3.1. Architecture of the system

The air traffic surveillance system illustrated in Figure 1 consists of a space segment, an access segment and a ground segment. The space segment is composed of a SCN, i.e., dozens of satellites in well-designed orbits. According to their altitude, SCNs can be roughly divided into three types: low earth orbit (LEO), medium earth orbit (MEO) and geosynchronous orbit (GEO). There are several reasons why the LEO rather than MEO/GEO constellation is considered for this surveillance project. First, the ADS-B receiver has a sensitivity limitation, a minimum threshold level of the receiver, that imposes a restriction on the communication range. As the communication range that was calculated before cannot extend to MEO/GEO altitude, the LEO constellation is chosen. Second, space-based ADS-B receivers experience signal collisions (Yu et al., Reference Yu, Chen, Li and Zhang2018), and it is even worse if the satellite detection coverage is larger, which means that the MEO/GEO satellites are not suitable for air traffic surveillance. Although the multi-beam antenna can solve the signal conflicts partially, the cost of it will also be much higher. Third, the latency of MEO/GEO to the ground station is longer than LEO. Air traffic surveillance requires a short data downloading time, so LEO satellites can transmit the ADS-B data instantly to the ground even with a multi-hop transmission path. In conclusion, the LEO satellite constellation is the best SCN to adopt. The access segment consists of globally distributed aeroplanes carrying ADS-B transmitters with them. The ground segment includes one or several ground stations (GSs). Although in some papers GSs are divided into two types, data collection center and operating control center, they are not distinguished here, for simplification The placement optimisation of GSs is a complex problem coupled with the SCN, as well as the number of the GSs, which is not discussed in this paper.

Figure 1. The architecture of the system.

Based on the system architecture, three aspects of the SCN design should be discussed. First, a global seamless coverage ability is needed to cover all the aeroplane routes in real time. Therefore, the payload performance needs to be analysed and a constellation type should be determined. As aeroplanes fly below 15 km, their altitude can be ignored compared with the Earth's radius. The coverage of aeroplane routes can be transferred to the coverage of the Earth's surface, resulting in an acceptable deviation of 0·1%.

Second, the SCN is required to maintain ISLs between adjacent satellites for as long as possible, and thus ISL type should be considered. ISLs are indispensable to ensure that ADS-B data can be relayed back to the GS in time, so the design of the SCN should make ISLs more stable and durable. As a result, a link establishment strategy should be proposed to reduce the probability of rerouting and data loss.

Third, the SCN should satisfy transmission requirements to ensure ADS-B data are routed back to the GS promptly, which means that at least one data packet from a certain plane is transmitted back successfully within the threshold time. As the signal collision problem always exists (Yu et al., Reference Yu, Chen, Li and Zhang2018), only part of the data packets of a plane can be decoded by receivers on-board. Thus, the network topology should be taken into consideration to achieve good transmission performance.

Finally, based on the analysis results, the design parameters should be determined to propose the SCN evaluation model.

3.2. Coverage of on-board ADS-B payload

Compared with a ground ADS-B receiver, an on-board receiver should satisfy the long-distance transmission link loss, and thus the sensitivity needs to be less than -95 dBm, with a medium-to-high gain antenna. The link budget in dB is given by

(1)$$P_t -L_t +G_t -L_f +G_r -L_r =P_r +\Delta _l$$

where each symbol's meaning and typical value are shown in Table 1. The free-space loss is computed by the formula in Table 1, in which d is the communication distance, and f is signal frequency and equals 1090 MHz. Using parameters in Table 1, the maximum communication distance can be calculated as 2,758 km.

Table 1. Link budget calculation symbols.

The coverage region of the payload is depicted in Figure 2. According to the geometric restriction, the half geocentric angle of coverage region α, the antenna half-beam angle β, the orbit altitude h and the ground elevation E have relations as follows:

Figure 2. Geometric relation of on-board ADS-B payload coverage region.

