite trail were independent, we would only need to extract the pixel
values within the contour to determine the trail brightness and easily
identify the most extreme points of the trail and their coordinates. However,
the recent increase in satellite trails often results in multiple trails
crossing each other, as shown in the first row of Fig 5.

To address this occurrence, for each independent contour identified, there is
an effective method to determine if the contour is actually the intersection
of two or more trails. This is achieved by running a clustering algorithm
based on DBSCAN (Density-Based Spatial Clustering of Applications with Noise,
Ester  _et al._ , 1996) on the angles of the Hough Transform segments that
compose the current contour. This can quickly identify as many clusters as
there are intersecting trails and provides an effective solution for
separating them. Once the trails are separated, features such as length,
location of start and end points, inclination, and brightness are easily
extracted.

##  IV Application to MeerLICHT Data

Figure 6: Matched (left) and unmatched (right) GEO/GES satellite trails in
celestial equatorial coordinates. The geostationary belt is visible at Dec=+6∘
, as observed from South Africa. The sinusoidal band are the geosynchronous
satellites. The unmatched trails show satellites in both the geostationary
belt and in geosynchronous orbit, as well as more extreme cases at slightly
higher and lower declinations.

In this section, we apply ASTA to MeerLICHT images collected since January
2020. We analyzed approximately 200,000 non-red-flagged (i.e., science-grade)
images, detecting both Low Earth Orbit (LEO) and Geostationary/Geosynchronous
(GEO/GES) satellites. LEO satellites typically cause streaks from edge to edge
of a full-field MeerLICHT image, as illustrated in Fig. 1, complicating their
identification. These will be highlighted in a forthcoming paper. Here, we
focus on the detection and identification of both GEO and GES satellites,
whose trails generally start and stop within one exposure with a 60-second
integration time.

GEO satellites orbit Earth at an altitude of approximately 35,786 kilometers,
matching the planet’s rotational period. This allows them to remain stationary
relative to a fixed point on Earth. GEOs are widely used for communication,
weather monitoring, and broadcasting. In astronomical images, GEOs typically
appear as short streaks with a length consistent with the integration time in
seconds due to the fact that the telescope tracks celestial objects and not
satellites. For MeerLICHT/BlackGEM, which use 60 s integration times, this
means a streak of 15 arcminutes in length, corresponding to $\sim$1600 pixels,
in the East-West direction (90∘ in the convention used here).

GES satellites, on the other hand, have orbits with the same period as the
Earth’s rotation but are inclined relative to the equator. This results in
their position in the sky tracing an analemma over time. These satellites can
appear as longer or shorter trails depending on their current position in
their orbit relative to the observer and are generally at an angle to the
East-West orientation on the detector.

While ASTA effectively detects satellite trails, some artefacts and image
contaminations can mimic satellite signatures. Diffraction spikes, which are
linear artefacts caused by bright stars, always align at 45∘ and can be
mistaken for satellite trails. Additionally, CCD defects such as dead or hot
pixels and reflections from the Moon and/or local light sources such as car
headlights can produce spurious linear patterns. Specifically for
MeerLICHT/BlackGEM, there is a class of reflections from bright stars just
outside the field of view that appear as short horizontal (North-South)
streaks on the edge of the detector due to an unwanted reflection in the
cryostat. However, for this application to GEOs and GES satellites, these
artefacts are less problematic since GEO trails are almost exactly in the
East-West direction and geosynchronous satellites are centred at
$90\hbox{${}^{\circ}$\,}$ with an inclination of $\pm
15\hbox{${}^{\circ}$\,}$.

Starting from ASTA’s results for over 200,000 MeerLICHT images, we selected
all trails away from the edges of the full-field images and matched them with
catalogs obtained for the day of the observation from celestrak.org. This
process involved cross-referencing the detected satellite trails with
cataloged positions and trajectories using Two-Line Elements (TLEs), ensuring
accurate identification of the satellites. We consider a detected trail to
match a cataloged trail if the difference in their inclinations is less than
0.4 degrees, reflecting the high parallelism of well-matched trails.
Additionally, the average distance between the ends of the two trails must be
within a maximum of 200 arcseconds, accommodating minor shifts due to timing
discrepancies between the known satellites and our exposure times.

For this initial study, we concentrated on GEO and GES satellites, applying a
declination cut between -15 and +25 degrees. Examining the distribution of the
matched satellites, we selected all detected trails, whether positively or
poorly matched, with trail lengths in pixels between 1440 and 1640, and
oriented between 72.5 and 107.5 degrees, as there is a clear cutoff in the
matched satellites at those degrees. Out of the remaining $9107$ detected
trails, $7365$ ($80.9\%$) were matched to known satellites, while $1742$
($19.1\%$) remained unmatched. In Fig. 6, we show both the matched and
unmatched satellites. These form a thin band around the (projected) celestial
equator. As seen from South Africa, part of the Virgo galaxy cluster lies
projected behind the geostationary belt, leading to a large number of
detections at RA$\sim$12hr, as these fields have been among the main targets
of the MeerLICHT telescopes.

