# An In-Depth Investigation of
LEO Satellite Topology Design Parameters

Wenyi Zhang  University of California, San Diego ,  Zihan Xu  Carnegie Mellon
University and  Sangeetha Abdu Jyothi  University of California, Irvine &
VMware Research

###### Abstract.

Low Earth Orbit (LEO) satellite networks are rapidly gaining traction today.
Although several real-world deployments exist, our preliminary analysis of LEO
topology performance with the soon-to-be operational Inter-Satellite Links
(ISLs) reveals several interesting characteristics that are difficult to
explain based on our current understanding of topologies. For example, a real-
world satellite shell with a low density of satellites offers better latency
performance than another shell with nearly double the number of satellites. In
this work, we conduct an in-depth investigation of LEO satellite topology
design parameters and their impact on network performance while using the
ISLs. In particular, we focus on three design parameters: the number of orbits
in a shell, the inclination of orbits, and the number of satellites per orbit.
Through an extensive analysis of real-world and synthetic satellite
configurations, we uncover several interesting properties of satellite
topologies. Notably, there exist thresholds for the number of satellites per
orbit and the number of orbits below which the latency performance degrades
significantly. Moreover, network delay between a pair of traffic endpoints
depends on the alignment of the satellite’s orbit (Inclination) with the
geographic locations of endpoints.

††copyright: none

##  1\. Introduction

Low Earth Orbit (LEO) satellite networks are rapidly becoming a significant
component of the backbone Internet infrastructure. They offer low latency and
connectivity to remote areas not served by terrestrial fiber/wireless
infrastructure. Today, several commercial providers are deploying massive LEO
satellite constellations, including Starlink (SpaceX, 2023), Kuiper (Henry,
2019), OneWeb (OneWeb, 2023), and Telesat (Telesat, 2023). Unlike
Geosynchronous Earth Orbit (GEO) and Medium Earth Orbit (MEO) satellite
networks, LEO networks require thousands of satellites to provide good
coverage globally. The most prominent LEO provider today, Starlink, has
already launched over 3,000 satellites and expects to complete its first phase
of 4408 satellites before 2024 (Sesnic, 2023). Eventually, Starlink is slated
to have over 12,000 satellites in their LEO constellation.

A distinguishing feature of LEO satellites, compared with GEO and MEO
satellites, is the presence of Inter-Satellite Links (ISLs). These links allow
LEO satellites to communicate with each other directly at the speed of light
without relying on an intermediate ground station. Each Starlink satellite,
for example, is equipped with four optical space laser generators and four
laser signal receiver surfaces, which can support optical laser communication
between neighboring satellites up to 80 Gbps capacity (SpaceX, 2023; Wang,
2021; Defence and Space, 2018; Handley, 2018; Hecht, 2020; Bhattacherjee et
al., 2020).

ISL-based communication is currently tested only in limited areas, and its
present use is limited to areas where ground stations are missing (Smutny et
al., 2009; Müncheberg et al., 2019; Zafar, 2018). However, it is expected that
a network backbone entirely based on ISLs for end-to-end communication will be
realized in the near future. In this context, the most studied topology is the
+Grid topology (Wood, 2001; Del Portillo et al., 2019; De Weck et al., 2004;
Bhattacherjee and Singla, 2019; Sidibeh, 2008) in which each satellite has
links to the two nearest satellites in its own orbit and to the two nearest
satellites on adjacent orbits on either side. Communication using the +Grid
inter-satellite topology is shown to reduce temporal variations in latency,
offer greater resilience to weather variations typically encountered by ground
stations, and achieve $3\times$ the throughput compared to comparable LEO
network without ISLs (Gavish and Kalvenes, 1997; Hauri et al., 2020;
Bhattacherjee and Singla, 2019; Kassing et al., 2020).

Although there are several ongoing constellations deployments, our
understanding of the impact of various design parameters on the performance of
LEO topologies is limited, particularly in the presence of soon-to-be
operational ISLs. Our preliminary analysis reveals several performance
characteristics that cannot be explained based on the current understanding of
LEO topology design. We analyze the latency characteristics of the five shells
of the Starlink constellation with +Grid topology and observe that a shell
with fewer satellites offers better latency performance than another shell
with nearly double the number of satellites. More specifically, Starlink shell
1 contains 1584 satellites, and shell 2 contains 720 satellites. However, both
the mean and tail latency of shell 2 is significantly lower than shell 1
(detailed in § 3), achieving nearly 13.5% lower latency with fewer than half
the number of satellites of shell 1. While past work has explored the topology
design space in terms of various configurations of ISL links (Bhattacherjee
and Singla, 2019; Zhang et al., 2022; Yang et al., 2022; Chen et al., 2021;
Pan et al., 2019; Werner et al., 2001; Chang et al., 1995), the peculiarities
discovered in our preliminary experiments underscore the need for systematic
investigation of LEO topology characteristics.

In this work, we conduct an in-depth investigation of LEO satellite topology
design parameters to help guide the design of future LEO topologies as well as
routing and traffic engineering schemes. In particular, we analyze the impact
of three design parameters—the number of orbits in a shell, the inclination of
orbits, and the number of satellites per orbit—on network performance. We also
analyze how the relationship between the geographical distribution of traffic
sources and the inclination of satellite orbits affects the performance of the
constellation. While past work has analyzed the performance characteristics of
a limited set of proposed commercial topologies (Kassing et al., 2020;
Perdices et al., 2022; Michel et al., 2022; Kassem et al., 2022), our work
presents the first comprehensive study across topology design parameters in a
broad design space.

We analyze topologies of real-world LEO constellations: Starlink, Kuiper,
Telesat, and a broad set of synthetic configurations. Since ISL usage is
currently limited in real-world constellations, we rely on simulations to
understand the performance of future ISL-enabled LEO backbone networks. We
generate the synthetic topologies by varying the design parameters (number of
orbits per shell, inclination of orbit, and number of satellites per orbit)
over a wide range. We evaluate each configuration under a synthetic traffic
matrix commonly used in prior work, a traffic matrix between the top 100 GDP
city pairs (Perrig et al., 2017; Nordhaus and Chen, 2016) spanning a
significant fraction of Internet users. Our findings reveal several insights
that are expected to hold across a broad range of traffic matrices.

Based on extensive experiments, we identify several important characteristics
of LEO topology design. We highlight two key observations here. First, there
exists a threshold for the number of satellites per orbit below which the
performance degrades significantly. For example, when the number of satellites
per orbit is below 28, there is a significant performance drop even when the
total number of orbits is increased to a very high value of 59. Second, we
show that the latency performance is better when the angle between the traffic
endpoints is closely aligned with the satellite orbit inclination. In this
case, most hops in the path follow intra-orbit links instead of inter-orbit
links, thereby significantly improving the latency performance.

In summary, we make the following contributions:

  * •

We conduct an in-depth investigation of the impact of three previously
unexplored topology design parameters in LEO constellations: the number of
orbits per shell, the number of satellites per orbit, and the inclination of
orbits.

  * •

We investigate three real-world constellations and a broad range of synthetic
constellations to understand the impact of design parameters on several
network performance metrics: RTT, number of path changes, hop count, etc.

  * •

We identify the threshold for various design parameters below which the
constellation performance degrades significantly. Using these observations, we
are able to explain the better network performance of real-world
constellations with fewer satellites.

  * •

We examine the impact of the relationship between traffic distribution and the
inclination of satellite orbits on network performance.

  * •

We will open-source the code and data in this paper.

This work does not raise any ethical concerns.