(2)$$\begin{align} \alpha &= \arccos [{Re}/{(h+Re)}\cos E]-E \end{align}$$

(3)$$\begin{align} \beta &=\arcsin [{Re}/{(h+Re)}\cos E] \end{align}$$

where Re is the equivalent radius of the Earth, and these two equations are deduced by the sine law $\sin \beta /Re=\cos E/( Re+h) $. Other relations can be obtained by using the geometric relations,

(4)$$\begin{align} E &= \arctan \{{[(h+Re)\cos \alpha -Re]}/{[(h+Re)\sin \alpha }]\} \end{align}$$

(5)$$\begin{align} d &= [ {Re^2+(h+Re)^2-2Re(h+Re)\cos \alpha}] ^{1/2} \end{align}$$

where d is the communication range at the beam coverage edge, in other words, the maximum ground–satellite communication range. The coverage width on the ground, i.e., w, is given by $w=2\alpha \cdot Re$.

In order to diminish the atmospheric scattering or diffraction effect as well as the shadowing and multipath by the Earth's surface, E needs a minimum value to ensure communication performance. When d is 2,758 km, the orbit altitude is approximately 800 km, and thus w can be as wide as 5,000 km. Compared with the value resulting from link budget, an orbit altitude below 800 km is suitable for the payload. Otherwise, satellites in orbit above 800 km need antennae with higher gain.

3.3. Constellation type and ISL establishment type

According to the global coverage requirements of air traffic surveillance, an LEO constellation in circular orbit is chosen, and coverage belt configuration (CBC) is adopted to analyse its coverage performance (Liang et al., Reference Liang, Xiao and Zhang2011). A CBC constellation utilises partially-overlapped coverage properties of mutual satellites in the same orbit to form a continual coverage belt on the ground, and on the basis of that, several coverage belts are tightly arranged to ensure global seamless coverage performance.

CBC can be divided into two types – co-rotating and counter-rotating – in terms of motion direction between adjacent orbits, as depicted in Figure 3, where $\Delta \omega_{f}$ is the phase difference of adjacent satellites in two adjacent orbits. The effective coverage region is between two dashed lines, and Δ₁, Δ₂ are the longtitude difference of the right ascension of ascending node (RAAN) between two adjacent orbits. The half geocentric angle width of the coverage belt, i.e., c, satisfies

Figure 3. Geometric relations of two types of CBC: co-rotating (left) and counter-rotating (right).

(6)$$c=\arccos [{\cos \alpha } /{(\pi/ S)}]$$

where S is the number of satellite in an orbit.

An ISL is defined as a link between two adjacent satellites established under a certain condition. Therefore, there are two types of ISLs among LEO satellites: constant and temporary. Apparently, adjacent satellites in the same orbit can establish constant ISLs with appropriate phase difference, while those in different orbits may establish temporary ISLs only. In a near-polar CBC constellation, temporary ISLs disconnect only when satellites are over a polar region and reconnect after passing it, which is a simple and consistent regularity subject to satellite orbit dynamics.

There are two types of establishment of intra-orbit ISLs – staggered and consistent – as shown in Figure 4. The staggered method establishes links with a changing phase difference (within a positive and a negative value) of neighbour satellites, while the consistent method establishes links with the same phase difference of neighbour satellites. These two types of link establishment, though, occasionally cause effects on ISL length and changing rate, which can influence the topology of the SCN. The staggered method is adopted due to its simple topology.

Figure 4. Two types of link establishment for intra-orbit ISLs: staggered method (left) and consistent method (right).

3.4. Network topology architecture

According to the architecture of the system, satellites can be treated as traffic sources regardless of the access segment. The GS regarded as the unique traffic sink is linked with one satellite above it by satellite-ground links (SGLs). Owing to satellites' regular movements, the virtual node (VN) (Ekici et al., Reference Ekici, Akyildiz and Bender2000; Henderson and Katz, Reference Henderson and Katz2000) is used to simplify the dynamic topology. Every VN represents its nearest satellite with a logic location address, and its location remains static relative to the inertial space. A satellite owns the logic location address while it is near a certain VN, and changes to another when it moves to the area near the next VN.