In addition to matched GEO and GES satellites, a substantial number of
detections, $19.1\%$, cannot be matched to any object in public catalogues. A
significant fraction of these unmatched satellites exhibit properties similar
to those of the matched ones. Fig. 7 shows the comparison in terms of
orientation and median brightness, while Fig. 8 presents a stacked histogram
of pixel lengths for both matched and unmatched trails, further supporting
this observation.

Figure 7: Comparison between matched and unmatched satellites in terms of
orientation and median brightness of GEOs/GESs. $90\hbox{${}^{\circ}$\,}$ is
E-W. Figure 8: Stacked histogram of pixels length for matched and unmatched
trails.

The similarity in properties between matched and unmatched satellites
indicates the presence of a high fraction of publicly unknown (former)
satellites or rocket stages. By identifying and cataloguing these unmatched
satellites, especially if they are observed in multiple images taken within a
short time frame and exhibit realistic orbital characteristics, we can
contribute to the maintenance and expansion of public satellite catalogues.
This effort is crucial for maintaining the safety and accuracy of future
astronomical observations and could aid in the management and mitigation of
space debris.

##  V Conclusions

In this study, we introduced ASTA (Automated Satellite Tracking for
Astronomy), a robust methodology combining U-Net and Probabilistic Hough
Transform to detect and analyze satellite trails in ground-based astronomical
observations. Using data from the MeerLICHT telescope, we demonstrated the
effectiveness of ASTA in identifying and characterizing satellite trails.

Importantly, the methodology developed in this study can be easily adopted by
other observatories. The use of LABKIT for manual annotation ensures a
straightforward and reproducible process, encouraging other researchers and
observatories to implement similar techniques. By sharing this approach, we
hope to foster a collaborative effort in addressing the challenges posed by
satellite trails, improving data quality across various astronomical
facilities.

The precision of ASTA in detecting satellite trails was validated against a
comprehensive test set, showcasing high accuracy and robustness. The
integration of deep learning and classical image processing techniques proved
effective in refining trail detection, even in the presence of bright stars
and other complex backgrounds.

ASTA was applied to approximately 200,000 MeerLICHT images, and the results
were cross-referenced with public satellite catalogs. We focused on
identifying geostationary and geosynchronous satellites, successfully matching
$81\%$ of the detected trails with known objects. Additionally, the discovery
of satellites not found in public catalogs underscores gaps in current
cataloging efforts and reveals the presence of previously uncatalogued
objects.

Future improvements could enhance artefact recognition and overall model
performance. One key improvement is the use of better and larger GPUs, which
would eliminate the need for patching. This would provide U-Net with more
context from larger images and nearby sources, reducing confusion caused by
small patches and minimizing gaps or inconsistencies in predictions. By
processing larger portions of the image at once, U-Net can maintain context
and continuity across the entire field, leading to more accurate and reliable
detections. Additionally, increased computational efficiency from advanced
GPUs will allow for faster processing times, making it feasible to analyze
large datasets of astronomical images more efficiently.

Future work will concentrate on a comprehensive statistical analysis of the
temporal and spatial components of all types of satellites, from GEOs to LEO
satellites, using MeerLICHT and BlackGEM data collected over the last five
years. This analysis will examine trends and patterns, improving our
understanding of satellite distributions and their impact on observational
data, providing a basis for developing more effective mitigation strategies.

###### Acknowledgements.

We thank Dr Marco Langbroek for the insightful discussions on satellite trails
and the various observational biases associated with them. PJG is partly
supported by SARChI Grant 111692 from the South African National Research
Foundation. MeerLICHT is designed, built and operated by a consortium of
universities and institutes, consisting of Radboud University, the University
of Cape Town, the South African Astronomical Observatory, the University of
Oxford, the University of Manchester and the University of Amsterdam.

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##  Appendix A Computation Time

To evaluate the efficiency of ASTA, we measured the computation time required
to process a full-field MeerLICHT image, including image loading, patch
creation, U-Net prediction, Hough Transform application, and satellite
information extraction. The tests were conducted on an Alienware Area 51M
equipped with an Intel Core i9-9900K processor, 32GB DDR4/2400 RAM, and an
Nvidia GeForce RTX 2080 GPU.

The processing times were measured using both CPU and GPU for each of the four
stages: Preprocessing, Prediction, Hough Transform, and Contour Analysis. The
results are summarized in Table 1.

These results demonstrate the substantial efficiency gains achievable through
GPU acceleration. The average total GPU processing time was significantly
lower than the total CPU processing time, highlighting the critical role of
GPU acceleration in handling large volumes of astronomical data effectively.

Table 1: Average computation time and standard deviation for each processing stage using CPU and GPU. Stage | CPU Time (s) | GPU Time (s)
---|---|---
Preprocessing | $0.66\pm 0.01$ | $0.66\pm 0.01$
Prediction | $21.97\pm 0.25$ | $2.56\pm 0.04$
Hough Transform | $0.13\pm 0.02$ | $0.13\pm 0.02$
Contour Analysis | $1.20\pm 0.26$ | $1.24\pm 0.26$
Total | $23.96\pm 0.53$ | $4.59\pm 0.33$

The Contour Analysis stage exhibits the highest variability in computation
time, which can be attributed to the complexity and number of detected
satellite trails in each image. The variability arises from the differing
number of contours that need to be processed and the presence of intersecting
trails, which require additional computation to separate accurately.


