Jamming Analysis between Non-Cooperative Mega-Constellations Based on Satellite Network Capacity

Abstract

Due to the openness of inter-satellite links (ISLs) in mega-constellations, the threat posed by jamming from non-cooperative constellations is becoming increasingly significant. Most of the existing approaches focus on the up/down link capacity between satellites and ground stations, which differs greatly from the situation whereby ISLs are subjected to non-cooperative jamming. Therefore, this work investigates the transmission rates of ISLs under jamming from non-cooperative mega-constellations. Based on this, a novel satellite network capacity calculation method is proposed to evaluate the mega-constellation network capacity when the transmission rates change dynamically. The simulation results show that the satellite number, jamming power and inclination of non-cooperative constellations have a significant influence on the network capacity. The optimal jamming efficiency occurs when the constellation inclinations are close.

1. Introduction

With the rapid development of satellite communication technologies, non-geostationary-orbit (NGSO) constellations have become an essential supplement for space–air–ground integrated networks (SAGINs) [1]. However, the huge increase in the satellite number and service requirements have led to great challenges for mega-constellations [2]. Due to the openness of inter-satellite links (ISLs), ISLs are no longer absolutely secure [3,4,5]. Microwave ISLs can be threatened by malicious jamming from non-cooperative constellations, as shown in Figure 1. ${\Psi }_A$ and ${\Psi }_B$ denote the jammed and jamming constellations, respectively. The red region inside the cone is the main jamming region (MJR). In non-cooperative constellations, each ISL may be subjected to jamming. As a result, the entire constellation network is under an all-encompassing jamming threat, which will affect the overall network capacity.

The analysis of the satellite network capacity under non-cooperative jamming can improve the anti-jamming ability and quality of service (QoS) [6] in terms of capacity, latency [7] and security [8]. However, the impact of the jamming parameters and constellation configurations on the satellite network capacity is not clear at present. In mega-constellations, ISLs switch frequently, leading to dynamic changes in the link capacity. Thus, it is difficult to apply terrestrial network capacity analysis methods directly to mega-constellations [9]. Meanwhile, most of the existing approaches focus on the up/down link capacity between satellites and ground stations [10,11,12], which differs greatly from the situation whereby ISLs are subjected to non-cooperative jamming.

To address these issues, we define a typical non-cooperative jamming scenario [13,14] with two mega-constellations to analyze the impact of potential non-cooperative jamming on the satellite network capacity. Against this background, two questions remain to be answered: (1) How can we determine the impact of the jamming parameters on a single ISL’s capacity? (2) How can we accurately measure the overall communication capacity of a mega-constellation network when the transmission rates of the ISLs are dynamically changing? To address the aforementioned challenges, we propose a calculation method for the satellite network capacity under non-cooperative jamming. We propose the main jamming region and propagation model to investigate the transmission rate under non-cooperative jamming. Based on this, a novel satellite network capacity calculation method is proposed to evaluate the network capacity of mega-constellations when the transmission rates change dynamically. The main determinants of the jamming effects are investigated to guide the anti-jamming design of mega-constellations. The main contributions of this paper are as follows:

• The concept of the main jamming region is defined in order to describe the jamming scene between mega-constellations;
• The transmission rates of ISLs in mega-constellations under jamming from non-cooperative constellations are calculated, and a novel satellite network capacity calculation method is propose to evaluate the network capacity of mega-constellations when the transmission rates change dynamically;
• The simulation results show that the jamming power, constellation scale and inclination of non-cooperative jamming constellations have a significant influence on mega-constellations’ network capacities.

The rest of the paper is organized as follows. In Section 2, we define the main jamming region and system model. In Section 3, we propose the satellite network capacity calculation method. In Section 4, the simulation results verify the effectiveness of our method. Section 5 concludes the paper.

2. System Model

In this section, the main jamming region is introduced. Then, we discuss the jamming scenario and propagation model. Finally, the transmission rate is calculated.

Table 1. Summary of mathematical notations.