We label the space segment as a graph G (N, L), where N is the set of VNs, and L is the set of ISLs. We define $p[ h, \; v] \in N_{G} $ as a VN with a logical address, i.e., [h, v], where h is the orbit number and its value range is from 1 to P; v is the number of satellites within an orbit and varies from 1 to S. The orbit is numbered from west to east, and the satellite number within an orbit is numbered along the moving direction, as shown in Figure 5. It should be pointed out that VN can only simplify the dynamic topology of satellites but can never eliminate the SGLs' switch. Due to the Earth's rotation, traffic associated with a VN still varies with time.

Figure 5. Network topology architecture and VN logical address.

3.5. SCN design parameters

Based on the analysis above, five parameters are needed for SCN design: orbit plane number P, satellite number per orbit S, orbit altitude h, orbit inclination i and co-rotating longitude difference of RAAN Δ₁. The value range of the design parameters is roughly determined as follows. First, an LEO constellation requires an orbit altitude varying from 500 km to 1,000 km. A satellite below 500 km might be more susceptible to atmospheric resistance and thus unable to maintain its altitude (Richharia, Reference Richharia2014). On the other hand, a satellite above 1,000 km would require higher sensitivity of ADS-B payloads, which would increase its complexity and incur more severe signal collision within the coverage. An orbit above 1,500 km can also result in intense radiation by the Van Allen Belt (Richharia, Reference Richharia2014). Second, a near-polar constellation requires that the orbit inclination should be from 80^° to 100^°. Third, as the orbit planes' RAANs spread from 0 to π while the satellites' phase angles in the same orbit distribute from 0 to 2π, it is reasonable to admit that S ≈ 2P. Hence, S could be in the interval of [2P − 2, 2P + 2] without loss of generality. Fourth, when h is from 500 km to 1,000 km, α is from 17^° to 26^°, which means that four orbits at least are needed for global coverage and that more than seven orbits can induce redundancy. Therefore, it is reasonable to set P from four to six. Finally, Δ₁ should be adjusted based on P, and is roughly set from 30^° to 60^°. In the following analysis, Δ₁ has a minimum value when tight global coverage is satisfied. The design parameters and their value ranges are listed in Table 2. We then establish the evaluation model of SCN performance in terms of three aspects: coverage, link and transmission. The evaluation model and simulation process of SCN performance are outlined in Figure 6.

Figure 6. Evaluation model and simulation process.

Table 2. SCN design parameters.

4.1. Constellation coverage model

The global coverage performance of a SCN can be evaluated by investigating the degree of coverage at the equator, because a constellation of near-polar orbit has a lower satellite density on the equator than other regions. As a result, the following deduction is based on coverage of the equator circle, as shown in Figure 7. Given a set of design parameters, a coverage performance evaluation function can be established. We first assume that i = 90^°, then $\Delta \omega_{f}$ is just half of the phase difference of two adjacent satellites of the same orbit, according to Figure 3, and is given by

Figure 7. Coverage relation of a satellite constellation (i = 90^°).

(7)$$\Delta \omega _f =\pi/S$$

Δ₁ and Δ₂ should satisfy Equation (8) to meet the global coverage requirement according to Figure 3.

(8)$$\begin{cases} \Delta _1 \le \alpha +c \\ \Delta _2 \le 2c \\ \end{cases}$$

Therefore, as P orbits are required to cover the equator circle, Δ₁ and Δ₂ should also satisfy Equation (9), according to Figure 7

(9)$$(P-1)\Delta _1 +\Delta _2 =\pi$$

From Equations (8) and (9) we can obtain

(10)$${(\pi -2c)}/{(P-1)}\le \Delta _1 \le \alpha +c$$

where α and c can be calculated by Equations (2) and (6), which are related to h and S. Hence, if i = 90^°, Equation (10) can be used to determine whether the design parameters, i.e. [P, S, h, Δ₁], can satisfy the global coverage performance.

Then, if i is near 90^° varying from 80^° to 100^°, Figure 3 can be depicted as Figure 8, where α, c and 2π/s are the same as they are in Figure 3 and are not labelled. $\Delta^{\prime} _{1}$, $\Delta^{\prime}_{2}$ and $\Delta \omega^{\prime}_{f}$ are those quantities in the inclined orbit case. By using the spherical trigonometry according to the geometric relations shown in Figure 8, the expression of $\Delta\omega^{\prime}_{f}$ becomes

Figure 8. Geometric relations of inclined orbits.