# Frequency enhancement and power tunability in tilted polarizer spin-torque
nano oscillator

R. Arun1<EMAIL_ADDRESS>R. Gopal2<EMAIL_ADDRESS>V. K.
Chandrasekar2<EMAIL_ADDRESS>M. Lakshmanan1<EMAIL_ADDRESS>1Department of Nonlinear Dynamics, School of Physics, Bharathidasan
University, Tiruchirapalli-620024, India
2Centre for Nonlinear Science & Engineering, School of Electrical &
Electronics Engineering, SASTRA Deemed University, Thanjavur- 613 401, India.

###### Abstract

In the absence of an applied magnetic field, a spin-torque nano
oscillator(STNO) with a tilted polarizer is studied using numerical simulation
of the associated Landau-Lifshitz-Gilbert-Slonczewski equation. We find
considerable enhancement of frequency by tilting the polarizer out-of-plane
appropriately. Also, we observe improved tunability of frequency of
oscillations from $\sim$15 GHz to $\sim$75 GHz and increment in the power
spectral density by current and tilt angle. In addition, our findings and
insights pave a simple way for nanoscale level microwave generators to be
implemented.

nonlinear dynamics,spintronics,synchronization

##  I Introduction

Spin-torque nano oscillator(STNO) is a nanosized device that consists of two
ferromagnetic layers, one with fixed magnetization and the other with variable
magnetization direction, separated by a thin non-magnetic conductive layer
kiselev:00 ; rippard:04 ; krivorotov:05 ; slavin:06 . When a current is passed
through the STNO the electrons get spin polarized and excert a spin-transfer
torque on the magnetization of the free layer slon:96 ; berger:96 ; slon:02 .
While the current density is above a threshold value, the spin transfer torque
oscillates the magnetization of the free layer continuously and these
oscillations are transformed into microwave voltage oscillations by
magnetoresistive effects sun:00 ; ralph:08 . Also, these STNOs are capable of
generating microwave signals with a broad range of frequencies in gigahertz
scale ralph:08 ; pufall:05 ; silva:08 . STNOs have many attractive features
such as ultra-small dimension slon:96 , wide tunability range bonetti:09 ,
compatibility during the semiconductor fabrication process and very high
modulation rates mudali:10 ; mudali:11 ; mudali:11:1 . Another important
feature of STNOs is the ability to tune their oscillation frequencies with
power enhancement, which is essential for wireless communication devices
Tare:19 . STNOs have shown tremendous growth towards experimental realization
with large operating frequency mohseni:11 . Nevertheless, STNO devices face
issues such as requirment of large static magnetic field and generation of
microwave with low frequency and output power.

A lot of techniques have been implemented by various researchers to achieve
the required power, which include utilizing magnetic tunnel junction in single
STNO deac:08 ; nazarov:08 ; pala  and synchronizing arrays of STNOs to produce
enhanced coherent microwave oscillations grollier:06 ; persson:07 ; li:11 ;
li:10 ; li:11:1 ; subash1 ; subash2 ; turtle ; gopal ; jensen . In addition to
the above, it has been observed that an STNO with perpendicularly polarized
pinned layer may drive an in-plane magnetization of free layer into an out-of-
plane precessional state even in the absence of applied magnetic field
redon:03 ; lee:05 . Additionally, the most widely adopted zero field
operations also include STNOs with perpendicular magnetic anisotrophy in free
and pinned layers rippard:10 ; mohseni:11 , dual spin polarizers lee:11 ,
vortex oscillators pribiag:07 ; pribiag:09 ; finocchio:10 ; locatelli:11 ;
dussaux:11 , current induced Oersted field in STNO bhoomeeswaran:18 , wavy
torque spin-torque oscillators boulle:07 , field-like torque tani:14 ,
achievement of doubling of output signal frequency and power with the help of
dual free STNOs pro:13 , study of synthetic-ferro magnetic spin-torque
oscillator in the absence of external field zhou:13 , and tilted polarizer
STNOs bhoomeeswaran:18 ; zhou1 ; zhou4 ; zhou5 ; he:09 ; he:10 ; wang:11 ;
zha:09 ; zha:09:01 ; zha:09:02 ; zhou:12 ; gang:15 ; cui .

A tilted polarizer STNO(TP-STNO) is a STNO where the magnetization of its
pinned layer can be tilted out of the film plane by an angle designated as
tilt angle. When the pinned layer is tilted, it exhibits two components of
magnetization, in-plane and out-of-plane. The spin current polarized by the
out-of-plane component of the magnetization drives the magnetization of the
free layer into steady state precessional motion in the absence of magnetic
field.

