Question

The mean diameters of X and Y, two planets in the same solar system, are 6.4 ? 10³ km and 1.8 ? 10⁴ km, respectively. The mass of X is 0.24 times the mass of Y. The value of g on Y is 8.4 m/s².

(a) What is the ratio of the mean density of X to that of Y?

?_X

?_Y

=  

(b) What is the value of g on X?
m/s²

(c) The mass of Y is 1.020 ? 10²⁵ kg. What is the escape speed on X?
m/s

Answer 1

R(Y) = 1.8 ? 10^4 km / 2 = 9 ? 10^3 km

R(X) = 6.4 ? 10^3 km / 2 = 3.2 ? 10^3 km

a) density goes as M/R^3

take ratio of X density to Y density:

rho(x)/rho(Y)=M(X)/M(Y) * (R(Y)/R(X))^3
rho(X)/rho(Y)=0.24*(9 ? 10^3/3.2 ? 10^3)^3= 5.34

b) g goes as M/R^2, so again take ratios

g(X)/g(Y)=M(X)/M(Y)*(R(Y)/R(X))^2 = 0.24*(9 ? 10^3/3.2 ? 10^3)^2 = 1.9  

so the value of g on X =1.9*8.4m/s/s = 15.95 m/s/s

c) esc speed goes as

vesc=sqrt[2GM/R]

ratios:

vexc(X)/vesc(Y)= sqrt[(M(x)/M(Y) * R(Y)/R(X)] =
sqrt[0.24*(9 ? 10^3/3.2 ? 10^3)]=
sqrt[0.675]=0.822

vesc(X)=0.822vesc(Y)
vesc(Y)=sqrt[2GM(Y)/R(Y)]= 12.296 km/s

so vesc(X)=0.822x12.296km/s= 10.12 km/s