Question

3) Consider the planet Mercury which has mass 3.34 x1023 kg, radius 2.44x10 m, semimajor axis 5.79x1010 m, and eccentricity .206. a) What is the acceleration due to gravity on the surface of Mercury? b) What is the escape velocity from the surface of Mercury? c) Calculate the force exerted on Mercury by the sun at its perihelion and appihelion. d) What are the perihelion and appihelion velocities of Mercury? e) What is Mercury's angular momentum at these two locations? f) If a Mercury probe of mass 5000 kg has a circular orbit around Mercury at a distance of 1.50 Mercury radii above the planet's surface, what is the velocity, energy and orbital period of the probe? g) If the probe in part f) increases its velocity by 20 %, what will be its new semimajor axis?

Answer 1

(a)

$g=\frac{GM}{r^2}=\frac{6.67\times3.34}{2.44^2}\times10^{-11+23-12}=3.74\textup{ m/s}^2$

(b)

By energy conservation,

$\frac{1}{2}mv_e^2-\frac{GMm}{r}=0$

$v_e=\sqrt{\frac{2GM}{r}}=\sqrt{2gr}\approx4.27\textup{ km/s}$

(c)

The forces are

$\frac{GM_sM}{a^2(1-e)^2}, \frac{GM_sM}{a^2(1+e)^2}$

at perihelion and appihelion respectively. Where Ms is sun's mass and a is semi-major axis.

Just put in the numbers with

$M_s=2\times10^{30}\textup{ kg}$

to get

$2.11\times10^{22}\textup{ N}, 9.15\times10^{21}\textup{ N}$

(d)

Conservation of angular momentum and energy

$Mv_pr_p=Mv_ar_a\Rightarrow v_p(1-e)=v_a(1+e)$

$\frac{1}{2}v_p^2-\frac{GM_s}{a(1-e)}=\frac{1}{2}v_a^2-\frac{GM_s}{a(1+e)}$

Solve the two equations to get

$v_p=\sqrt{\frac{GM_s}{a}\left (\frac{1+e}{1-e} \right )},v_a=\sqrt{\frac{GM_s}{a}\left (\frac{1-e}{1+e} \right )}$

(e)Angular momentum at any position is same

$L=M\sqrt{GM_sa\left (1-e^2 \right )}$