Initially, the TP-STNO has been studied by Zhou $et~{}al.$ and the maximum
operating frequency of TP-STNO up to 29 GHz as a function of drive current has
been reported for Py material zhou1 . Further, the specific magnetization
dynamics of TP-STNO as a function of the device’s drive current and fixed
layer angle as well as its related microwave frequency and efficient
magnetoresistance have been studied zhou4 ; zhou5 . In continuation of the
above studies, an intense interest has also been shown on studying such TP-
STNO based devices with the aid of design and experimental fabrication process
zha:09 ; zha:09:01 ; zha:09:02 . In particular, Zhou and Gang have reported a
full-scale micromagnetic study on the dynamics of TP-STNO zhou:12 ; gang:15 .
Based on the analytical theory and macrospin simulations Zhou $et~{}al.$ have
reported the hysteretic switching between bistable states in TP-STNO zhou:12 .
Gang $et~{}al.$ have shown the dependence of precession frequency on material
parameters of TP-STNO and acheived the frequencies ranging from 1.8 GHz to
41.2 GHz gang:15 . The range of drive current density for which the TP-STNO
exhibits stable oscillations in the presence of thermal noise for different
tilt angles has been identified and reported by Cui $et~{}al.$ cui . The above
mentioned studies show the significant advantage of TP-STNO, functioning
without external magnetic field, in order to maintain or generate microwave
signal with high frequency and power. In the earlier studies on TP-STNO, the
tunability of high frequency in the zero-field operation has been addressed by
the spin torque amplitude with in-plane asymmetric angular dependence zhou1 ;
zhou4 ; zhou5 . Further enhancement of frequency(above 50 GHz) and power in
TP-STNO along with a systematic study on both static and dynamical aspects of
TP-STNO is required to be addressed for the development of nanoscale
structured current driven TP-STNO devices.

Figure 1: (Color online) (a)Illustration of the typical structure of TP-STNO
and (b) Time evolution of $m_{x}$ and $m_{z}$ when $I$ = -2.5 mA and
$\theta_{p}$ = 45∘. (c) Magnification of the evolution of $m_{x}$ and $m_{y}$
in a short range of time $t\in$[15.0 ns, 15.2 ns]

Therefore, in our present study, we report the high frequency magnetization
oscillations with large tunability and power enhancement in a TP-STNO having
inplane magnetized free layer by considering the spin torque amplitude with
both in-plane and out-of-plane asymmetric angular dependence. We have derived
the relation between the current and the magnetization precession frequency of
the TP-STNO under zero applied field. The validity of the numerical simulation
is closely confirmed with the analytical solution. The paper is organized as
follows. Section II addresses a detailed geometry of the TP-STNO and its
governing equation, namely the Landau-Lifshitz-Gilbert-Slonczewski(LLGS)
equation, for the dynamics of magnetization precession. The dynamics of the
free layer magnetization for frequency tunabilty by varying tilt angles
through numerical and analytical studies are presented in Sec III. Finally,
concluding remarks are made in Section IV. In the Appendices, we obtain the
analytic form of steady states of magnetization for different values of spin
current, impact of damping, saturation magnetization and thermal noise on the
frequency of oscillations.

Figure 2: (Color online) (a) Tunability of frequency at different currents
with respect to tilt angle and (b) Tunability of frequency at different tilt
angles with respect to current. Figure 3: (Color online) Trajectory of
magnetization motion at (a) -2.2 mA, (b) -3.0 mA and (c) -5.0 mA when
$\theta_{p}=45^{0}$.

##  II Model

We consider a model structure of TP-STNO which is shown in Fig.1(a). It
consists of a free layer, a fixed layer and a spacer layer between them. The
free and fixed layers are the ferromagnetic layers, where the former is
allowed to change the direction of its magnetization and the later is not.
Spacer layer is a nonmagnetic but conducting layer. The unit vector of the
free layer’s magnetization is given by ${\bf m}=m_{x}{\bf e}_{x}+m_{y}{\bf
e}_{y}+m_{z}{\bf e}_{z}$. ${\bf e}_{x}$,${\bf e}_{y}$ and ${\bf e}_{z}$ are
considerd as the unit vectors along positive $x$,$y$ and $z$ directions
respectively. The plane of the free layer is taken perpendicular to ${\bf
e}_{z}$. The LLGS equation that governs the dynamics of the free layer’s
magnetization is given by

$\displaystyle\frac{d{\bf m}}{dt}=-\gamma~{}{\bf m}\times{\bf
H}_{eff}+\alpha~{}{\bf m}\times\frac{d{\bf m}}{dt}+\gamma~{}H_{S}~{}{\bf
m}\times({\bf m}\times{\bf p}),$ (1)

where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the Gilbert damping
parameter, ${\bf H}_{eff}$ is the effective field and the term $H_{S}$ is the
strength of the Slonczewski spin-transfer torque slon:02  with magnitude

$H_{S}=\frac{\hbar\eta I}{2eM_{s}V(1+\lambda{\bf m}.{\bf p})}.$ (2)

In Eq.(2) $\hbar=h/2\pi$ is the reduced Planck’s constant ($h$ \- Planck’s
constant), $V$ is the volume of the free layer, $M_{s}$ is the saturation
magnetization, $e$ is the electron charge, $I$ is the total current passing
through the free layer, $\eta$ and $\lambda$ are dimensionless parameters
which determine the magnitude of the spin polarization and the angular
dependence of the spin transfer torque, respectively.