Notation	Description
${\Psi }_A$	The jammed constellation
${\Psi }_B$	The jamming constellation
$A(i,j)$	The jth satellite in the ith orbit
${\theta }_{max}$	The semi-cone angle of the main jamming region
$G_t$	The transmitting antenna peak gain
$G_r$	The receiving antenna peak gain
$L_{(i,j)}^{(i,j+1)}$	The ISL from satellite $A(i,j)$ to satellite $A(i,j+1)$
$R_{i,j,t}^{\nearrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i,j+1)}$ with no jamming
$R_{i,j,t}^{\swarrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i,j-1)}$ with no jamming
$R_{i,j,t}^{\nwarrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i-1,j)}$ with no jamming
$R_{i,j,t}^{\searrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i+1,j)}$ with no jamming
$d_{i,j,t}$	The distance between $A(i,j)$ and $A(i,j+1)$ at time slot t
${\tilde{R}}_{i,j,t}^{\nearrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i,j+1)}$ under jamming
${\tilde{R}}_{i,j,t}^{\swarrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i,j-1)}$ under jamming
${\tilde{R}}_{i,j,t}^{\nwarrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i-1,j)}$ under jamming
${\tilde{R}}_{i,j,t}^{\searrow }$	The maximum transmission rate of ISL $L_{(i,j)}^{(i+1,j)}$ under jamming
$R_{(i,j)}^{(i,j+1)}$	The maximum transmission rate of intra-plane ISL $L_{(i,j)}^{(i,j+1)}$
$R_{(i,j)}^{(i,j-1)}$	The maximum transmission rate of intra-plane ISL $L_{(i,j)}^{(i,j-1)}$
$R_{(i,j)}^{(i+1,j)}$	The maximum transmission rate of inter-plane ISL $L_{(i,j)}^{(i+1,j)}$
$R_{(i,j)}^{(i-1,j)}$	The maximum transmission rate of inter-plane ISL $L_{(i,j)}^{(i-1,j)}$
$R_o$	The transmission rate of intra-plane ISLs
$R_h$	The transmission rate of inter-plane ISLs
$C_A$	The network capacity of ${\Psi }_A$
H	The total hop count of ${\Psi }_A$

gives the summary of the mathematical notations.

2.1. Main Jamming Region

The jamming effect between non-cooperative constellations is affected by the angle $\theta$ between the jamming and jammed satellites. $\theta$ is the angle between the jammed satellite’s pointing antenna and the jamming satellite–jammed satellite connection line. For an arbitrary ISL, the jamming effect will decrease as $\theta$ increases. Therefore, we mainly consider the jamming effect within a certain pinch range (e.g., ${\theta }_{max}$), i.e., the main jamming region (MJR). The MJR is the cone formed with ${\theta }_{max}$ as the half-cone angle. The MJR in a geocentric rectangular coordinate system is shown in Figure 2. For jammed ISLs, the MJR is the region where the jamming satellites can effectively interfere with the target ISLs.

In mega-constellations, $A(i,j)$ denotes the jth satellite in the ith orbit. The MJR is constructed as follows: take $A(i,j)$ in the jammed constellation ${\Psi }_A$ as the vertex, take the jammed ISL $L_{(i,j)}^{(i,j+1)}$ as the central vertical line and form a cone with ${\theta }_{max}$ as the semi-cone angle. The region where the cone intersects the orbits of jamming constellation ${\Psi }_B$ is defined as the MJR. The satellites in the MJR are the corresponding jamming satellites of $L_{(i,j)}^{(i,j+1)}$. According to the above definition, we can construct the MJR for each ISL in ${\Psi }_A$. For ISLs in the same orbit, if ${\theta }_{max}$ is small enough, the MJRs of each ISL will not overlap. Taking a single ISL as an example, we analyze the influence of the jamming parameters on the transmission rate.

2.2. Jamming Scenario

Assume that both the jammed constellation ${\Psi }_A$ and non-cooperative jamming constellation ${\Psi }_B$ are Walker constellations. Since the jamming effect is closely related to the relative positions of the satellites, we first define the satellite coordinates of ${\Psi }_A$ and ${\Psi }_B$ in the Cartesian coordinate system.

Let the radius of Earth be $R_e$. The Walker constellation determines the positions of satellites by seven parameters, i.e., $Z/N/F/h/\mu /{\Omega }_0/{\phi }_0$, where Z denotes the satellite number in the constellation, N denotes the satellite orbit number, F is the phase factor, h is the orbital altitude, $\mu$ denotes the orbital inclination, ${\Omega }_0$ is the initial right ascension of the ascending node (RAAN) and ${\phi }_0$ is the initial true anomaly. On this basis, we define the spatial dynamic coordinates of all satellites as well as Earth stations.