##  2\. Background & Related Work

This section provides an overview of LEO satellite networks and the recent
work on LEO topology design and LEO satellite network measurement.

LEO Constellation Structure. The Low Earth Orbit (LEO) satellite network
relies on a group of satellites orbiting the Earth at low altitudes, ranging
from 500 to 1500 km, much lower than traditional geostationary satellites at
an altitude of 36,000 km. A large LEO satellite constellation may contain
thousands of satellites that are distributed across a number of orbits, where
each orbit is a circular path around the globe.

Figure 1. Parameters of LEO constellation.

In order to fully describe the LEO mega-constellation, several parameters are
required, as shown in Fig. 1: (a) The orbital altitude of a satellite is its
height above sea level, which, in turn, determines its velocity and coverage
area; (b) The number of orbits in a shell determines the coverage. (c) The
number of satellites per orbit in a constellation determines the density of
the entire constellation. The density of the constellation affects the
availability and redundancy of the network service; (d) The inclination of an
orbital plane is the angle between the Equator and a satellite in that orbit
traveling northward. The inclination of orbits determines the coverage region
of the constellation. A lower inclination provides better coverage in
equatorial regions while a higher inclination provides better coverage in
polar regions; (e) Phase shifts of a satellite constellation determines the
difference in position between one orbital plane and another. Existing
constellations such as Starlink (SpaceX, 2023), Telesat (Telesat, 2023), and
Kuiper (Henry, 2019) apply uniform phase shift deployment, which evenly places
the orbital planes in a full circle.

Inter-Satellite Connectivity. Traditionally, satellite communication relied on
Ground Station-Satellite Links (GSLs) using a bent-pipe space-ground
architecture, where satellites only act as relays that transmit data to the
next ground station (Zhang et al., 2022; Angeletti et al., 2008). However,
this architecture’s performance may be compromised by several factors. The
signal transmission time between the satellite and the ground stations can
result in significant latency. Also, GSLs are vulnerable to interference from
other communication signals, weather conditions, and other sources (Leyva-
Mayorga et al., 2020; Del Portillo et al., 2019; Motzigemba et al., 2019). In
addition, the satellites in LEO constellations are highly dynamic, resulting
in limited time windows to communicate with ground stations.

Today, satellites are equipped with lasers that enable Inter-Satellite Links
(ISLs) to provide a promising improvement over GSLs. ISL-based communication
architecture allows the satellites to communicate directly with each other
and, in turn, reduces the reliance on ground stations. For example, each
satellite in Starlink (SpaceX, 2023) has the hardware capabilities for four
ISL links. The most studied topology is the +Grid topology (Bhattacherjee and
Singla, 2019; Hu et al., 2020; Wang et al., 2006; Mclaughlin et al., 2023),
where each satellite connects to two adjacent satellites in the same orbit and
the two nearest satellites on each of its neighboring orbits. The topology of
the three existing major constellations is shown in Appendix C.

LEO Constellation Performance Analysis. Several past works measure the
performance of existing satellite constellations. A recent study (Perdices et
al., 2022) investigates the performance of GEO satellite networks,
highlighting the importance of ground station selection. Michel et al. (Michel
et al., 2022) compare the performance of Starlink with GEO networks and show
that the LEO constellation can achieve higher throughput and lower data
transmission latency. Kassem et al. (Kassem et al., 2022) examine the
connectivity of Starlink, suggesting that the current bent-pipe connection
results in high performance variability during bad weather conditions.

Due to limited practical deployments, several works resorted to simulation-
based performance analysis of LEO networks. Kassing et al. (Kassing et al.,
2020) developed a satellite network simulator and compared three prominent LEO
satellite constellations (Starlink, Kuiper, and Telesat). Izhikevich et al.
(Izhikevich et al., 2023) introduced LEO HitchHiking to democratize LEO
satellite measurements. Bhosale et al. (Bhosale et al., 2023a) developed a
model that captures the RTT variability exhibited in LEO networks.

Some recent works analyze the impact of satellite constellation parameter
settings on performance, albeit under limited scope. Bhosale et al. (Bhosale
et al., 2023b) characterized route variability in LEO satellite networks,
focusing on route churn and RTT variability. Basak et al. (Basak et al., 2023)
focused on designing constellations under a specified budget for target
markets. All prior work on simulation-based LEO constellation analysis
(Peluso, 2022; Bhosale et al., 2023b; Basak et al., 2023) explore the
performance of constellations over a limited range of parameter settings and
provide a high-level overview of a small subset of satellite network
characteristics. Moreover, certain performance characteristics of the
constellation are inexplicable based on our current understanding. Therefore,
a comprehensive investigation of the impact of parameter settings on
constellation performance is currently needed. This work aims to fill this
gap.

##  3\. Motivation

Figure 2. Distribution of maximum RTT (ms) across top 100 city pairs using
Starlink shells, S1 and S2.

To underscore the need for design parameter analysis, we compare the latency
performance of two shells of Starlink (shell 1 and shell 2). Shell 1 consists
of 72 orbits, with 22 satellites per orbit, and an inclination of 53°, while
shell 2 consists of 20 orbits, with 36 satellites per orbit and an inclination
of 70°. We use these shell configurations with +Grid topology and traffic
between the top 100 city pairs for evaluation (detailed experiment setup in §
4). In Fig. 2, we plot the ECDF of maximum RTT across all city pairs when
using each of the shells independently. We observe that shell 2 offers a lower
mean and tail latency. This is counter-intuitive since shell 1, with nearly
double the number of satellites (1584 in S1 vs. 720 in S2), offers worse
latency performance.

Shell 1 and shell 2 differ across all design parameters: the number of
satellites per orbit (22 in S1 vs. 36 in S2), the number of orbits (72 in S1
vs. 20 in S2), and inclination (53° in S1 vs. 70° in S2). Based on our current
understanding, it is difficult to pinpoint the factor that contributed to the
poor performance of shell 1. These observations inspire us to evaluate each of
the design parameters in depth.

##  4\. Experiment Design

In this section, we give an overview of the evaluation metrics and experiment
configurations that we use for understanding the impact of various satellite
topology design parameters.

###  4.1. Evaluation Metrics

The major factors that affect the user experience of data transmission over a
satellite network are latency and network stability. In order to quantify
these factors, we organize the evaluation metrics into three categories: (1)
Round-trip time (RTT), (2) Path changes, and (3) Hop count.

Round-Trip Time (RTT). The round-trip time denotes the propagation latency,
i.e., the time taken for a packet of data to travel from a sender to a
receiver and back to the sender. The RTT depends on the total distance between
the sender and receiver, which includes 1) the line-of-sight distance from the
sender to the first-hop satellite, 2) the sum of distances between satellites
along the routing path, and 3) the distance from the last-hop satellite to the
receiver. The RTT is computed as the total distance divided by the speed of
light in vacuum (3*$10^{8}$ m/s). For satellite network evaluation, we
consider the following characteristics of RTT:
(a) Max. (Min.) RTT, which denotes the maximum (minimum) RTT observed between
the sender and the receiver during a specified period. RTT variations occur in
satellite networks due to the changes in topology caused by satellite motion.
(b) Max. RTT - Min. RTT, is the difference between the maximum RTT and minimum
RTT observed between the sender and the receiver during a specified period.
This metric reflects the fluctuations in RTT experienced by a connection.

Since RTT measurements are dependent on the distance between the sender and
the receiver, we define two additional metrics that facilitate comparison of
performance across different pairs of endpoints.

