Question

A spacecraft flies over Mars at a geopotential altitude of 80 kilometres. Determine its geometric altitude (in metres).

Answer 1

The geopolitical altitude refers to the height of an object above a reference potential surface, often approximated using a planet's mean sea level. For Mars, the geopotential altitude can be converted to geometric altitude by considering the differences in gravity between ground level and higher altitudes.

The gravitational acceleration on Mars is approximately 3.72076 m/s² at the surface and is slightly lower at higher altitudes. To convert from geopotential altitude to geometric altitude, we can use the formula:

\[
h_g = \frac{h}{1 - \frac{h}{R}}
\]

Where:
- \( h_g \) is the geometric altitude,
- \( h \) is the geopotential altitude,
- \( R \) is the radius of Mars (approximately 3,389.5 km or 3,389,500 m).

Using \( h = 80 \) km = 80,000 m, we have:

1. Convert the radius of Mars to meters: \( R = 3,389,500 \) m.
2. Apply the formula:

\[
h_g = \frac{80,000}{1 - \frac{80,000}{3,389,500}}
\]

First, calculate the fraction:

\[
\frac{80,000}{3,389,500} \approx 0.0236
\]

So,

\[
1 - 0.0236 \approx 0.9764
\]

Now, substitute this back into the formula:

\[
h_g = \frac{80,000}{0.9764} \approx 81,793.73 \, \text{m}
\]

Therefore, the geometric altitude of the spacecraft is approximately **81,794 metres**.