Assume that the parameters of the jammed constellation ${\Psi }_A$ are $Z_A/N_A/F_A/h_A/{\mu }_A/$${\Omega }_{A0}/{\phi }_{A0}$. The coordinates of the jth satellite in the ith orbit of ${\Psi }_A$ can be expressed as

$$A(i,j,t)=(x_A(i,j,t),y_A(i,j,t),z_A(i,j,t))$$

(1)

$$\left\{\begin{array}{c}{x}_A(i,j,t)=(R_e+h_A)(cos{\Omega }_{A}cos({\phi }_A+{\omega }_{A}t)\\ -sin{\Omega }_{A}sin({\phi }_A+{\omega }_{A}t)cos{\mu }_A)\\ y_A(i,j,t)=(R_e+h_A)(sin{\Omega }_{A}cos({\phi }_A+{\omega }_{A}t)\\ +cos{\Omega }_{A}sin({\phi }_A+{\omega }_{A}t)cos{\mu }_A)\\ z_A(i,j,t)=(R_e+h_A)sin({\phi }_A+{\omega }_{A}t)sin{\mu }_A\end{array}\right.$$

(2)

where ${\omega }_A=\sqrt{\frac{GM}{{(R_e+h_A)}^3}}$, G denotes the gravitational constant, M denotes the mass of the Earth and ${\Omega }_A={\Omega }_{A0}+i\times (2\pi /N_A)$, ${\phi }_A={{\phi }_A}_0+2\pi (i\times \frac{F_A}{Z_A}+j\times \frac{N_A}{Z_A})$.

Assume that the parameters of ${\Psi }_B$ are $Z_B/N_B/F_B/h_B/{\mu }_B/{\Omega }_{B0}/{\phi }_{B0}$; the satellite coordinates in ${\Psi }_B$ are similar to those of ${\Psi }_A$.

2.3. Propagation Model

The free space propagation model is utilized to calculate the channel gain between satellites. Assume that the distance between the jamming satellite and jammed satellite is d, and the free space attenuation of the signal is

$$L_d={\left(\frac{\lambda }{4\pi d}\right)}^2$$

(3)

According to ITU-R S.1528 [15], when the satellite antenna’s off-axis angle is $\phi$, the receiving antenna peak gain $G_r\left(\phi \right)$ is

$$G_r\left(\phi \right)=\left\{\begin{array}{cc}{G}_r^{max},\hfill & \mathrm{if}\phi \le {\phi }_b\hfill \\ G_r^{max}-3{\left(\frac{\phi }{{\phi }_b}\right)}^2,\hfill & \mathrm{if}{\phi }_b<\phi \le Y\hfill \\ G_r^{max}+L_s-25log\left(\frac{\phi }{Y}\right),\hfill & \mathrm{if}Y<\phi \le X\hfill \\ L_F,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(4)

where $G_r^{max}$ is the maximum receiving antenna peak gain. $X=Y{\left(10\right)}^{0.04(G_r^{max}+L_s-L_F)}$, $L_F=0$. For LEO satellites, $L_s=-6.75$, $Y=1.5{\theta }_b$. According to the reciprocity theorem, the transmitting antenna peak gain $G_t\left(\phi \right)$ equals $G_r\left(\phi \right)$.

2.4. Transmission Rate Calculation

Denote the transmission rate of ISL $L_{(i,j)}^{(i,j+1)}$ from satellite $A(i,j)$ to satellite $A(i,j+1)$ in ${\Psi }_A$ at time slot t as $R_{i,j,t}^{\nearrow }$. When $L_{(i,j)}^{(i,j+1)}$ is not jammed, $R_{i,j,t}^{\nearrow }$ is calculated as follows:

$$R_{i,j,t}^{\nearrow }={log}_2(1+\frac{p_A{G}_t^{max}{G}_r^{max}{\left(\frac{\lambda }{4\pi d_{i,j,t}^{\nearrow }}\right)}^2}{KTW})$$

(5)

where $d_{i,j,t}^{\nearrow }=∥A(i,j,t)-A(i,j+1,t)∥$.