The effective field ${\bf H}_{eff}$ can be expressed as ${\bf
H}_{eff}=H_{k}m_{x}{\bf e}_{x}-4\pi M_{s}m_{z}{\bf e}_{z}$ which includes
magnetocrystalline anisotropy field $H_{k}$ and demagnetization field $4\pi
M_{s}$. Further, the magnetization of the pinned layer is considered to be
tilted in the $xz$-plane by the tilt angle $\theta_{p}$ from positive $x$-axis
and the unit vector of the pinned layer’s magnetization is given by ${\bf
p}=~{}\cos\theta_{p}~{}{\bf e}_{x}+\sin\theta_{p}~{}{\bf e}_{z}$.The current
in Eq.(2) is considered as positive or negative when it flows across the STNO
from pinned layer to free layer or free layer to pinned layer respectively.
The material parameters $M_{s}=1448.3$ emu/c.c., $H_{k}=18.6$ kOe, $\eta$ =
0.54, $\lambda$ = $\eta^{2}$, $\gamma$ = 17.64 Mrad/(Oe s), $\alpha$ = 0.005,
and $V=\pi\times 60\times 60\times 2$ nm3 for cobalt material have been
adopted from Ref.tani:18 .

##  III Results and Discussion

The oscillations of ${\bf m}$ around the z-axis due to tilted polarizer is
confirmed in Fig.1(b) by plotting $m_{x}$ and $m_{z}$ with respect to time for
$I$ = -2.5 mA and $\theta_{p}$ = 45∘. The inset figure in Fig.1(b) confirms
the oscillations of $m_{x}$ and $m_{z}$ with respect to time. It has been
numerically verified that the oscillations are not exhibited when the current
is applied along the positive direction. The frequency of oscillations in
terms of $I$ and $\theta_{p}$ is determined from Eq.(1) after transforming it
into spherical polar co-ordinates ${\bf
m}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ as follows:

$\displaystyle\frac{(1+\alpha^{2})}{\gamma}\frac{d\theta}{dt}~{}=~{}H_{k}\sin\theta\cos\phi(\alpha\cos\phi\cos\theta-\sin\phi)$
$\displaystyle+4\pi M_{s}\alpha\sin\theta\cos\theta-
H_{S}\cos\theta_{p}~{}[\cos\phi\cos\theta+\alpha\sin\phi]$
$\displaystyle+H_{S}\sin\theta_{p}~{}\sin\theta,$ (3)
$\displaystyle\frac{(1+\alpha^{2})}{\gamma}\sin\theta\frac{d\phi}{dt}~{}=~{}-H_{k}\sin\theta\cos\phi(\alpha\sin\phi+\cos\theta\cos\phi)$
$\displaystyle-4\pi
M_{s}\cos\theta\sin\theta+H_{S}\cos\theta_{p}~{}[\sin\phi-\alpha\cos\theta\cos\phi]$
$\displaystyle+H_{S}\sin\theta_{p}~{}\alpha\sin\theta.$ (4)

Since the amplitude of $m_{z}$ is small(see Fig.1(b)), $\theta$ can be
approximated as constant and $d\phi/dt=2\pi f=2\pi/T$, where $f$ is the
frequency and $T$ is the time period of the oscillations. By integrating
Eq.(4) with respect to time from 0 to $nT$, where $n$ is the number of
oscillations, we can derive the frequency as,

$\displaystyle
f=~{}\frac{\gamma}{2\pi(1+\alpha^{2})}\left\\{-\left(\frac{H_{k}}{2}+4\pi
M_{s}\right)\cos\theta\right.$ $\displaystyle\left.+\alpha
H_{S}\left[Q~{}{\sin\theta_{p}}-\frac{\cos\theta}{\lambda\sin^{2}\theta}\left(1-Q~{}({1+\lambda\sin\theta_{p}\cos\theta})\right)\right]\right\\},$
(5)

where
$Q=[{1+2\lambda\sin\theta_{p}\cos\theta+\lambda^{2}(\sin^{2}\theta_{p}-\sin^{2}\theta)}]^{-1/2}$.

Equation (5) qualitatively denotes the dependence of frequency $f$ with
respect to tilt angle $\theta_{p}$ and flow of current $I$. Figs.2(a) and (b)
show the variation of frequency $f$ versus $\theta_{p}$ and $I$ respectively.
In Figs.2, the solid line corresponds to numerically computed frequency and
bullets correspond to analytically computed frequency using Eq.(5). From
Figs.2(a) and (b) it is clearly observed that the frequency of the
magnetization precession enhances by increasing the values of $\theta_{p}$ and
$I$ respectively. Here, the initial conditions for ${\bf m}$ are chosen to be
very close to ${\bf e}_{z}$. In addition to this choise of initial condition,
we find that oscillations are also possible when the initial state of the
magnetization is taken upto 45∘ from positive z-direction in the xz-plane as
well as near negative y-direction. For other choises of initial conditions the
magnetization reaches steady state along the positive x-direction without
showing any stable oscillations. Since the frequencies corresponding to
$\theta_{p}$ from 90∘ to 180∘ are symmetrical with the frequencies
corresponding to $\theta_{p}$ from 0∘ to 90∘, they are not shown in Fig.2(a).
Further, Figs.2(a) and (b) show that the current required for the maximum
frequency can be reduced when the pinned layer’s magnetization is tilted out
of the plane. The small discrepancy between the analytical(bullets) and
numerical(lines) frequencies in Figs.2 arises due to the small amplitude
approximation of $m_{z}$. From Figs.2 we can clearly observe that the
frequency can be tuned from $\sim$25 GHz to $\sim$75 GHz by varying the tilt
angle and the magnitude of current.