(c) Max. to Min. Slowdown captures the extent of variation in RTT observed
between a pair of endpoints. It is defined as
${S}_{m}=\frac{max.RTT}{min.RTT}$, where $max.$ $(min.)$ $RTT$ is the maximum
(minimum) RTT during a period of time.
(d) Geodesic Slowdown is the slowdown with respect to the best possible path.
It is defined as ${S}_{g}=\frac{max.RTT}{ideal.RTT}$, where $ideal.RTT$ is the
RTT obtained when directly transmitting data between two endpoints through the
shortest Euclidean distance on the earth’s surface at the speed of light.

Path Changes. The dynamic nature of LEO networks with constantly moving
satellites results in changes in the routing paths over time. Frequent changes
in these paths can lead to jitter in data transmission, high computational
overhead for updating routing tables, and loss of connectivity.
(e) The number of path changes between two endpoints over a period of time
denotes the extent of network stability.

Hop count. While using ISLs, data is transmitted between satellites over a
series of hops. Each hop will result in additional transmission delay as well
as energy consumption. If the number of hops increases significantly, it may
indicate high latency in transmitting data. Hence, we measure the following
characteristics of hops between two endpoints:
(f) Average hop count denotes the average number of hops between a pair of
endpoints during a specified period of time.
(g) Max. hop count - Min. hop count is the difference in the number of hops
across paths during a specified period.
(h) Max. hop count / Min. hop count denotes the relative change in the number
of hops between a pair of endpoints during a specific period of time.

###  4.2. Experiment Configurations

The end-to-end network performance over LEO constellations can vary widely
based on a variety of configuration parameters. Under a given budget of the
number of satellites, it is unclear what set of configurations would lead to
optimal performance. Therefore, it is essential to understand the impact of
each configuration parameter on network performance. What are the most
important parameters affecting LEO mega-constellation network performance? In
what ways and along what dimensions do they influence the performance of
satellite networks? How do we design an optimal LEO mega-constellation? In
order to scientifically and rigorously answer the above questions, we begin
with the existing commercial LEO mega-constellations and expand to an in-depth
analysis of synthetic LEO mega-constellations generated across a wide range of
parameter settings. We design three key experiments to explore the impact of
LEO mega-constellation parameter settings.

(i) Exploring different shells of the existing LEO mega-constellations. We
first simulate three leading existing LEO mega-constellations and analyze the
network performance of their shells: five shells for Starlink Phase I, three
shells for Kuiper, and two shells for Telesat. Detailed parameter settings are
given in § 5.1.1. We focus on four of the most representative metrics—Max.
RTT, Geodesic Slowdown, (Max. RTT - Min. RTT), Number (#) of Path Changes—to
evaluate these shells, detailed experimental results are in § 5.3.1.

(ii) Exploring different synthetic configurations of LEO mega-constellations.
In order to explore the impact of parameters on network performance, we
analyze different settings for three parameters—Orbit Number, Satellite Number
per Orbit, and Inclination. We vary each parameter independently while fixing
others and generate 20 synthetic LEO mega-constellations (the settings are
shown in § 5.1.2). We select the three most representative metrics: Max. RTT,
Geodesic Slowdown, and Number (#) of Path Changes for evaluation, (detailed
results in § 5.3.2).

(iii) Understanding the Impact of User Endpoints Location. Visualization of
LEO paths reveals varying levels of zig-zag in paths between various user
endpoint pairs (Appendix G). We conjecture that the degree of alignment
between the Inclination of the satellite orbit and the Geographic Angle
(wikipedia, [n. d.]) of the plane between source and destination endpoint
locations could influence the latency performance. To verify this conjecture,
we divide the traffic between user endpoints into nine categories based on the
Geographic Angle of the plane connecting them (detailed in Appendix F) and
evaluate the network performance of these nine groups under different
Inclination values of synthetic constellations. We choose the two most
representative indicators, average hops and Geodesic Slowdown, for analysis.
Max. RTT is not a suitable metric in this analysis since endpoint pairs in the
nine groups have widely different distance distributions.

##  5\. Evaluation

In this section, we provide an overview of the datasets used in our analysis,
the implementation of our analysis framework, and our key observations while
answering the following questions:

  * •

What are the key parameters that influence the performance of an LEO network?

  * •

Are there thresholds for each parameter above/below which the performance
degrades significantly?

  * •

Why does a shell with a low number of satellites outperform another shell with
nearly double the number of satellites (S1 vs. S2 of Starlink)?

  * •

How does the difference between satellite orbit inclination and the
geographical angle of the plane containing the source and destination location
affect performance?

###  5.1. Datasets

Name | H (km) | Orb. | Sats/Orb | Total Sats | Incl.
---|---|---|---|---|---
S1 | 550 | 72 | 22 | 1584 | 53.0°
S2 | 570 | 20 | 36 | 720 | 70.0°
S3 | 560 | 6 | 58 | 348 | 97.6°
S4 | 540 | 72 | 22 | 1584 | 53.2°
S5 | 560 | 4 | 43 | 172 | 97.6°
K1 | 630 | 34 | 34 | 1156 | 51.9°
K2 | 610 | 36 | 36 | 1296 | 42°
K3 | 590 | 28 | 28 | 784 | 33°
T1 | 1015 | 27 | 13 | 351 | 98.98°
T2 | 1325 | 40 | 33 | 1320 | 50.88°
Table 1. Shell configurations for Starlink’s first phase (S1-S5), Kuiper
(K1-K3), and Telesat (T1-T2).

####  5.1.1. Commercial LEO Mega-constellations

In Table 1, we detail the configurations of the three leading commercial LEO
mega-constellations used in our evaluation: Starlink (SpaceX, 2023), Kuiper
(Henry, 2019), and Telesat (Telesat, 2023).

####  5.1.2. Synthetic LEO Mega-constellations

We generate multiple synthetic constellations to understand the impact of
parameters better. We fix each parameter to a constant value and vary others.
First, we fix the inclination angle to 53°, covering more than 90% of the
cities on the earth and almost all the Internet user population, and
synthesize satellite shells with different number of orbits (20, 33, 46, and
59) and number of satellites per orbit (20, 28, 36, and 44). To understand the
impact of inclination, we fix the number of orbits to 33 and the number of
satellites per orbit to 28 and vary the inclination across four values (45°,
55°, 65°, and 75°). Thus, we have a total of 20 synthetic shells. We use a
fixed orbital altitude of 570km for all synthetic shells. Note that we analyze
the design parameter, the altitude of the satellite orbit, and find that
variations in altitude had minimal impact on network performance in the
altitude range of LEO satellites (Appendix B).

####  5.1.3. Traffic Matrix

To evaluate the performance of the LEO satellite-based networks, based on the
known correlation between Internet penetration and GDP (Amiri and Reif, 2013),
we choose the Top 100 Gross Domestic Product (GDP) cities as a representative
to generate the traffic matrices for Internet use requests, following the
methodology of past work on satellite network performance (Bhattacherjee and
Singla, 2019). The top 100 cities with the highest GDPs encompass a
significant portion of the global Internet population (Amiri and Reif, 2013).
We collect the latitude and longitude of these Top 100 GDP cities and generate
traffic matrices with one city as the source and another as the destination,
with a total of 9,900 city pairs. We maintain constant traffic between the
9900 pairs of cities during the period of evaluation. For each time step in
the simulation, we take into account the communication load between all pairs
of cities and estimate their instantaneous network performance. (detailed in
Appendix F.2). Note that several observations in this paper hold independent
of the traffic matrix.

