Question

Consider an artificial satellite moving around the Moon in an elliptic orbit. The altitude of the satellite from the Moon's surface at the perigee is 25 km and at the apogee is 134 km. Assume the Moon to be spherical with a radius of 1737 km. The trajectory is considered with reference to a coordinate system fixed to the center of mass of the Moon. The ratio of the speed of the satellite at the perigee to that at the apogee is .......... (rounded off to 2 decimal places).

Hint:

In elliptical orbits, use the vis-viva equation for velocity. The ratio simplifies to depend only on $r_p, r_a$ and $a$, without requiring $\mu$ (gravitational parameter).

Answer 1

Step 1: Convert altitudes into distances from the Moon's center.
Moon's radius = \(1737 \, \text{km}\). \[ r_p = 1737 + 25 = 1762 \, \text{km} \] \[ r_a = 1737 + 134 = 1871 \, \text{km} \] Step 2: Use vis-viva equation.
The orbital velocity at a distance \(r\) is: \[ v = \sqrt{\mu \left( \frac{2}{r} - \frac{1}{a} \right)} \] where \(a\) = semi-major axis = \(\frac{r_p + r_a}{2}\). \[ a = \frac{1762 + 1871}{2} = 1816.5 \, \text{km} \] Step 3: Ratio of perigee to apogee speeds.
\[ \frac{v_p}{v_a} = \sqrt{ \frac{ \frac{2}{r_p} - \frac{1}{a} }{ \frac{2}{r_a} - \frac{1}{a} } } \] Substitute values (in km): \[ \frac{v_p}{v_a} = \sqrt{ \frac{ \frac{2}{1762} - \frac{1}{1816.5} }{ \frac{2}{1871} - \frac{1}{1816.5} } } \] Numerator: \[ \frac{2}{1762} - \frac{1}{1816.5} = 0.0011357 - 0.0005503 = 0.0005854 \] Denominator: \[ \frac{2}{1871} - \frac{1}{1816.5} = 0.001069 - 0.0005503 = 0.0005187 \] \[ \frac{v_p}{v_a} = \sqrt{\frac{0.0005854}{0.0005187}} = \sqrt{1.1285} \approx 1.06 \] \[ \boxed{1.06} \]