In order to depict the detailed magnetic precession dynamics of the TP-STNO,
we carried out macromagnetic simulations for different values of current which
is shown in Fig.3. Figures 3(a), (b) and (c) confirm that the magnetization
exhibits steady state motion towards ${\bf e}_{x}$, steady precession about
$z$-axis and steady state towards ${\bf e}_{z}$ at different currents -2.2 mA,
-3.0 mA and -5.0 mA respectively. Here, one may note that the magnetization
switches between oscillatory and steady state depending upon the value of
current. Also it indicates that the magnetization precession is possible only
when the magnitude of the current is between two critical values, denoted as
$I^{c}_{max}$ and $I^{c}_{min}$. Beyond this region of current the
magnetization exhibits steady state. When the magnitude of current is below
$I^{c}_{min}$, the magnetization flips to the $xy$-plane and reaches steady
state near ${\bf e}_{x}$(or $-{\bf e}_{x}$). Similarly, when the magnitude of
current is above $I^{c}_{max}$ the magnetization approaches the steady state
close to ${\bf e}_{z}$. For instance, the values of $I^{c}_{max}$ and
$I^{c}_{min}$ are identified numerically as -3.9 mA and -2.3 mA respectively
when $\theta_{p}$ is 45∘.

To obtain a deeper insight into the stable precessions and steady state, the
current required for the oscillation corresponding to $\theta$ can be
estimated by incorporating the condition
$\frac{1}{nT}\int_{0}^{nT}\frac{d\theta}{dt}~{}dt=0$ on Eq.(3) as,

$\displaystyle
I=\frac{(2eM_{s}V/\hbar\eta)\alpha\cos\theta(\frac{H_{k}}{2}+4\pi
M_{s})}{\frac{\cos\theta}{\lambda\sin^{2}\theta}[1-(1+\lambda\sin\theta_{p}\cos\theta)R]-\sin\theta_{p}~{}R},$
(6)

where,
$R=[(1+\lambda\sin\theta_{p}\cos\theta)^{2}-\lambda^{2}\cos^{2}\theta_{p}\sin^{2}\theta]^{-1/2}$.
From Eq.(6) we can derive the $I_{max}^{c}$ by taking $\theta\rightarrow 0$ as

$\displaystyle
I_{max}^{c}=\frac{-2eM_{s}V\alpha(1+\lambda\sin\theta_{p})(\frac{H_{k}}{2}+4\pi
M_{s})}{\hbar\eta\left[\sin\theta_{p}+\frac{\lambda\cos^{2}\theta_{p}}{2(1+\lambda\sin\theta_{p})}\right]}.$
(7)

Eq.(7) gives the magnitude of current above which TP-STNO exhibits steady
state only.

The possible values of $\theta_{p}$ and $I$ corresponding to the steady state
and oscillatory motions of magnetization are depicted by plotting the
frequency in Fig.4 with respect to $\theta_{p}$ and $I$. Fig.4 confirms that
the magnitude of current required for a given frequency reduces when the tilt
angle is increased. Also, we can understand that the frequency can be enhanced
when the tilt angle(current) is increased while the current(tilt angle) is
fixed. And the frequency of oscillations is enhanced up to $\sim$50 GHz by
tuning the tilt angle and current. The black circles in Fig.4 represent the
values of $I_{max}^{c}$ obtained from Eq.(7).

Figure 4: (Color online) The region of current and tilt angle for which the
oscillations are possible. The white regions are corresponding to steady
state. The black circles are the values of $I_{max}^{c}$ plotted from Eq.(7).

Since the magnetization settles into the xy-plane and aligns closely with
positive or negative $x$ direction when the current is below $I_{min}^{c}$,
the steady state values can be identified from Eqs.(3) and (4) as (see
Appendices A and B)

$\displaystyle\theta^{*}=$ $\displaystyle\frac{\pi}{2}$ (8)
$\displaystyle\phi^{*}=\frac{H_{S}\sin\theta_{p}}{H_{k}(1+\lambda\cos\theta_{p})}~{}~{}$
$\displaystyle(\textrm{or})~{}~{}\pi+\frac{H_{S}\sin\theta_{p}}{H_{k}(1-\lambda\cos\theta_{p})},$
(9)

where, $H_{S}=\hbar\eta I/2eM_{s}V$. Similarly, the magnetization aligns
closely with positive $z$ direction and settles in the fourth quadrant of
$xy$-plane when the magnitude of current is above $I_{max}^{c}$ and the
corresponding steady state points are also obtained from Eqs.(3) and (4) as

$\displaystyle\theta^{*}$ $\displaystyle=\frac{-H_{S}\cos\theta_{p}}{4\pi
M_{s}(1+\lambda\sin\theta_{p})},$ (10) $\displaystyle\phi^{*}$
$\displaystyle=\frac{3\pi}{2}-\frac{H_{S}\sin\theta_{p}}{(H_{k}+4\pi
M_{s})(1+\lambda\sin\theta_{p})}.$ (11)

Eqs.(8) and (9) represent the steady state points when the magnitude of
current is below $I_{c}^{min}$ and the Eqs. (10) and (11) represent the steady
state points when the magnitude of current is above $I_{max}^{c}$.