Satellites in the MJR corresponding to $L_{(i,j)}^{(i,j+1)}$ satisfy the following conditions:

$$\left\{\begin{array}{c}B\left(t\right)=\left(x_B\left(t\right),y_B\left(t\right),z_B\left(t\right)\right)\hfill \\ 〈A(i,j,t)-A(i,j+1,t),B\left(t\right)-A(i,j+1,t)〉<\theta \hfill \end{array}\right.$$

(6)

Assume that, at time slot t, the number of satellites satisfies Equation (6) is $J\left(t\right)$. Denote the above satellite coordinates as $B_j\left(t\right),j=1,2,\cdots ,J\left(t\right)$. Thus, we obtain the data transmission rate ${\tilde{R}}_{i,j,t}^{\nearrow }$ of $L_{(i,j)}^{(i,j+1)}$ under non-cooperative jamming.

$$\begin{array}{c}{\tilde{R}}_{i,j,t}^{\nearrow }={log}_2\left(1+\frac{p_A{G}_t^{max}{G}_r^{max}{\left(\frac{\lambda }{4\pi d_{i,j,t}^{\nearrow }}\right)}^2}{{\displaystyle \sum _{k=1}^{J\left(t\right)}}{p}_B{G}_r^{max}{G}_t\left({\theta }_{i,j,t}^{\nearrow }\left(k\right)\right){\left(\frac{\lambda }{4\pi D_{i,j,t}^{\nearrow }\left(k\right)}\right)}^2+KTW}\right)\hfill \end{array}$$

(7)

where

$$\begin{array}{c}{D}_{i,j,t}^{\nearrow }\left(k\right)=∥B_k\left(t\right)-A(i,j+1,t)∥\hfill \\ {\theta }_{i,j,t}^{\nearrow }\left(k\right)=〈A(i,j,t)-A(i,j+1,t),B_k\left(t\right)-A(i,j+1,t)〉\hfill \end{array}$$

(8)

Similarly, we can calculate the transmission rate $R_{i,j,t}^{\swarrow },R_{i,j,t}^{\nwarrow },R_{i,j,t}^{\searrow }$ of $L_{(i,j)}^{(i,j-1)}$, $L_{(i,j)}^{(i-1,j)}$ and $L_{(i,j)}^{(i+1,j)}$ under jamming-free conditions, as well as the transmission rate ${\tilde{R}}_{i,j,t}^{\swarrow },{\tilde{R}}_{i,j,t}^{\nwarrow },{\tilde{R}}_{i,j,t}^{\searrow }$ after jamming.

In summary, for satellite $A(i,j)$ in ${\Psi }_A$, the inter-satellite link rates of $A(i,j)$ are $(R_{i,j,t}^{\nearrow },R_{i,j,t}^{\swarrow },$$R_{i,j,t}^{\searrow },R_{i,j,t}^{\nwarrow })$ and $\left({\tilde{R}}_{i,j,t}^{\nearrow },{\tilde{R}}_{i,j,t}^{\swarrow },{\tilde{R}}_{i,j,t}^{\searrow },{\tilde{R}}_{i,j,t}^{\nwarrow }\right)$ under jamming-free and non-cooperative jamming conditions, respectively.

3. Satellite Network Capacity Calculation Method

Assume that the number of orbits and the number of satellites in each orbit in ${\Psi }_A$ are N and M, respectively. The ISLs are connected in the “+Grid” mode, as shown in Figure 3. In ${\Psi }_A$, the link rates of $A(i,j)$ are $\left(R_{i,j,t}^{\nearrow },R_{i,j,t}^{\swarrow },R_{i,j,t}^{\searrow },R_{i,j,t}^{\nwarrow }\right)$, respectively. $A(i,j)$ sends data equally to all other satellites, i.e., the traffic rates sent by $A(i,j)$ along the four directions are $\frac{R_{i,j,t}^{\nearrow }}{NM-1},\frac{R_{i,j,t}^{\swarrow }}{NM-1},\frac{R_{i,j,t}^{\searrow }}{NM-1},\frac{R_{i,j,t}^{\nwarrow }}{NM-1}$, which can be denoted as ${\tilde{R}}_{i,j,t}^{\nearrow },{\tilde{R}}_{i,j,t}^{\swarrow },{\tilde{R}}_{i,j,t}^{\searrow },{\tilde{R}}_{i,j,t}^{\nwarrow }$, respectively. The network capacity $C_A$ of ${\Psi }_A$ equals the total transmission rate R divided by the average hop count $E\left(H\right)$.