###  5.2. Implementation

####  5.2.1. LEO Mega-constellations Generation

Using an open-source LEO satellite network analysis platform, Hypatia (Kassing
et al., 2020), we generate the dynamic states of the LEO constellations under
various configurations. Each experiment is conducted for a duration of 400
seconds. We snapshot the link pattern of LEO mega-constellations in 1-second
increments and evaluate the network performance characteristics at each time
step. Note that the topology patterns of constellations exhibit cyclical
behavior with an approximate time period of 200 seconds. Hence, an evaluation
period of 400 seconds is sufficient to cover the complete span of topology
configurations and comprehensively reflect the performance patterns.

We use the NetworkX (Hagberg et al., 2008) library to generate a network graph
of the LEO satellite-based network for each time interval, considering the
satellite positions and link lengths between satellites and ground stations.
We calculate the forwarding state for each node based on a range of routing
strategies. Our implementation employs shortest-path routing, computed using
the Floyd-Warshall algorithm (Hougardy, 2010). The latest forwarding state is
read into static routing table entries during each change event in the
network. Based on the forwarding state in each snapshot, we estimate the
various evaluation metrics using the city pairs-based traffic matrix. In order
to simulate the dynamic states and compute the evaluation metrics, we rely on
three files: the TLE file, the ISL file, and the TM file.

TLE File:  The TLE format is a two-line element (TLE) standard format for
representing the trajectory of an Earth-orbiting object (Kauderer and
Dismukes, 2011). It is essential for satellite mobility modeling. Using the
configuration parameters of real-world and synthetic constellations, we
calculate the movement of satellites in space and save their positions in the
TLE file.

ISLs File:  We connect each satellite using the +Grid topology and build the
network for the LEO mega-constellations. We store these edges between
satellites in an ISLs file, which contains the network graph.

TM File:  To balance the coverage of users and computational overhead, we
selected 100 cities with the highest Gross Domestic Product (GDP) as traffic
endpoints. We consider all pairs of cities in this set and store the traffic
endpoints in the Traffic Matrix file (TM file). This file contains the
latitude, longitude, and city ID of all pairs of traffic endpoints.

####  5.2.2. Geographic Angle Between Traffic Endpoints Implementation

To investigate the impact of the alignment of the plane containing the source
and destination of traffic with the satellite orbit inclination on network
performance, we group the traffic endpoint pairs into nine categories based on
their geographic angle (intervals of 10° ranging from 0°-10° to 80°-90°). The
geographic angle is the angle between the plane connecting the endpoint pair
and the equator (Appendix F.1).

###  5.3. Results

####  5.3.1. Real-World LEO Constellations’ Performance

Max. RTT:  We evaluate the maximum RTT observed between city pairs over a
period of 400s. In Fig. LABEL:starlink_max_rtt, we observe that different
shells of Starlink have significant variations in maximum RTT. This behavior
can be explained based on the shell configurations. Shells 1,2 and 4 have a
large number of satellites and offer good coverage. However, shells 3 and 5
are sparse with many endpoints away from the poles unreachable during
significant durations, causing large fluctuations in Max. RTT. In Fig.
LABEL:kuiper_&_telesat_max_rtt, we observe that both Kuiper and Telesat with
shells that have large coverage also have an even distribution of Max. RTT.

Geodesic Slowdown:  We investigate the geodesic slowdown of the three
constellations and find that over 80% of the endpoint pairs have a maximum RTT
less than 2.5$\times$ geodesic RTT, showing good latency reduction compared to
today’s terrestrial Internet networks (terrestrial fiber paths are often long-
winded, and the speed of light in fiber is roughly two-thirds of the speed of
light in air leading to poor latency performance on land (Bozkurt et al.,
2017)). However, some shells have a long tail for the geodesic slowdown. In
Fig. LABEL:starlink_geodesic_slowdown, shells 2, 3, and 5 have longer tails
than shells 1 and 4. Fig. LABEL:kuiper_&_telesat_geodesic_slowdown also shows
that when the total number of satellites decreases, the slowdown tends to have
a longer tail. Prior work (Kassing et al., 2020) indicated that a drastic
increase in the ratio might be caused by short-distance data transmission—the
latency overhead of ground station-satellite link resulting in significant
inflation of geodesic slowdown. Besides the overhead, the sparsity of the
satellite constellation also causes an increase in geodesic slowdown. Routing
across a limited number of satellites results in a longer path than a fully
covered satellite shell.

Max. RTT - Min. RTT:  To examine the latency variation, we further explore the
difference between maximum and minimum RTT over time. Fig.
LABEL:starlink_max_minus_min_rtt shows that for Starlink, over 90% of the
endpoint pairs have latency variation less than 25ms. Fig.
LABEL:kuiper_&_telesat_max_minus_min_rtt shows that for Kuiper, over 90% of
the endpoint pairs have latency variation less than 20ms. We observe that
shell 2 of Telesat has the lowest latency variation, with almost 100% of the
endpoint pairs having less than 10ms variation. Similar to the geodesic
slowdown, the total number of satellites affects the tail of the latency
variation.

Number of path changes:  We investigate the number of path changes to
understand the stability of LEO mega-constellations. Fig.
LABEL:kuiper_&_telesat_path_changes shows that with fewer satellites, the
shell has less number of path changes. However, in Fig.
LABEL:starlink_path_changes, we observe that with the smallest number of
satellites, shells 3 and 5 have the greatest number of path changes. When the
coverage of a satellite constellation is above a threshold, the number of path
changes increases as the satellite density increases due to frequent shifts in
the best available shortest path in a dense and dynamic network. However,
below this threshold, the number of path changes is high due to poor coverage
and loss of paths (detailed results in Appendix D).

Real-world constellation analysis key takeaways:

  * •

Very sparse shells can lead to a high variance in RTT.

  * •

Path stability is affected by the number of satellites in the shell. The
stability offered by a shell depends on its coverage. For shells with good
coverage, the number of path changes increases as shell density increases.

  * •

Typically, shells with a larger number of satellites and higher coverage offer
a lower RTT, lower geodesic slowdown, and lower variance in RTT. However,
there are some caveats. When the constellation parameters are in a certain
range, the performance may be poor (for example, shell 1 vs. shell 2 in
Starlink). This calls for a more thorough investigation of design parameters.

####  5.3.2. In-Depth Analysis of Synthetic Configurations

Number of Satellites Per Orbit:  At an inclination angle of 53°, we measure
the network performance of synthetic LEO constellations by varying the number
of satellites per orbit across (20, 28, 36, 44) and the number of orbits
across (20, 33, 46, 59). The measurement results are shown in Fig.
LABEL:RTT_Sats/Orbits, Fig. LABEL:Slowdown_Sats/Orbits, and Fig.
LABEL:Path_Sats/Orbits. Note that the tails are very long in certain cases,
and the ECDF is cropped in some plots for better visualization. More detailed
experimental results are in Appendix E.1.

In Fig. LABEL:RTT_Sats/Orbits and Fig. LABEL:Slowdown_Sats/Orbits, we observe
that the distribution of Max. RTT and Geodesic Slowdown demonstrate similar
behavior irrespective of the number of orbits. As the number of satellites per
orbit (Sats/Orbit) increases, Max. RTT and Geodesic Slowdown decrease for a
given orbit number. We also find that when the number of satellites per orbit
is 20, the Max. RTT and Geodesic Slowdown are much higher than others (e.g.,
$2\times$ higher than 28 Sats/Orbit in Fig. LABEL:RTT_Sats/Orbits_1 and Fig.
LABEL:Slowdown_Sats/Orbits_1). This effect is especially pronounced when the
Orbit number is less than 33. The long tail effect is also more pronounced
when Sats/Orbit = 20 (tail at 250ms+ vs. 150ms when Sats/Orbit = 28). When
Sats/Orbit increases beyond 28, the performance gains are marginal. For
example, a 33% increase in satellite density from 36 to 48 results in a less
than 2% reduction in Max. RTT and less than 1% reduction in Geodesic slowdown.

