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Example 10.8, Orbital Mechanics for Engineering Students, 2nd Edition.

A satellite is to be completely despin using a two-mass yo-yo device with tangential release. Assume the spin axis of moment of inertia of the satellite is C = 200 kg · m^2 and the initial spin rate w0 = 5 [rad/s]. The total yo-yo mass is 4 kg, and the radius of the spacecraft is 1 meter.

```clc;
C  = 200;       % [kg*m^2]
m  = 4;         % [kg]
R  = 1;         % [m]
K = 1 + C/(m*R^2); %     Nondimensional factor K```

The required cord length l

```l = R*(K)^0.5       %[m]
% Time t to despin
t = (K)^0.5/w0      %[s]
% Tension in the yo-yo cables
t = 0:0.001:5;
N = 2*K*C*w0^3*t./(R*(K + w0^2*t.^2).^2);
plot(t,N);
xlabel('time [s]');
ylabel('Tension N [N]');
grid on;
[maxN,i] = max(N);```
```l =

7.1414

t =

1.4283```

The maximum tension in the yo-yo cables [N];

```
maxN
% which ocurs at
t = i*0.001  % [s]```
```maxN =

454.7542

t =

0.8260```

The speed of the masses at release;

`v = R*w0*(K)^0.5  % [m/s]`
```v =

35.7071```

The angle rotated by the satellite during the despin;

`theta = (pi/2 - 1)*(K)^0.5   % [rad]`
```theta =

4.0763```

Cord length required for complete despin

`l = R*(K^0.5 - 1)            % [m]`
```l =

6.1414```

Published with MATLAB® 7.10

Contents

Consider the following example of a spacecraft telemetry link for which we wish to find the maximum range. Suppose we have a 1 W telemetry transmitter onboard a spacecraft and that the data rate requires a channel capacity corresponding to a signal-to-noise ratio of at least unity in a 1 Hz bandwidth. This link uses a frequency of 3 GHz (10 cm wavelength). The transmitting antenna on the spacecraft is a 2 m diameter dish. The ground station antenna is a 10 m diameter dish. Assume that both these dish antennas have an eective area equal to 60% of their physical apertures. Assume also that there is no pointing error, that is, the antennas always point directly at each other and that the system temperature of the ground station receiver is 25 K. (The system temperature is the sum of the equivalent receiver noise temperature, the antenna noise temperature and the sky noise temperature.) Boltzmann constant k = 1.38*10-23 W/(Hz*K)

```clc;
clear all;
k    = 1.38e-23;
c    = 3e8;         % Speed of Light [m/s]
F    = 3;           % Operating Frequency[GHz]
lm   = c/(F*10^9);  % Wavelength at the operating frequency  [m/s]
E_gs = 10;          % Ground Station antenn diameter [m]
E_sc = 2 ;          % Spacecraft antenn diameter [m]
eff  = 0.6;         % 60[%] Aperture Efficiency```

1. What is the equivalent input Noise Power of the receiver[dBm]?

```%N = 10*log10()
Te = 25;  % System (receiver) noise temperature in [K];
B = 1;    % System noise bandwidth in              [Hz]
N = 10*log10(k*Te*B); % or
N = -228.6 + 10*log10(Te) + 10*log10(B);         % [dBW]
N = N + 30;                                      % [dBm]
fprintf('Equivalent input Noise Power of the receiver %4.2f [dBm] \n\n',N);```
`Equivalent input Noise Power of the receiver -184.62 [dBm]`

2. What is the Effective Area of the receiving antenna [m2]?

```A_gs = pi*E_gs^2/4;        % Area of antenna aperture
G_gs = 20*log10(E_gs) + 20*log10(F) + 20*log10(10*pi/3*sqrt(eff));
% or
G_gs = 10*log10(4*pi*A_gs*eff/lm^2); % [dBi]
Aeff_gs = 10^(G_gs/10)*lm^2/(4*pi);          % [m^2]
fprintf('Gain of the receiving antenna = %4.2f [dBi] \n',G_gs);
fprintf('Effective Area of the receiving antenna = %4.2f [m^2]\n\n',Aeff_gs);```
```Gain of the receiving antenna = 47.72 [dBi]
Effective Area of the receiving antenna = 47.12 [m^2]```

3. What is the gain of the transmitting antenna [dBi]?

```A_sc = pi*E_sc^2/4;                  % Area of SC antenna aperture
G_sc = 10*log10(4*pi*A_sc*eff/lm^2); % [dBi]
Aeff_sc = 10^(G_sc/10)*lm^2/(4*pi);          % [m^2]
fprintf('Gain of the transmitting antenna = %4.2f [dBi] \n',G_sc);
fprintf('Effective Area of the transmitting antenna = %4.2f [m^2]\n',Aeff_sc);```
```Gain of the transmitting antenna = 33.75 [dBi]
Effective Area of the transmitting antenna = 1.88 [m^2]```

4. What is the maximum range, R [km] for the spacecraft to maintain the required signal-to-noise ratio?

```L = -(N-30); % dBw maximum path Loss
Power = G_sc + G_gs; %dBi
R = 10^((Power + L - 92.4 - 20*log10(F))/20);            %[km]
fprintf('Maximum range, R = %4.2d [km]\n\n',R);```
`Maximum range, R = 5.10e+009 [km]`

## Coherent-Frequency Turnaround

Consider a spacecraft transponder which operates in frequency-coherent mode. The transmitted uplink frequency is 2 GHz (S-Band), and the downlink frequency is in X-Band. The turn-around ratio r inside the spacecraft transponder is 880/221 (downlink-freq/uplink-freq). The assumed velocity of the spacecraft relative to the groundstation is 10km/sec. Assume further that the downlink telemetry (TM) rate of 10 kbit/sec is modulated onto a sub-carrier with frequency 150 kHz.

1. Calculate the Doppler shift[kHz] on the uplink and downlink carriers, which are received by the transponder and the groundstation,respectively.

```f_uplink   =  2;                            %[GHz]
f_sub = 150;                                %[kHz]
v_sat = 10000;                              %[m/s]
dr = 10;                                    %[kbit/sec]
```Doppler shift Uplink Carrier  =  66.67 [kHz]
Doppler shift Downlink Carrier = 265.46 [kHz]```

2. Calculate the Doppler shifts on the received downlink sub-carrier[Hz], and the TM data rate[bit/s]

```df_sub= v_sat/c*f_sub*10^3;                %[Hz]
fprintf('Doppler shift Sub-Carrier = %4.2f [Hz] \n',df_sub);
% Assuming that the spacecraft was opserved during the recession
% Shifted frequency
f_shif = f_sub - df_sub;
%TM data shift rate[bit/s]
dr_sub= v_sat/c* dr*1000;
fprintf('Downlink telemetry (TM) rate shift = %4.2f [bit/sec] \n',dr_sub);```
```Doppler shift Sub-Carrier = 5.00 [Hz]
Downlink telemetry (TM) rate shift = 0.33 [bit/sec]```

Published with MATLAB® 7.10