(11)$$\Delta {\omega }'_f =\Delta \omega _f -\arctan (\cos i\tan {\Delta }'_1 )$$

and Equation (8) will thus become

(12)$$\begin{cases} {\Delta }'_1 \le \arcsin [{\sin \Delta _1 }/ {\sin i}]\\ {\Delta }'_2 \le \arccos [{(\cos \Delta _2 -\cos ^2i)}\ {\sin ^2i}] \\ \end{cases}$$

For simplicity, we substitute Equations (7) and (8) into (11) and (12), and we still use quantities without an apostrophe to represent all the cases including polar orbits as well as near-polar orbits. Combined with Equation (9), which is still true, Equation (8) can be deduced to

(13)$$\{\pi -\arccos [({\cos 2c-\cos ^2i)}/ \sin^2i]\}/(P-1)\le \Delta _1 \le \arcsin [{\sin (\alpha +c)} /{\sin i}]$$

which implies the following inequation

(14)$$(P-1)\arcsin [{\sin (\alpha +c)}/ {\sin i}]+\arccos [({\cos 2c-\cos ^2i)} / {\sin ^2i]}\}\ge \pi$$

Therefore, the coverage performance evaluation function can be defined as

(15)$$f_{cov} (P,S,h,i)=(P-1)\arcsin [{\sin (\alpha +c)} /{\sin i}]+\arccos [({\cos 2c-\cos ^2i)}/{\sin^2i]}\}$$

where α and c are determined by S, h and i. If $f_{cov} ( P, \; S, \; h, \; i) \ge \pi $ and Δ₁ satisfies Equation (13), the SCN is considered to have global coverage performance.

4.2. ISL performance model

The ISL's performance is difficult to quantify but can be evaluated by range d, elevation angle θ, and azimuth angle ϕ, as well as their changing rates $\dot{d}$, $\dot{\theta }$ and $\dot{\phi}$. Range mainly affects signal attenuation and latency, and thus the data transmission rate. Range changing rate can induce a Doppler effect that might reduce the quality of the received signal. Elevation angle and azimuth angle themselves do not affect the ISL's performance, though their changing rates may bring about complex problems; excessive $\dot {\theta }$ and $\dot {\phi }$ make it hard to track and aim the target for the antenna and precisely control the attitude for the satellite platform.

ISL between satellite A and satellite B, which are arbitrarily chosen, is depicted in Figure 9, where γ is the geocentric angle from A to B, and $\Delta \lambda$ is the longitude difference of RAAN between A and B. We suppose that the orbit eccentricity is 0; orbit inclination is i; and orbit period is T. The initial orbit phase angles of A and B are set to be ω₁ and ω₂, respectively.

Figure 9. Geometric relation of inter-orbit ISL.

As intra-orbit ISLs are almost constant and have slight effects on SCN end-to-end link paths, they will not be discussed further and the two satellites are deemed to be in different orbits. When it comes to inter-orbit ISLs, the longitude difference of RAAN between two adjacent satellites of adjacent orbits satisfies $\Delta \lambda = \Delta _{1}$, and ω₁, ω₂ have a relation of $\omega _{2} -\omega _{1} = \pm \Delta \omega _{f}$ as they defined, which means we can set $\omega _{1} = 0, \; \omega _{2} = \Delta \omega _{f}$. We can obtain the expression of γ in Liang (Reference Liang2006),

(16)$$\begin{align} \cos \gamma (t) & =[\cos ^2(\Delta _1 /2)-\cos ^2i\sin ^2(\Delta _1/2)]\cos \Delta \omega _f \\ &\quad +\cos i\sin \Delta _1 \sin \Delta \omega _f -\sin ^2i\sin^2(\Delta _1 /2)\cos (2ft+\Delta \omega _f ) \\ \end{align}$$

which is related to i, Δ₁, f and $\Delta \omega_{f}$. In the equation, f is the satellite orbit angular rate and thus f = 2π/T. According to Liang (Reference Liang2006), the ISL range can then be given by