Note that while the differences between performance curves may be minimal in
some experiments, we are interested in trends in performance changes as we
vary each parameter. Hence, we highlight the performance trends by comparing
plots across multiple parameter settings in our analysis.

The variations in path change reflect a different behavior. When the number of
orbits increases, the path stability of constellations with a lower number of
satellites per orbit improves. For example, in Fig. LABEL:Path_Sats/Orbits_1,
Sats/Orbits = 20 leads to the highest number of path changes (across over 80%
of endpoint pairs); however, in Fig. LABEL:Path_Sats/Orbits_4, Sats/Orbits =
20 leads to the least number of path changes (across over 90% of endpoint
pairs). Also, in Fig. LABEL:Path_Sats/Orbits_2, for a fixed number of Orbit =
33, the median number of path changes increases from 6 to 10 as the number of
satellites per orbit increases.

These experiments reveal a strong correlation between the number of satellites
per orbit (Sats/Orbit) and the network performance of the constellations. The
Sats/Orbit will determine the distance between satellites in each orbit, with
a lower distance between satellites as Sats/Orbit increases. With the gridded
satellite network, a higher value of Sats/Orbit can lead to the routing path
in the +Grid LEO constellation being more closer to the optimal geodesic
transmission path. However, with the high level of dynamism under a high
density of satellites, the optimal shortest paths will change at a faster
rate. Therefore, the trade-off between latency and path stability needs to be
considered during constellation design. 20 Sats/Orbit offers poor network
performance, and hence, 28 or more Sats/Orbit are needed to achieve the
combination of relatively low delay and a lower number of path changes.

Number of Orbits:  With a constant inclination of 53° and constant satellites
per orbit in the set (20, 28, 36, 44), we evaluate the variation in
performance by varying the number of orbits across 20, 33, 46, and 59. More
detailed experimental results are in Appendix E.3. In Fig. LABEL:RTT_Orbits_1,
we see that when the Sats/Orbit = 20 (i.e., a large distance between
satellites), the performance improves significantly as the number of orbits is
increased from 20 to 59. Compared to 20 Orbits, a 65%, 120%, and 195% increase
in orbits densities to 33, 44, and 59 Orbits leads to a 20%, 50%, and 50%
decrease in the median value of Max. RTT. Similarly, significant improvement
is observed in Geodesic Slowdown with Sats/Orbit = 20 in Fig.
LABEL:Slowdown_Orbits_1. Moreover, from Fig. LABEL:RTT_Orbits and Fig.
LABEL:Slowdown_Orbits, we observe marginal benefits when Sats/Orbit density
increases beyond a threshold. Beyond 28 Sats/Orbit, an increase in the number
of orbits has a minimal effect on the maximum RTT, reducing the maximum RTT
from 175ms to 150ms.

In Fig. LABEL:Slowdown_Orbits, we see that when Sats/Orbit is greater than 28,
we achieve minimal improvement by increasing the number of orbits.
Interestingly, most endpoint pairs have a Geodesic Slowdown in the range
1.4-2.5 $\times$. With the number of path changes, we observe similar behavior
as previously noted, i.e., when the number of satellites per orbit increases,
the path stability is higher with a lower number of orbits. For Sats/Orbits =
28, the median number of path changes increases from 6 to 9 when the number of
orbits increases.

The above results align with our predictions for the LEO mega-constellation
network. At a given number of satellites per orbit, as the number of orbits
increases, the density of the satellite network also increases. This causes
the inter-satellite routing route to be closer to the optimal geographical
distance but leads to a higher number of shortest path changes. Also shown in
Fig. LABEL:RTT_Orbits and Fig. LABEL:Path_Orbits, when the number of
satellites per orbit is low, i.e., the satellite network has a low density, 20
Orbits cannot offer the ideal satellite network performance, and at least 33
Orbits are required to achieve relatively low delay and the number of path
changes.

Inclination:  With a fixed number of Sats/Orbits = 28 and a number of orbits
of 33, we measure the variations in network performance with varying
Inclinations (45°, 55°, 65°, and 75°) and show the results in Fig.
LABEL:Inclination. More detailed experimental results are in Appendix E.2. In
Fig. LABEL:Inclination_1, the variations in maximum RTT across inclinations
are minimal, although Inclination = 75° has the largest Max. RTT. Moreover,
the difference between maximum and minimum RTT, shown in Fig.
LABEL:Inclination_3, is below 16 ms for nearly 90% of endpoint pairs; however,
at 45°, the tail can be as high as 60 ms. With Geodesic Slowdown in Fig.
LABEL:Inclination_2, the lower the inclination angle, the larger the fraction
of endpoint pairs with a slowdown factor below 1.8$\times$. In addition, as
shown in Fig. LABEL:Inclination_4, an inclination of 55° has the lowest median
value of the number of path changes. The median value for inclinations 75° and
65° are similar, while the number of path changes at an inclination of 45° is
the largest, with a median of 9. Overall, it is difficult to identify clear
trends in network performance associated with variations in inclination. We
conclude that the influence of Inclination on network performance is multi-
dimensional and defer a detailed study at the intersection of patterns in the
traffic matrix and inclination to § 5.3.3.

Synthetic constellation analysis key takeaways:

  * •

The number of satellites per orbit is a key determinant of network
performance. As the number of satellites per orbit increases, the latency and
geodesic slowdown in the network decreases (Fig. LABEL:RTT_Sats/Orbits &
LABEL:Slowdown_Sats/Orbits).

  * •

We identify a threshold for the number of satellites per orbit, 28, below
which the network performance degrades significantly (Fig.
LABEL:RTT_Sats/Orbits & LABEL:Slowdown_Sats/Orbits).

  * •

Increasing the number of orbits is beneficial for performance only when the
number of satellites per orbit is very low, i.e., below the identified
threshold of 28 (Fig. LABEL:RTT_Orbits & LABEL:Slowdown_Orbits).

  * •

Inclination has a complex relationship with network performance and needs to
be evaluated in conjunction with the traffic matrix for a better understanding
(Fig. LABEL:Inclination).

Why does Starlink Shell 2 offer better performance compared to Shell 1? We
circle back to the observation that led to our analysis: S2 offers better
latency performance compared to S1 in spite of having only nearly half the
number of satellites. Based on our synthetic constellation analysis, we
identify that the poor performance of S1 is because its number of satellites
per orbit (20) is lower than the minimum threshold required for good
performance (28). When the number of satellites is below this threshold, it is
difficult to compensate for the drop in performance, even with a much larger
number of satellites. S2, on the other hand, has all its key design parameters
above the performance threshold. Hence, S2 has better performance than S1.

####  5.3.3. Impact of Traffic Matrix

Traffic Distribution:  We use the geographic angle of endpoint pairs to
investigate the impact of traffic matrices (user distribution) on performance.
The geographic angle is defined as the angle between the plane connecting the
endpoint pair and the equator. We divide endpoint pairs into nine groups
according to their geographic angle. For example, Fig. LABEL:gs_pairs_example
shows endpoint pairs with geographic angles in the range 40-50°(detailed
information in Appendix F.2). Fig. LABEL:gs_pairs_distribution shows the
distribution of geographic angles of all city pairs. The majority of endpoint
pairs are distributed within the range 20-90°, with the highest number of
pairs in the range 30-40°.