Finally, the power spectral densities corresponding to the current $-3.0$ mA
and tilt angle 30∘ are plotted in Figs.5(a) and (b) respectively. Fig.5(a)
shows that the peak height increases when the magnetization of the pinned
layer is tilted out-of-the plane, which confirms that the power can be
enhanced by tilted polarizer. The enhancement of peak height at large
frequencies in Fig.5(b) when the magnitude of current is increased implies
that the power can be enhanced at small as well as large frequencies by
tilting the magnetization of the pinned layer.

Also in Appendix C we point out the impact of the damping parameter $\alpha$
and saturation magnetization $M_{s}$ for cobalt material. Further in Appendix
D we point out that thermalization does not lead to any appreciable change in
the frequency of oscillations.

Figure 5: (Color online) The power spectral densities at different (a)
$\theta_{p}$ when $I$ = -3 mA and (b) $I$ when $\theta_{p}$ = 30∘

##  IV Conclusions

In summary, spin-transfer torque induced magnetization dynamics in a TP-STNO
consisting of an in-plane magnetized free layer and a pinned layer with the
tunability of its magnetization direction has been analytically and
numerically studied by solving the governing LLGS equation in the absence of
external magnetic field. We find that the frequency of oscillations can be
tuned from $\sim$15 GHz to $\sim$75 GHz by tilting the magnetization of the
pinned layer into out-of-plane. Also, we have found that the current required
for the maximum frequency can be reduced when the pinned layer’s magnetization
is tilted. Further, we have observed the switching of magnetization between
precessional state and steady state takes place depending upon the value of
current. Therefore, the present study of the TP-STNO yields the combined
advantage of zero field operation and high frequency signal which can give
potential applications for future wireless communications.

## Acknowledgements

The work of V.K.C. forms part of a research project sponsored by CSIR Project
No. 03/ 1444/18/EMR II. M.L. wishes to thank the Department of Science and
Technology for the award of a SERB Distinguished Fellowship under Grant
No.SB/DF/04/2017 in which R. Arun is supported by a Research Associateship.

## Appendix

###  IV.1 Steady states when $I<I_{min}^{c}$

When the current is below $I_{min}^{c}$, the magnetization flips to the
$xy$-plane and may settle near positive or negative $x$ direction depending
upon the initial conditions. The steady state points are identified as
$\theta^{*}=\pi/2$ and $\phi^{*}=\delta\phi~{}\rm{or}~{}\pi+\delta\phi$, where
$\delta\phi\approx 0$. The quantity $\delta\phi$ can be derived by
substituting $\frac{d\phi}{dt}=\frac{d\theta}{dt}=0$, $\cos\phi=\pm 1$ and
$\sin\phi=\pm\delta\phi$ in Eqs.(3) and (4) as

$\displaystyle-
H_{k}\delta\phi\mp\frac{H_{S}\alpha\cos\theta_{p}\delta\phi}{1\pm\lambda\cos\theta_{p}}+\frac{H_{S}\sin\theta_{p}}{1\pm\lambda\cos\theta_{p}}$
$\displaystyle=0,$ (12) $\displaystyle-\alpha
H_{k}\delta\phi\pm\frac{H_{S}\cos\theta_{p}\delta\phi}{1\pm\lambda\cos\theta_{p}}+\frac{H_{S}\alpha\sin\theta_{p}}{1\pm\lambda\cos\theta_{p}}$
$\displaystyle=0.$ (13)

From Eqs.(12) and (13) we can derive,

$\displaystyle\delta\phi=\frac{H_{S}\sin\theta_{p}}{H_{k}(1\pm\lambda\cos\theta_{p})}.$
(14)

Hence,

$\displaystyle\phi^{*}=\frac{H_{S}\sin\theta_{p}}{H_{k}(1+\lambda\cos\theta_{p})}~{}~{}\rm{(or)}~{}~{}\pi+\frac{H_{S}\sin\theta_{p}}{H_{k}(1-\lambda\cos\theta_{p})}$
(15)

###  IV.2 Steady states when $I>I_{max}^{c}$

Similarly, the steady states are identified as $\theta^{*}\approx 0$ and
$\phi^{*}=\frac{3\pi}{2}+\delta\phi^{\prime}$ when the current is above
$I_{max}^{c}$. $\theta^{*}$ can be derived by substituting
$\frac{d\phi}{dt}=\frac{d\theta}{dt}=0$, $\cos\phi=\delta\phi^{\prime}$ and
$\sin\phi=-1$ in Eqs.(3) and (4) and after omitting terms with
${\delta\phi}^{2}$ as