(17)$$d=2(h+Re)\sin (\gamma (t)/2)$$

The elevation angle and azimuth angle are also given by

(18)$$\begin{align} \theta &=-\gamma (t)/2 \end{align}$$

(19)$$\begin{align} \tan \phi & =[\sin i\sin \Delta _1 \cos (ft+\Delta \omega _f )-\sin (2i)\sin ^2(\Delta _1 /2)\sin (ft+\Delta \omega _f )] \nonumber\\ & /\{\sin ^2i\sin ^2(\Delta _1 /2)\sin (2ft+\Delta \omega _f )+\cos i\sin \Delta _1 \cos \Delta \omega _f \nonumber\\ & +[\cos ^2(\Delta _1 /2)-\cos ^2i\sin ^2(\Delta _1 /2)]\sin \Delta \omega _f \} \end{align}$$

Thus, $\dot {d}, \; \dot {\theta }, \; \dot {\phi }$ can be deduced by Equations (17)–(19). ISL range d can be deduced by combining Equations (16) and (17), and thus will be related to SCN design parameters, i.e. [P, S, h, i,Δ ₁]. As link range directly influences the network latency, it will be further discussed in the following sections.

4.3. Transmission performance model

Based on the VN method, the network transmission model can be established. In general, network evaluation index includes throughput, end-to-end latency etc. The throughput is often affected by traffic congestion and thus is computed by simulation. The end-to-end latency consists of transmission latency, processing latency and queuing latency, and both the processing latency and the queuing latency are subject to actual condition of the SCN, which relies on the network protocols. As the transmission latency is only related to network topology and is determined by ISL range given by Equation (17), it is reasonable to adopt an end-to-end link range composed of ISL ranges as the evaluation index for network transmission performance. However, the link range between two non-adjacent satellite nodes can be calculated in different ways, because there are several effective paths with multiple hops between them. As a result, the shortest path's link range is chosen to measure the node-to-node distance and hence the network transmission performance.

We assume that p ₁ and p ₂ are two VNs with logic address $[ {h_{1}, \; v_{1}}] $, $[ {h_{2}, \; v_{2} }] $ respectively, and hop(p ₁, p ₂) denotes minimum hops from p ₁ to p ₂. We then have $hop( p_{1}, \; p_{2}) =hop( p_{2}, \; p_{1}) $, and the minimum hops are given by Ekici et al. (Reference Ekici, Akyildiz and Bender2000) if we regard the orbit as a round ring and thus v ₁ = 1 and v ₂ = S are neighbours. Therefore, we have the following equation, according to Figure 5,

(20)$$hop(p_1, p_2 )=\vert {h_1 -h_2 } \vert +\min \{ {\vert {v_1 -v_2 } \vert, S-\vert {v_1 -v_2 } \vert } \} $$

Let p ₁ denote the source VN, and p ₂ denote the access VN of the GS. As the GS changes its access VN with time, the minimum hops from p ₁ to p ₂ are time varying because of the Earth's rotation. We assume that the logic address of p ₁ is $[ {i, \; j}] $, and that of p ₂ is $[ {h( t), \; v( t) } ] $, then

(21)$$hop_{i,j} (t) = hop_{i,j}^H (t)+hop_{i,j}^V (t)$$

where $hop_{i,j}^H (t) \buildrel \Delta \over = \left| {i-h(t)} \right|$, $hop_{i,j}^V (t)\buildrel \Delta \over = \min \left\{ {\left| {j-v(t)} \right|,S-\left| {j-v(t)} \right|} \right\}$. The sum of $hop_{i,j} (t)$. The sum of hop _{i, j} (t) throughout the network is given by

(22)$$\sum_{j=1}^S {\sum_{i=1}^P {hop_{i,j} (t)} } = \sum_{j=1}^S {\sum_{i=1}^P {[ {hop_{i,j}^H(t)+hop_{i,j}^V (t)} ] } }$$

which is the global minimum hops of all the satellites to GS at time t, if we ignore the final SGL hop.

