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Starship-class 2000kg satellite
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Economic equations
Required laser pulse energy
Hancock et al (2021) provide the equation below for the required laser pulse energy that has to be transmitted from a LiDAR system to achieve a sufficient return.
$$
E_{shot} = \frac{E_{det}}{Q} \frac{\pi h^{2}}{A} \frac{1}{\rho \tau^2}
$$
Where:
- $E_{shot}$ is the required laser pulse energy.
- $E_{det}$ is the required energy that reaches the detector = 2.8110^-16 J. We go with this figure to match GEDI, which can reliably retrieve ground data in forests with 96% canopy cover by day with 0.281 fJ (2.8110^-16 J).
- $Q$ is the detector efficiency = 0.7. While Icesat-2 had a detector efficiency of 0.15 and GEDI 0.45, a detector efficiency of 0.7 should be doable in space.
- $h$ is the instrument altitude = 450,000m for the Falcon-class satellite and 300,000m for the Starship-class satellite.
- $A$ is the telescope area = 0.196sqm, which is the area of a 50cm circle, for the Falcon-class satellite. We use the area of a 1m circle, or 0.785sqm for the Starship-class satellite.
- $\rho$ is the surface reflectance = 0.4.
- $\tau$ is the atmospheric transmittance = 0.8.
Laser safety
It is important that our LiDAR satellites do not cause any harm to people on the ground. For the sake of simplicity, we will assume there is no overlap in any points on the ground as our satellites scan the Earth. As such, we will only consider the eye safety implications of a single pulse hitting the eye.
We use the linked resources to build our equations. We start with the ground radiant exposure per pulse:
$$
H_1 = \frac{E_{shot} \tau \eta}{A}
$$
Where:
- $H_1$ is the ground radiant exposure per pulse.
- $E_{shot}$ is the laser pulse energy. This will depend on the satellite architecture.
- $\tau$ is the atmospheric transmittance = 1, conservatively.
- $\eta$ is a term for any other losses = 1, conservatively.
- $A$ is the area of the ground spot = ~0.785m^2.
We then look at the Maximum Permissible Exposure (MPE). A commonly cited exposure limit calculation for laser pulses between 10ps to 1ns is as follows:
$$
MPE_{single} = 5.5 t^{3/4}
$$
Where:
- $MPE_{single}$ is the maximum permitted exposure for a single pulse, measured in J/cm^2
- $t$ is the pulse width = 2ns for 30cm range resolution (Falcon-class satellite), and 0.67ns for 10cm range resolution (Starship-class satellite). While 2ns is greater than the 1ns pulse length limit for this equation, using this equation results in a more conservative calculation.
From there, we compare $H_1$ with $MPE_{single}$. If $H_1$ is greater than $MPE_{single}$, the LiDAR is considered eye safe. Vice versa, if $H_1$ is less than $MPE_{single}$, the LiDAR is considered not eye safe. Ideally, $H_1$ should be much less than $MPE_{single}$.
Resulting swath width
Hancock et al (2021) also provides the equation for the average swath width a satellite can achieve, given power available for the LiDAR payload. The actual swath can be wider, though it then means that the satellite will need to store up power to operate the LiDAR. For example, a 1m average swath implies that the satellite can save up power for 999s to achieve 1000m swath for 1 second.
$$
s = \frac{P_{pay} L_e}{E_{det}} \frac{A}{ \pi h^2} Q \rho \tau^2 \frac{r^2 (R+h)^{3/2}}{R \sqrt{GM}}
$$
Where:
- $s$ is the resulting swath width
- $P_{pay}$ is the power available to the LiDAR payload = 400W for the Falcon-class satellite, and 10,000W for the Starship-class satellite.
- $L_e$ is the wall-to-plug efficiency of the LiDAR’s laser/s = 15%, which was believed to be feasible by NASA in 2010 and should be feasible today or in the very near future with fibre lasers.
- $R$ is the radius of the Earth = 6,371,000m.
- $G$ is the gravitional constant = 6.67E-11 m^3/kg/s^2.
- $M$ is the mass of the Earth = 5.97E+24 kg
- $r$ is the resolution = 1m, allowing us to reach resolution similar to that of aerial LiDAR.