3.1. Transmission Rate

In the inclined-orbit constellation ${\Psi }_A$, the intra-plane ISLs and inter-plane ISLs have similarities; therefore, we only give the proof of the intra-plane ISLs’ transmission rate here.

When M is odd, for any intra-plane ISL $L_{(i,j)}^{(i,j+1)}$, $L_{(i,j)}^{(i,j+1)}$ will be passed through when $A(i,j)$ sends data to $A(i,(j+1)modM)$, $A(i,(j+2)modM)$, …, $A(i,(j+\frac{M-1}{2})modM)$. Therefore, $L_{(i,j)}^{(i,j+1)}$ is used $\frac{M-1}{2}$ times. The maximum transmission rate of $A(i,j)$ is ${\tilde{R}}_{i,j,t}^{\nearrow }$. Meanwhile, $L_{(i,j)}^{(i,j+1)}$ will also be passed through when $A(i,(j-1)modM)$ sends data to $A(i,(j+1)modM)$, $A(i,(j+2)modM)$, …, $A(i,(j+\frac{M-1}{2}-1)modM)$. $L_{(i,j)}^{(i,j+1)}$ is passed through $\frac{M-1}{2}-1$ times. The maximum transmission rate under the jamming condition from $A(i,j-1)$ to $A(i,j)$ is ${\tilde{R}}_{i,(j-1)modM,t}^{\nearrow }$. By analogy, $L_{(i,j)}^{(i,j+1)}$ will still be passed through until $A(i,(j-(\frac{M-1}{2}-1))modM)$ sends data to $A(i,j+1)modM)$. Under this condition, $L_{(i,j)}^{(i,j+1)}$ is used once. The maximum transmission rate is ${\tilde{R}}_{i,(j-\frac{M-1}{2})modM,t}^{\nearrow }$. Therefore, the actual maximum transmission rate of $L_{(i,j)}^{(i,j+1)}$ is

$$\begin{array}{cc}\hfill R_{(i,j)}^{(i,j+1)}& ={\displaystyle \sum _{k=0}^{\left(M-1\right)/2-1}}\left(\frac{M-1}{2}-k\right){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow }\hfill \\ \hfill & ={\displaystyle \sum _{k=0}^{\left(M-1\right)/2}}\left(\frac{M-1}{2}-k\right){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow }\hfill \end{array}$$

(9)

When M is even, $L_{(i,j)}^{(i,j+1)}$ will be passed through when $A(i,j)$ sends data to $A(i,(j+1)modM)$, $A(i,(j+2)modM)$, …, $A(i,(j+\frac{M-2}{2})modM)$. $L_{(i,j)}^{(i,j+1)}$ is utilized $\frac{M-2}{2}$ times. When $A(i,j)$ sends data to $A(i,(j+\frac{M}{2})modM)$, both forward and reverse ISLs can be used; therefore, $L_{(i,j)}^{(i,j+1)}$ is passed through $\frac{1}{2}$ times. The maximum transmission rate is ${\tilde{R}}_{i,j,t}^{\nearrow }$. When $A(i,(j-1)modM)$ sends data to $A(i,(j+1)modM)$, $A(i,(j+2)modM)$, …, $A(i,(j+\frac{M}{2}-1)modM)$, $L_{(i,j)}^{(i,j+1)}$ will be passed through $\frac{M-1}{2}-1$ times. The maximum transmission rate is ${\tilde{R}}_{i,(j-1)modM,t}^{\nearrow }$. By analogy, $L_{(i,j)}^{(i,j+1)}$ will still be passed through until $A(i,(j-\frac{M}{2})modM)$ sends data to $A(i,(j+1)modM)$. Under this condition, $L_{(i,j)}^{(i,j+1)}$ is used $\frac{M}{2}-\left(\frac{M}{2}-1\right)-\frac{1}{2}=\frac{1}{2}$ times. The maximum transmission rate is ${\tilde{R}}_{i,(j-\frac{M-2}{2})modM,t}^{\nearrow }$. Therefore, the actual maximum transmission rate of $L_{(i,j)}^{(i,j+1)}$ is

$$\begin{array}{cc}\hfill R_{(i,j)}^{(i,j+1)}& ={\displaystyle \sum _{k=0}^{\left(M-2\right)/2}}(\frac{M-2}{2}+\frac{1}{2}-k){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow }\hfill \\ \hfill & ={\displaystyle \sum _{k=0}^{\left(M-2\right)/2}}(\frac{M-1}{2}-k){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow }\hfill \end{array}$$