Geodesic Slowdown across endpoint pairs at different Inclinations: Fig.
LABEL:inclination_geodesic_slowdown shows the geodesic slowdown of four groups
of endpoint pairs from 40° to 80° in intervals of 10° under four values of
orbit inclination (45°, 55°, 65°, and 75°). The performance of endpoint pairs
varies based on the geographic angle. The plots for all nine bins (0°-90°) are
given in Fig. LABEL:appendix:inclination_geodesic_slowdown (Appendix F.3). We
observe that the latency performance is the best when the geographic angle of
the endpoint pairs is aligned with the orbital Inclination. The shell with a
45° inclination offers the lowest slowdown for endpoint pairs with a
geographic angle in the range 0-50°. The remaining shells also perform the
best across endpoint pairs which have a geographic angle closest to the
orbit’s Inclination. We surmise that when the orbital inclination aligns with
the geographic angle, the shortest path between endpoints follows more intra-
orbital hops within the same orbit, leading to a better latency performance.
We observe that this pattern holds across all ranges of geographic angles in
Fig. LABEL:appendix:inclination_all_pairs.

Figure 6. Evaluation of Average Hop Count of nine groups of endpoint pairs
under varying Inclination.

Average Hop Count for different endpoint pairs under varying Inclination: We
evaluate the average hop count for all nine groups of endpoint pairs under
different inclinations and present the results in Fig. 6. Across all endpoint
pairs, an inclination of 55° offers the best performance. Most commercial
constellations today use an orbital angle very close to this value. Another
interesting observation is that the inclination that offers the smallest
average hop count is the one closest to the geographic angle of endpoint
pairs. For example, for endpoint pairs with a geographic angle in the range of
40-50°, the constellation with an inclination of 45° offers the least average
hop count. Similarly, 50-60° endpoint pairs fare best under an inclination of
55° and so on. When endpoint pairs have a geographic angle greater than 70°,
the average hop count gradually decreases with an increase in inclination from
45° to 75°. Similarly, for endpoint pairs with a geographic angle of less than
40°, the average hop count gradually decreases as inclination decreases from
75° to 45°.

Figure 7. Heatmap of ”Distance” vs. ”Number of Average Hops” for 50-60°
Endpoints Pairs under different LEO mega-constellation inclinations (a) 45°,
(b) 55° (c) 65° and (d) 75°. The 50-60° city pairs are further divided into
bins based on their geodesic distance (along the x-axis). Comparing the same
bin across (a) - (d) helps us understand the change in the number of average
hops for the same city pairs under different inclinations. For example,
50°-60° city pairs have a lower average hop count in (b) when the inclination
angle of the orbit is more closely aligned with their geographic angle.

We present a more fine-grained analysis of the variation of average hop count
across endpoint pairs based on both geographic angle and geographic
distance.(Full experimental results in Appendix F.4) We choose four
representative values of 50-60° endpoints pairs under different orbital
inclination, 45°, 55°, 65° and 75°, in Fig. 7, respectively. Note that across
both figures, the set of endpoint pairs that belong to each bin along the
x-axis remains the same (since the distance between endpoint pairs is fixed).
However, their average hop count (y-axis) can vary based on the inclination of
the orbit.

By comparing the same column in Fig. 7(a)-(d), we observe that the number of
average hops changes with the inclination. For example, consider endpoint
pairs within the distance interval of 10000-12500km and a geographic angle of
50-60° (i.e., the sixth column in each subplot of Fig. 7). Under the
inclination of 45° (Fig. 7(a)), more than 120 endpoint pairs have an average
hop count in the range 10-11, with only a tiny fraction with an average hop
count of 9-10 or 11-12. However, under the inclination of 55° (Fig. 7(b)),
most of the endpoint pairs have a smaller average hop count than in Fig. 7(a).
Less than 40 endpoint pairs have an average hop count in the range of 10-11
and the remaining have an average hop count in the range of 8-9, thereby
decreasing the overall average hop count. In contrast, for the sixth column’s
color block, as we increase the orbital angle (Fig. 7(b) vs. Fig. 7(c) and
Fig. 7(d)), the average hop count increases. This is because the orbital angle
of 55° (Fig. 7(b)) is closer to the geographic angle for 50-60° endpoint
pairs. The same pattern holds for all other groups of endpoint pairs and their
more aligned inclinations (Appendix F.4).

From these two experiments, we confirm that alignment between the inclination
or orbit and the geographic angle of endpoint pairs is positively correlated
with a reduced average hop count and improved performance. When an orbit is
more aligned with the geographic angle of endpoint pairs, the satellite path
uses more intra-orbit hops and reduces inter-orbit zig-zag hops, which in turn
reduces the hop count. Therefore, if endpoint traffic distributions can be
known a priori, designing orbital angles that align closely with endpoint
traffic distributions can improve performance.

Traffic Matrix Analysis Key Takeaways:

  * •

The network delay is closest to optimal when the geographic angle of the
traffic endpoints aligns closely with the orbital angle of the constellation.

  * •

Average hop count is lower when the geographic angle of the traffic endpoints
aligns closely with the orbital angle of the constellation.

###  5.4. Summary of Results

  * •

The design of most commercial constellations (except S3 and S5) is well-suited
to support communication between the top 100 cities. Considering both delay
and link stability, S2, K2, and T2 offer the best performance in each of the
three constellations, respectively.

  * •

The two design parameters—the number of satellites per orbit and the number of
orbits—strongly influence network performance. For high performance, the
number of satellites per orbit should be above a threshold, which we
empirically estimate as 28. Similarly, the performance is high when the number
of orbits is at least 33.

  * •

Network performance is not strongly correlated with the total number of
satellites. S2 offers better latency performance than S1 in spite of much
fewer satellites because both its key design parameters are above the
performance threshold. The poor performance of S1 is due to its low number of
satellites per orbit (20 vs. threshold of 28).

  * •

The inclination of the LEO mega-constellation is important. The alignment
between the inclination and the geographic angle of endpoint pairs is
positively correlated with network performance. When the endpoint pairs are
more aligned with the orbital angle, they experience lower delay and lower
average hop count.

##  6\. Discussion

In this section, we provide suggestions for the design of LEO mega-
constellations and identify avenues for future work.

Suggestions for the design of LEO mega-constellations: We identify the
thresholds for optimal performance as 33 for the number of orbits and 28 for
the number of satellites per orbit. Moreover, network delay is minimized when
the geographic angle between endpoint pairs is aligned with the orbital angle.
Based on the known user distribution on Earth, the range between 45° to 55° is
the ideal orbital inclination.

Impact of topology variations:  The recommendations of this study are based on
the assumption of LEO mega-constellations relying on ISL links and the +Grid
topology. Although the +Grid topology is the most intuitive and popular
structure, several recent papers investigate designs for better ISLs
topologies (Soret et al., 2019; Beech et al., 1999; Akyildiz et al., 2002; Bai
et al., 2004; Song et al., 2014; Taleb et al., 2008). Our evaluation needs to
be repeated under novel topologies.

Routing and traffic engineering:  In this work, we assume shortest path based
routing scheme and ideal network conditions. In practice, the observed delay
and hop count can vary based on the routing and traffic engineering schemes
employed by the constellation operator. Moreover, the network load may not be
uniformly distributed, with high congestion closer to heavily populated areas.
We do not make assumptions about any intelligent routing or TE schemes.