As polar region can affect the link range, we define Φ_p as the threshold of polar region. The GS is not located in a polar region, so only the source the VN's location needs to be considered. If p ₁ is not in a polar region, its latitude denoted by Φ_j satisfies $\Phi _{j} \notin [ \Phi _{p}, \; \pi /2] \cup [ -\pi /2, \; -\Phi _{p} ] $, and the minimum path's link range is given by

(23)$$len_{i,j} (t)=hop_{i,j}^H (t)\cdot L_H \min \{ {\cos \Phi_j, \cos \Phi_d } \} +hop_{i,j}^V (t)\cdot L_V$$

where Φ_d are the latitude of p ₂; L _H is inter-orbit ISL range over the equator; L _V is intra-orbit ISL range and remains constant.

On the other hand, when p ₁ is in a polar region, len _{i, j} (t) becomes

(24)$$len_{i,j} (t)=hop_{i,j}^H (t)\cdot L_H \cos \Phi _p +hop_{i,j}^V (t)\cdot L_V$$

Equations (23) and (24) can be combined by defining a latitude factor

(25)$$f_{_j }^\Phi = \begin{cases} {\min \{ {\cos \Phi_j, \cos \Phi_d } \}, \vert {\Phi_j } \vert \le \Phi _p } \hfill \\ {\cos \Phi _p, \mbox{ }\vert {\Phi_j } \vert >\Phi _p } \hfill \\ \end{cases}$$

and thus len _{i, j} (t) is given by

(26)$$len_{i,j} (t)=hop_{i,j}^H (t)\cdot L_H \cdot f_{_j }^\Phi +hop_{i,j}^V (t)\cdot L_V$$

To evaluate network performance, the average of len _{i, j} (t) of all the satellites over time should be calculated. We have this following equation about the average path link range (APLR),

(27)$$\frac{1}{S}\sum\limits_{j=1}^S {\frac{1}{P}\sum\limits_{i=1}^P {\langle{len_{i,j} (t)} \rangle } } =\frac{L_H }{S\cdot P^2}\sum\limits_{j=1}^S {\sum\limits_{i=1}^P {\sum\limits_{k=1}^P {f_j^\Phi \vert {i-k} \vert } } } \mbox{+}\frac{L_V }{S}\sum\limits_{r=1}^S {\min \{ {r,S-r} \} }$$

where $\langle\bullet \rangle $ is defined as $\langle{f( t) } \rangle \buildrel \Delta \over = \frac{1}{T} \int_{0}^{T} {f( t) dt} $, in which f(t) is a periodic function with a period T. Equation (27) can be proved by proposing a network access stationary process (NASP) theorem, which is illustrated in detail in Appendix A below.

5.1. Coverage performance evaluation

Under the constraint of minimum ground elevation E, if P, S and i are given, the minimum h can be obtained by solving $f_{cov} ( P, \; S, \; h, \; i) =\pi $ by iteration, and thus Δ₁ has the only value determined by Equation (12). Figure 10 shows that h varies with i when minimum E is 5^°. It is apparent that for a given set of P and S, h equals maximum when i is 90^°, and it does not change much with i varying from 80^° to 100^°.

Figure 10. Relationship between minimum h and i under restriction of global coverage (E = 5^°). (a) orbit number P = 4. (b) orbit number P = 5. (c) orbit number P = 6.

According to Figure 10(a), when P = 4, h is always greater than 1,000 km for every S, which is beyond the value range of h. If S is greater than 10, the altitude can decrease to below 1,000 km but the total number of satellites increases relatively. Therefore, the following sections will not consist of that case: in Figure 10(b), when P = 5, h is in the required interval for every S except S = 8; and in Figure 10(c), all of the design parameters are acceptable.

As i has just a slight effect on coverage performance, it is set to be 90^° temporarily for simplicity. We take E to be 5^°, 8^° and 10^°, and then obtain the relationship between minimum h and total number of satellites as depicted in Figure 11, where the Iridium-Next is also plotted with satellite number of 66 and altitude of 780 km. As the total number of satellites increases, the orbit altitude needed to ensure global coverage decreases. Furthermore, E increases as the orbit altitude is bigger if the total satellite number is set to constant. On the other hand, E also increases as the total number of satellites increases if the orbit altitude is set to constant. The discontinuity is caused by different configurations of satellite constellation, which means that there are two configurations of different P and S, but having the same total satellite number, i.e., 60.