(10)

In summary, the maximum transmission rate of intra-plane ISL $L_{(i,j)}^{(i,j+1)}$ is

$$R_{(i,j)}^{(i,j+1)}=\left\{\begin{array}{c}{\displaystyle \sum _{k=0}^{\frac{M-2}{2}}}(\frac{M-1}{2}-k){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow },M=2n\hfill \\ {\displaystyle \sum _{k=0}^{\frac{M-1}{2}}}(\frac{M-1}{2}-k){\tilde{R}}_{i,(j-k)modM,t}^{\nearrow },otherwise\hfill \end{array}\right.$$

(11)

Similarly, the maximum transmission rates of intra-plane ISL $L_{(i,j)}^{(i,j-1)}$, inter-plane ISL $L_{(i,j)}^{(i+1,j)}$ and $L_{(i,j)}^{(i-1,j)}$ are $R_{(i,j)}^{(i,j-1)}$, $R_{(i,j)}^{(i+1,j)}$ and $R_{(i,j)}^{(i-1,j)}$, respectively.

$$R_{(i,j)}^{(i,j-1)}=\left\{\begin{array}{c}{\displaystyle \sum _{k=0}^{\frac{M-2}{2}}}(\frac{M-1}{2}-k){\tilde{R}}_{i,(j+k)modM,t}^{\swarrow },M=2n\hfill \\ {\displaystyle \sum _{k=0}^{\frac{M-1}{2}}}(\frac{M-1}{2}-k){\tilde{R}}_{i,(j+k)modM,t}^{\swarrow },otherwise\hfill \end{array}\right.$$

(12)

$$R_{(i,j)}^{(i+1,j)}=\left\{\begin{array}{c}{\displaystyle \sum _{k=0}^{\frac{N-2}{2}}}(\frac{N-1}{2}-k){\tilde{R}}_{(i-k)modN,j,t}^{\searrow },N=2n\hfill \\ {\displaystyle \sum _{k=0}^{\frac{N-1}{2}}}(\frac{N-1}{2}-k){\tilde{R}}_{(i-k)modN,j,t}^{\searrow },otherwise\hfill \end{array}\right.$$

(13)

$$R_{(i,j)}^{(i-1,j)}=\left\{\begin{array}{c}{\displaystyle \sum _{k=0}^{\frac{N-2}{2}}}(\frac{N-1}{2}-k){\tilde{R}}_{(i+k)modN,j,t}^{\nwarrow },N=2n\hfill \\ {\displaystyle \sum _{k=0}^{\frac{N-1}{2}}}(\frac{N-1}{2}-k){\tilde{R}}_{(i+k)modN,j,t}^{\nwarrow },otherwise\hfill \end{array}\right.$$

(14)

Therefore, the transmission rates of intra-plane ISLs $R_o$ and inter-plane ISLs $R_h$ are, respectively,

$$R_o={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}\left(R_{(i,j)}^{(i,j+1)}+R_{(i,j)}^{(i,j-1)}\right)$$

(15)

$$R_h={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}\left(R_{(i,j)}^{(i+1,j)}+R_{(i,j)}^{(i-1,j)}\right)$$

(16)

The total transmission rate R of ${\Psi }_A$ is

$$\begin{array}{c}\hfill R=R_o+R_h={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}\left(R_{(i,j)}^{(i,j+1)}+R_{(i,j)}^{(i,j-1)}\right)\\ \hfill +{\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}\left(R_{(i,j)}^{(i+1,j)}+R_{(i,j)}^{(i-1,j)}\right)\end{array}$$

(17)

3.2. Average Hop Count

The topology of an inclined orbit constellation has symmetry. Take the center node as the source node (e.g., $\left(\frac{N}{2},\frac{M}{2}\right)$ if N and M are even). Either of the other nodes is equivalent to $\left(\frac{N}{2},\frac{M}{2}\right)$ because either node can be rotated to the center by rotation. Take the example that both N and M are even. When $\left(\frac{N}{2},\frac{M}{2}\right)$ sends data to other nodes, the total hop count is

$${\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)$$

(18)

The topology of ${\Psi }_A$ can be divided into four regions, i.e., the lower left, lower right, upper left and upper right, as shown in Figure 3. The hop counts for each region are $H_{ll}$, $H_{lr}$, $H_{ul}$ and $H_{ur}$, respectively.