$\displaystyle[(H_{k}\delta\phi^{\prime}+4\pi
M_{s}\alpha)P+H_{S}\sin\theta_{p}]\theta^{*}=H_{S}(\delta\phi^{\prime}-\alpha)\cos\theta_{p},$
(16) $\displaystyle[(\alpha H_{k}\delta\phi^{\prime}-4\pi
M_{s})P+H_{S}\alpha\sin\theta_{p}]\theta^{*}=H_{S}(1+\alpha\delta\phi^{\prime})\cos\theta_{p},$
(17)

where,
$P=1+\lambda\theta^{*}\cos\theta_{p}\delta\phi^{\prime}+\lambda\sin\theta_{p}$.
By solving Eqs.(16) and (17) with the approximation
$\lambda\theta^{*}\cos\theta_{p}\delta\phi\approx 0$, we can get

$\displaystyle\theta^{*}=\frac{-H_{S}\cos\theta_{p}}{4\pi
M_{s}(1+\lambda\sin\theta_{p})}.$ (18)

By using Eq.(18) in Eqs.(16) and (17), we can get

$\displaystyle\delta\phi^{\prime}=-\frac{H_{S}\sin\theta_{p}}{(H_{k}+4\pi
M_{s})(1+\lambda\sin\theta_{p})}.$ (19)

Hence,

$\displaystyle\phi^{*}=\frac{3\pi}{2}-\frac{H_{S}\sin\theta_{p}}{(H_{k}+4\pi
M_{s})(1+\lambda\sin\theta_{p})}.$ (20)

$\theta^{*}=\pi/2$, Eq.(15) and Eq.(18), Eq.(20) represent the steady states
of the magnetization when the current is below $I_{min}^{c}$ and above
$I_{max}^{c}$ respectively.

###  IV.3 Impact of $\alpha$ and $M_{s}$ on the frequency

To investigate the impact of the material parameters on STNO, the frequency of
the oscillations is plotted for different values of saturation
magnetization($M_{s}$) and damping parameter($\alpha$) against current in
Fig.6(a) and (b) respectively. Since the present work deals with cobalt
material for the free layer, Figs.6 are plotted for three different values of
$M_{S}$(1348.3, 1448.3 and 1548.3 emu/cc) and three values of $\alpha$(0.004,
0.005 and 0.006). From Figs.6, it can be understood that for fixed values of
current the frequency increases with a decrease of $\alpha$ and $M_{s}$.
Fig.6(a) confirms that high frequency is acheived at large current when the
saturation magnetization is large. On the other hand, Fig.6(b) implies that
high frequency can be achieved at low currents when the valueof damping
parameter is low. We have also observed corresponding impact with other
parameters such as spin polarization efficiency($\eta$) and anisotropy
field($H_{k}$).

Figure 6: (Color online) Variation of the frequency of STNO’s oscillations
with respect to current for different values of (a) $M_{s}$ when $\alpha$ =
0.005 and (b) $\alpha$ when $M_{s}$ = 1448.3 emu/cc in the absence of thermal
noise.

###  IV.4 Impact of thermal noise

To investigate the impact of thermal noise on the STNO, the dynamics of
magnetization is studied by the effective field with thermal noise as ${\bf
H}_{eff}=H_{k}m_{x}{\bf e}_{x}-4\pi M_{s}m_{z}{\bf e}_{z}+{\bf H}_{th}$, where
the thermal noise ${\bf H}_{th}$ is given by roma ; smirnov ; Hahn

$\displaystyle{\bf H}_{th}=\sqrt{D}~{}{\bf G},~{}~{}~{}D={\frac{2\alpha
k_{B}T}{\gamma M_{s}\mu_{0}V\triangle t}}.$ (21)

Here, ${\bf G}$ is the Gaussian random number generator vector of the STNO
with components $(G_{x},G_{y},G_{z})$, which satisfies the statistical
properties $<G_{m}(t)>=0$ and
$<G_{m}(t)G_{n}(t^{\prime})>=\delta_{mn}\delta(t-t^{\prime})$ for all
$m,n=x,y,z$. $k_{B}$ is the Boltzmann constant, $T$ is the temperature,
$\triangle t$ is the step size of the time scale used in the simulation and
$\mu_{0}$ is the magnetic permeability at free space.

Figure 7: (Color online) (a) Time evolution of (a) $m_{x}$ and $m_{z}$ for
$T=300$ K and $I$ = -2.5 mA. (b) Time evolution of $m_{x}$ in the presence(red
solid line) and absence(blue solid line) of thermal noise. (c) Frequency with
respect to current in the absence(red solid line) and presence(black open
circle) of thermal noise when $\theta_{p}=45^{\circ}$.

Fig.7(a) shows the evolution of $m_{x}$ and $m_{z}$ with respect to time for
$T=300$ K and $I$ = -2.5 mA. The irregularities in the time evolution arises
due to the thermal noise (see Fig.1). Fig.7(b) shows the time evolution of
$m_{x}$ in the presence (solid red line) and absence (solid blue line) of
thermal noise for the same initial conditions, which confirms a small phase
shift due to the thermal noise. From Fig.7(c) it is verified that there is no
appreciable variation in the frequency of oscillations of magnetization due to
thermal noise.

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