Figure 11. Required minimum orbit altitude varies with total number of satellites.

5.2. Link performance evaluation

On the basis of coverage performance evaluation, Equations (16) and (17) are used to calculate the ISL range varying with time. As the intra-orbit ISL remains constant, only the inter-orbit ISL is considered and is simplified as ISL for brevity.

The maximum hops between two satellites in a period can be calculated by Equation (20), and are given by

(28)$$hop_{\max } (p_1, p_2)= P+\lfloor{S/2} \rfloor$$

where $\lfloor\bullet \rfloor$ means downward rounding. The path link ranges of the maximum hops are

(29)$$L_{\max } =P\cdot l_h +\lfloor{S/2} \rfloor \cdot l_v$$

and the relationship of L _max and satellite number is illustrated in Figure 12, where Iridium-Next is also plotted. As shown, the greater S is, the shorter L _max is.

Figure 12. Relationship between the maximum path link range and the number of satellites.

When the total number of satellites is given, the greater P is, the longer L _max is. Therefore, the constellation of P = 5 and S = 11 has the best performance considering both the link range and satellite number.

We take the constellation of P = 5 and S = 11 as an example to evaluate the effect of inclination on ISLs. The statistical results of ISL range, including the maximum, the mean and the minimum, are shown in Figure 13. As the orbit inclination angle i increases, the maximum ISL range increases at first and then decreases, and it reaches the peak when i = 92^°. In contrast, the mean and the minimum ISL range always increase with i. Because ISL range affects signal attenuation and latency, smaller orbit inclination with shorter ISL range can have better performance.

Figure 13. Effect of inclination on ISL range: (a) maximum ISL range, (b) mean ISL range, (c) minimum ISL range.

The maximum changing rates of ISL range, azimuth and elevation are shown in Figure 14 to illustrate better the effect of orbit inclination. As i increases, the maximum changing rate of ISL azimuth decreases while that of ISL range and elevation increases slightly and then decreases. The maximum changing rates of range, azimuth and elevation vary quite little for less than 5%, as the orbit inclination increases if we ignore the ISL's change in polar regions. It can be concluded that the orbit inclination has small effects on ISL performance in our design, and smaller inclination can decrease ISL range to some extent according to Figure 13.

Figure 14. Effect of inclination on ISL changing rates: (a) ISL azimuth rate, (b) ISL elevation rate, (c) ISL range rate.

5.3. Transmission performance evaluation

Based on the former analysis results, we use APLR to evaluate the transmission performance. As the latitude of the GS can affect the APLR, we choose four different sites and the results are shown in Figure 15, where the Iridium-Next is also plotted.

Figure 15. APLR varies with constellation parameters for different GS latitudes: (a) GS latitude = 0^°, (b) GS latitude = 10^° N, (c) GS latitude = 30^° N, (d) GS latitude = 50^° N.

If the GS is in low latitudes, APLR generally decreases with satellite number, increasing for each P as shown in Figure 15(a) and (b). As for configurations of 60 satellites, P = 5 has a lower mean route distance than P = 6, which is due to the mean hops and the ISL range. On the other hand, if the GS is in higher latitudes such as 30^° N and 60^° N, the relation between the APLR and the number of satellites becomes more complex, as shown in Figure 15(c) and (d). The APLR fluctuates as the number of satellites increases, and more satellites cannot ensure lower mean route distance. Therefore, it becomes an interesting problem to select the proper position for GS when considering the APLR. For the constellation of the same satellite number, the APLR decreases as the GS latitude increases, because inter-orbit ISLs in higher latitudes are more likely to be chosen for transmission downward to the ground when GS is at higher latitude. Besides, it is apparent that the Iridium-Next always has a much more APLR than our design prototypes. If we choose Changsha (113^° E, 28·2^° N) as the GS site, constellations of P = 5, S = 11 and of P = 5, S = 12 have shorter APLR than others, and then have better transmission performance. Furthermore, it is obvious that the Iridium-Next always has a greater mean route distance than the configuration of nearly the same number of satellites.