$$\left\{\begin{array}{c}{H}_{ll}={\displaystyle \sum _{i=0}^{N/2}}{\displaystyle \sum _{j=0}^{M/2}}\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)\hfill \\ H_{lr}={\displaystyle \sum _{i=N/2}^{N-1}}{\displaystyle \sum _{j=0}^{M/2}}\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)\hfill \\ H_{ul}={\displaystyle \sum _{i=0}^{N/2}}{\displaystyle \sum _{j=M/2}^{M-1}}\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)\hfill \\ H_{ur}={\displaystyle \sum _{i=N/2}^{N-1}}{\displaystyle \sum _{j=M/2}^{M-1}}\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)\hfill \end{array}\right.$$

(19)

Therefore, the total hop count H is

$$\begin{array}{cc}\hfill H& =H_{ll}+H_{lr}+H_{ul}+H_{ur}\hfill \\ \hfill & ={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|)\hfill \\ \hfill & ={\displaystyle \sum _{i=0}^{N-1}}\left|i-\frac{N}{2}\right|+{\displaystyle \sum _{j=0}^{M-1}}\left|j-\frac{M}{2}\right|\hfill \\ \hfill & =\frac{N{M}^2+N^{2}M}{4}\hfill \end{array}$$

(20)

Thus, the average hop count $E\left(H\right)$ of each ISL is

$$E\left(H\right)=\frac{N{M}^2+N^{2}M}{4(NM-1)}$$

(21)

The above is the case when both N and M are even. Other cases are similar. Thus, the average hop count of each ISL is, respectively,

$$E\left(H\right)=\left\{\begin{array}{c}\frac{N{M}^2+N^{2}M}{4(NM-1)},N=2n,M=2n\hfill \\ \frac{N(M^2-1)+N^{2}M}{4(NM-1)},N=2n,M=2n+1\hfill \\ \frac{N{M}^2+(N^2-1)M}{4(NM-1)},N=2n+1,M=2n\hfill \\ \frac{N(M^2-1)+(N^2-1)M}{4(NM-1)},N=2n+1,M=2n+1\hfill \end{array}\right.$$

(22)

Therefore, we obtain the network capacity $C_A$ of ${\Psi }_A$ by dividing the transmission rate R by the average hop count $E\left(H\right)$.

$$C_A=\frac{R}{E\left(H\right)}$$

(23)

Thus, when the link rates in ${\Psi }_A$ change dynamically, the network capacity $C_A$ is defined as follows:

$$C_A=\left\{\begin{array}{c}\frac{4}{N+M}(R_o+R_h),N=2n+1,M=2n+1\hfill \\ \frac{4(NM-1)}{N{M}^2+N^{2}M}(R_o+R_h),N=2n,M=2n\hfill \\ \frac{4(NM-1)}{N{M}^2+(N^2-1)M}(R_o+R_h),N=2n+1,M=2n\hfill \\ \frac{4(NM-1)}{N(M^2-1)+N^{2}M}(R_o+R_h),N=2n,M=2n+1\hfill \end{array}\right.$$

(24)

where $R_o$ and $R_h$ denote the sums of the actual transmission rates of intra-plane and inter-plane ISLs, respectively.

$$R_o={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}{\displaystyle \sum _{k=0}^{\frac{M-1}{2}}}(\frac{M-1}{2}-k)(P+Q)$$

(25)

$$R_h={\displaystyle \sum _{i=0}^{N-1}}{\displaystyle \sum _{j=0}^{M-1}}{\displaystyle \sum _{k=0}^{\frac{N-1}{2}}}(\frac{N-1}{2}-k)(U+V)$$

(26)

where $P={\tilde{R}}_{i,(j-k)modM,t}^{\nearrow }$, $Q={\tilde{R}}_{i,(j+k)modM,t}^{\swarrow }$, $U={\tilde{R}}_{(i-k)modN,j,t}^{\searrow }$, $V={\tilde{R}}_{(i+k)modN,j,t}^{\nwarrow }$.