Multi-Shell Analysis:  Our analysis focuses on the network performance of a
single shell. Currently deployed satellites do not support inter-shell
communication even within the same LEO mega-constellation; their ISLs are
designed to communicate with satellites in the same shell. Understanding the
aggregate network performance of multiple shells, each with different
configuration settings, is an open problem.

##  7\. Conclusion

We evaluate three key design parameters—the number of orbits per shell, the
inclination of orbit, and the number of satellites per orbit—on real-world
topologies of Starlink, Kuiper, Telesat, and a broad set of synthetic
topologies. Our analysis reveals interesting insights and answers to
previously inexplicable behavior patterns. We identify the parameter
thresholds for optimal network performance, the causes for the poor
performance of a high-density shell (S1) of Starlink compared to one with
lower density (S2), and determine the impact of traffic endpoint locations on
performance by analyzing their geographic angles. Our interesting observations
underscore the need for a systematic analysis of the impact of design
parameters in the context of the emerging domain of LEO mega-constellations.
We highlight several avenues for future research in this space, including the
impact of topology in relation to design parameters and multi-shell analysis.

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##  Appendix A Ethics

This work does not raise any ethical issues. All the code and data used in
this work were available in open source. This work does not rely on any data
that compromises the privacy of end users or the security of enterprises.

##  Appendix B Altitude

Altitude is one of the potential parameters that could affect the network
performance of the LEO mega-constellation. However, altitude may not be a
factor affecting the absoluteness of the LEO mega-constellation. From the
parameter setting of the LEO mega constellation, we can find that although
different commercial satellites have different altitude Settings, the most
extensive range ranges from 540km for Starlink to 1325 km for Telesat.
However, a change of a few hundred kilometers is only a hundredth of a change
for an Earth over ten thousand kilometers across. It is reasonable to assume
that the change in distance of several hundred kilometers may make altitude
not a factor that can significantly affect the LEO mega-constellation. Our
experiment confirms our hypothesis. We estimate the LEO mega-constellations
with altitude = 540, 740, 940, and 1140 km, respectively. The other
configurations are Orbits = 33, Sats/Orbits = 28, Inclination = 53°.

Altitude is not the critical parameter. The experiment result of a) Max. RTT,
b) Min. RTT, c) Max. RTT - Min. RTT, d) Max. RTT / Min. RTT, e) Geodesic
Slowdown, f) # of Path Change, g) Max. Hop / Min. Hop, h) Max. Hop - Min. Hop
are shown in Fig. LABEL:Altitude. From the result, we can see the difference
in Max. RTT and Min, RTT between different altitude constellations are
negligible. The Geodesic Slowdown, # of Path Change, Max. Hop / Min. Hop, Max.
Hop - Min. Hop follows the same rules. Only the Max. RTT - Min. RTT, Max. RTT
/ Min. RTT shows little difference between different altitude constellations:
when the altitude increases, the fluctuation of RTT becomes less. However, the
RTT fluctuation of most endpoints pairs is still within 10ms. From what has
been discussed above, we may conclude that altitude is not the critical
parameter that could significantly determine the network performance of LEO
mega-constellations.

##  Appendix C Visualization of Multishells

Fig. LABEL:VisShell demonstrates the topology of the three existing major
constellations.

##  Appendix D Commercial LEO mega-constellations

As mentioned in § 5.3.1, we estimate the network performance of different
shells of existing commercial LEO mega-constellations. Shown in Fig.
LABEL:all_Commercial, we plot all the shells’ curves in one figure to compare
their performance intuitively. Noted that we cut the long tail in Fig.
LABEL:all_Commercial_2 and Fig. LABEL:all_Commercial_4. We can see that Kuiper
Shell 1, Shell 2, Starlink Shell 2, and Telesat shell 2 all perform well in
Max. RTT, Geodesic Slowdown, and RTT fluctuation. For the # of Path Changes,
Telesat shell 2 performs the best because of its fewer satellite number.

##  Appendix E SYNTHETIC LEO MEGA-CONSTELLATIONS

In this section, we show more experiment result for the synthetic LEO mega-
constellations.

###  E.1. Satellites Per Orbit

At an inclination angle of 53°, we measured the synthetic LEO mega-
constellations’ network performance with the Satellite Number per Orbit = 20,
28, 36, 44 under the same Orbit Number = 20, 33, 46, 59, respectively. The
extension measurement results of Min. RTT, Max. RTT - Min. RTT, Max. RTT to
Min. RTT, Max. hop - Min. hop, Max. hop to Min. hop are show in Fig.
LABEL:Min_RTT_Sats/Orbits, Fig. LABEL:minus_RTT_Sats/Orbits, Fig.
LABEL:To_RTT_Sats/Orbits, Fig. LABEL:minus_hop_Sats/Orbits, Fig.
LABEL:to_hop_Sats/Orbits, respectively. Because the long-tail effect of some
parameter configurations is too significant, we are forced to select only part
of the ECDF curves and ignore the excessively long tails to facilitate the
presentation. From Fig. LABEL:Min_RTT_Sats/Orbits, we can see that the
difference in changing the Sats/Orbits number is not that much, and it also
follows the rule that when Sats/Orbits number increases, the Min. RTT is fewer
than before. For Fig. LABEL:minus_RTT_Sats/Orbits and Fig.
LABEL:To_RTT_Sats/Orbits, we can find that with the Sats/Orbits increases, the
RTT fluctuation is less. Also, when the Orbits number achieves 33+, the
decrease of RTT fluctuation becomes saturation, although Sats/Orbits
increases. The same rules apply to hop fluctuation, shown in Fig.
LABEL:minus_hop_Sats/Orbits and Fig. LABEL:to_hop_Sats/Orbits. Moreover, we
can see that at least 20 Sats/Orbits is far from the ideal satellite network
performance, and 28+ Sats/Orbits are needed to achieve relatively low delay,
low network churn, and stable network connection.

###  E.2. Inclination

The results of Min. RTT, Max. RTT to Min. RTT, Max. hop - Min. hop, Max. hop
to Min. hop are shown in Fig. LABEL:A_Inclination. From the result, we can see
that the change in Inclination is not apparent. For the Min. RTT, the
Inclination 75° one has the largest RTT, while the Inclination 55° one has the
best performance of the median value of RTT. However, the difference between
the best and the worst is still negligible. This is especially true for the
maximum to minimum ratio of RTT and hop. The only point that can be found is
that 30% Endpoints pairs of constellation Inclination 45° have more than two
hop fluctuations, the group with the worst performance among the four
parameters. From the results of these images, it is still difficult to see the
impact of Inclination change on LEO mega-constellation’s network performance,
so further experiments are necessary.