In summary, network capacity $C_A$ is a function of the satellite transmission rate. Denote the network capacity function as $\Gamma$; then, $C_A=\Gamma (R_{i,j,t}^{\nearrow },R_{i,j,t}^{\swarrow },R_{i,j,t}^{\searrow },R_{i,j,t}^{\nwarrow })$. Therefore, the network capacities under jamming-free conditions and non-cooperative jamming conditions are $C_A=\Gamma (R_{i,j,t}^{\nearrow },R_{i,j,t}^{\swarrow },R_{i,j,t}^{\searrow },R_{i,j,t}^{\nwarrow })$ and ${\tilde{C}}_A=\Gamma ({\tilde{R}}_{i,j,t}^{\nearrow },{\tilde{R}}_{i,j,t}^{\swarrow },{\tilde{R}}_{i,j,t}^{\searrow },{\tilde{R}}_{i,j,t}^{\nwarrow })$, respectively. $\left|C_A-{\tilde{C}}_A\right|$ portrays the effect of non-cooperative jamming on the network capacity of ${\Psi }_A$. Based on the network capacity obtained from the calculation method, we analyze the trend of the jamming effects of various jamming parameters of ${\Psi }_B$.

4. Simulation Results

In the simulation, we consider a fundamental scenario with two mega-constellations: ${\Psi }_A$ and ${\Psi }_B$. The altitude of ${\Psi }_A$ is 500 km, which is jammed by ${\Psi }_B$ at an altitude of 1000 km. The simulation ranges from 1 January 2024 04:00:00 to 10 January 2024 04:00:00. The simulation results are averaged over this period. The antenna models of ${\Psi }_A$ and ${\Psi }_B$ are both ITU-R S.1528. The parameter settings in the simulation are shown in

Table 2. Simulation parameters [16,17,18].

Parameters	ΨA	ΨB
Orbit prediction model	Two body	Two body
Altitude h/km	500	1000
Inclination $\mu /{(}^{\circ })$	60	70
Number of orbits N	30	30
Satellites per orbit M	40	20
Phase factor F	1	1
Satellite antenna model	ITU-R S.1528	ITU-R S.1528
Transmitting power/dBW	7	7
Communication frequency/GHz	27.075	27.075
Communication bandwidth/MHz	50	50
Transmitting antenna peak gain/dBi	39.9	39.9
Receiving antenna peak gain/dBi	37	37
Transmitting antenna HPBW/${(}^{\circ })$	5	3.5
Receiving antenna HPBW/${(}^{\circ })$	5	3.5
Temperature of terminal/K	200	200

.

We analyze the impact of the jamming power $P_J$ and constellation scale $Z_B$ of ${\Psi }_B$ on the network capacity of ${\Psi }_A$, as shown in Figure 4. An increase in both the jamming constellation scale and jamming power can lead to an increased jamming intensity, which will finally decrease the overall network capacity.

The constellation configurations of ${\Psi }_B$ also have a significant impact on the jamming effects. We analyze the network capacity of ${\Psi }_A$ when the inclination ${\mu }_B$ of ${\Psi }_B$ changes within $\left[{50}^{\circ },{68}^{\circ }\right]$, as shown in Figure 5. In our simulation, ${\mu }_A={60}^{\circ }$. The network capacity of ${\Psi }_A$ is minimized when ${\mu }_A$ and ${\mu }_B$ are close. The probability of satellites in ${\Psi }_B$ appearing in the MJR of ${\Psi }_A$ will increase due to the proximity of ${\mu }_A$ and ${\mu }_B$, leading to an increased jamming intensity.

Besides the inclination, we also analyze the impact of the altitude on the network capacity $C_A$, as shown in Figure 6. It can be seen that there is no clear trend in $C_A$ when $h_B$ varies. As $h_B$ varies, so does the jamming angle. Both of the above factors have an effect on the jamming intensity. We also investigate the influence of the phase factor on $C_A$. Compared to ${\mu }_B$ and $P_J$, the effect of the phase factor is negligible, as shown in Figure 7.

5. Conclusions

In this work, we investigate the transmission rates of ISLs in mega-constellations under jamming from non-cooperative constellations and propose a novel satellite network capacity calculation method to evaluate the network capacity of mega-constellations when the transmission rates change dynamically. The experimental results illustrate that the jamming power, constellation scale and inclination of non-cooperative jamming constellations have a significant influence on mega-constellations’ network capacities. The optimal jamming efficiency occurs when the constellation inclinations are close. It should be noted that the proposed method is also applicable to free space optical (FSO) communication. The difference is that the beamwidth of FSO is relatively narrow; thus, the MJR needs to be set at a smaller ${\theta }_{max}$.