###  E.3. Orbit Number

With the inclination of 53°, the satellite network performance is measured
when the Orbits number = 20, 33, 46, 59 under the same Sats/Orbits = 20, 28,
36, 44. The extension result of Min. RTT, Max. RTT - Min. RTT, Max. RTT to
Min. RTT, Max. hop - Min. hop, Max. hop to Min. hop are show in Fig.
LABEL:Min_RTT_Orbits, Fig. LABEL:minus_RTT_Orbits, Fig. LABEL:To_RTT_Orbits,
Fig. LABEL:minus_hop_Orbits, Fig. LABEL:to_hop_Orbits, respectively. Because
the long-tail effect of some parameter configurations is too significant, we
are forced to select only part of the ECDF curves and ignore the excessively
long tails to facilitate the presentation. From Fig. LABEL:Min_RTT_Orbits, we
can see that the Min. RTT does not differ much between different orbits
numbers for 28+ Sats/Orbits. This phenomenon means that LEO mega-constellation
with 20+ Sats/Orbits and 20+ orbits are close enough to the upper limit
performance of Min. RTT. Moreover, by observing the tail of the ECDF curve in
Fig. LABEL:Min_RTT_Orbits, we can find that with the increase of the number of
Sats/Orbits, the maximum value of Min. RTT gradually decreases from nearly
200ms to less than 150ms. From Fig. LABEL:minus_RTT_Orbits and Fig.
LABEL:To_RTT_Orbits, we can see the fluctuation amplitude of RTT. When
Sats/Orbits = 20, the 33- orbits constellation has a large RTT fluctuation.
However, for Sats/Orbits ¿ 28 LEO mega-constellations, when the orbits number
increases, the influence of the number of Orbits on the network RTT
fluctuation reaches saturation and can no longer significantly improve the
stability of RTT fluctuation.As shown in Fig. LABEL:minus_hop_Orbits and Fig.
LABEL:to_hop_Orbits, hop fluctuations follow the same pattern. Combining the
above experiments result, we can conclude that when the orbits number
increases, both the latency and hop performance will improve. We also notice a
relative saturation state value for Sat/Orbits between 20 and 28. To
understand this phenomenon better, we further generated different satellite
constellations with smaller intervals of Sats/Orbits between 20 and 28, hoping
to find one Sats/Orbits number that could help the LEO mega-constellation
reach the saturation state no matter the change in the orbits number, the
reults are shown in § E.4.

###  E.4. Further Exploration

We further explore performance threshold of number of satellites per orbit.
According to Fig. LABEL:RTT_Orbits_(2), Fig. LABEL:slowdown_RTT_Orbits_(2),
Fig. LABEL:Min_RTT_Orbits_(2), Fig. LABEL:path_Orbits_(2), the improvement of
increasing number of satellites per orbit converges at 22 Satellites per
orbit. When exceeding 22 satellites, the performance does not greatly improve.

##  Appendix F Endpoints pairs and Inclination

Figure 15. Geographic Angle between City A and B.

###  F.1. Geographic Angle (Bearing Angle)

Illustrated by Fig. 15, the Geographic Angle, also known as the Bearing angle,
is a measure of the angle between the plane that contains two geographic
locations (such as cities A and B) and the center of the Earth, and the plane
that represents the Earth’s equator. It is a fundamental concept for
understanding and describing the relative positions of locations on the
Earth’s surface, which provides information about the orientation or direction
from one point to another on the Earth’s surface, with a range of 0 to 90
degrees.

###  F.2. Visualization of User Endpoints

As mentioned in § 5.2.2, we divided the endpoints pairs into nine groups based
on their geographic angle. The visualization of these nine groups and the
positions of the top 100 GDP cities on the earth is shown in Fig.
LABEL:Vis_City_pairs.

###  F.3. Geodesic Slowdown

Fig. LABEL:appendix:inclination_geodesic_slowdown shows the geodesic slowdown
of nine endpoints pairs groups from 0° to 90° under inclinations = 45°, 55°,
65°, and 75° LEO mega-constellations shells.

We can observe that the LEO mega-constellations’ Inclination aligned with the
geographic angle of the endpoints pairs performs better. Shell with a 45°
inclination has the lowest slowdown over endpoints pairs with a 0-50°
geographic angle. The rest of the shells also perform the best in the
endpoints pairs with the closest geographic angle to Inclination.

Fig. LABEL:appendix:inclination_all_pairs shows the geodesic slowdown of nine
groups of user points pairs in constellations with different Inclinations. We
can see that solid line in each graph and conclude that for the median value
of Geodesic Slowdown, the endpoints pairs aligned most with the inclination
angle of LEO mega-constellations have the least value.

Therefore, the optimal range of the LEO mega-constellation should be more
aligned with the endpoints pairs’ geographic angle.

###  F.4. Average Hop

As mentioned in § 5.3.3, we plot the heatmaps according to the distance of the
endpoints in each Geographic Angles group and the corresponding number of
average hops, showing inclination changes Intuitively. We only put the result
of Inclination 45° and Inclination 55° in the main text. Here we also plot the
heatmaps of Inclination 45°, 55°, 65° and 75°, shown in Fig. LABEL:heatmap and
Fig. LABEL:65_75_heatmap.

Let us call endpoints within the distance interval of 6000-8000km and a
geographic angle of 40-50° (i.e., the fourth column in the heatmap (5) at the
center of each figure) as bin A and endpoints within the distance interval of
8000-10000km and a geographic angle of 50-60° (i.e., the fifth column in the
heatmap (6) of each figure) as bin B. Under the inclination of 45° (Fig.
LABEL:heatmap_1), more than 160 pairs of bin A have an average hop count in
the range 6-7, with only a tiny fraction with an average hop count of 5-6 or
7-8. However, under the inclination of 55° (Fig. LABEL:heatmap_2), only about
60 pairs in bin A have an average hop count in the range of 6-7. The remaining
have an average hop count in the range of 7-8, thereby increasing the overall
average hop count. In contrast, for bin B, as we increase the orbital angle
(Fig. LABEL:heatmap_1 vs. Fig. LABEL:heatmap_2), the average hop count
decreases. This is because the orbital angle of 55° in Fig. LABEL:heatmap_2 is
closer to the geographic angle for bin B, while the orbital angle of 45° is
closer to the geographic angle for bin A. The same pattern holds for all other
groups of endpoint pairs.

As shown in Fig. LABEL:65_75_heatmap_1 and Fig. LABEL:65_75_heatmap_2, we can
observe the changes in average hops of endpoints pairs with different
Geographic Angles and different distances. For user points pairs between
60-70°, under the inclination of 65°, more than 160 pairs of endpoints of
12000-14000 km have an average hop of 10-11. Only a tiny percentage‘s average
hop is 11-12 or 10. However, under the inclination of 75°, the same endpoints
pair with 12000-14000 km and Geographic Angles of 60-70° only has an average
hop of 12-13, close to 140 pairs. The rest are distributed in the higher
average hops 13-14, and the overall number of average hops is improved. In
contrast, the opposite result is shown for endpoints pairs with Geographic
Angles 70-80°. Under the 45° inclination LEO mega-constellation, 12000-14000
km endpoints pairs have more than 120 pairs with an average hop of 12.
However, at Inclination 55° one, the average hop of endpoints within the same
range decreases significantly, and most of the inclination is concentrated at
eleven average hops. The same pattern holds for the other groups of endpoints
pairs.

Combined these two experimental results with the result of Inclination 45° and
55°, shown in Fig. LABEL:heatmap_1 and Fig. LABEL:heatmap_2, we can confirm
that the congruency between the inclination and distribution of endpoints
pairs Geographic Angles is positively correlated with the performance of the
average hop number of the LEO mega-constellation. When the LEO mega-
constellation’s orbits are more aligned to the lines between endpoints pairs,
the route between source and destination will favor the use of in-orbit
satellites, avoiding more inter-track zig-zag and bringing less hop count.
Therefore, when we design the LEO mega-constellation, adjusting the
inclination of LEO mega-constellations as far as possible to make it more
consistent with the distribution of endpoints can bring less hop and network
delay.

##  Appendix G Visualization of routing path

Fig. LABEL:vis_path shows the routing path of different Endpoints pairs, and
shows the changes of routing path of one particular Endpoints pairs. We can
observe that when the data is transmitted to north or south direction, the
routing path tends to go through intra-orbit hops. When transmitting to east
or west direction, the routing path tends to go through inter-orbit hops,
resulting a zig-zag pattern.