Question

An Asteroid with its own Moon: The asteroid 243 Ida has its own small moon, Dactyl. (See the photo on p 394) Another such system, asteroid 624 Hektor, has a mass of 7.9 x 10^18, with its moon orbiting at a radius of 623.5 km and a period of 2.965 days. A) Given that the orbital radius of Dactyl is 108 km, and its period is 1.54 days, is the mass of 243 Ida greater than, less than ore equal to the mass of 624 Hektor? B) Calculate the mass of 243 Ida

The answer provided by the book is:

A): less than

B): 1.5 x 10 ^17 kg

Please provide direction as to how to get to this solution. Thanks

Answer 1

Please approach as following.

From the data of 624 Hektor, calculate G, universal gravitational constant. To do that, apply Kepler's 3rd law of planetary motion, which is written as follows.

T is period, r is orbiting radius, G is universal gravitational constant and M is mass.

Therefore, ........(1)

Substitute the above quantity values for 624 Hector.

Given, M =7.9*10^18 kg , r =623.5 km=623.5*10³ m, T=2.965 days = 2.965*24*60*60 sec

Putting these values in equation (1) and after calculations, we get   

Now, you have to use this value of G to calculate mass of 243 lda.

We need to use equation 1 again.

For 243 lda, given quantities are , r (dactyl) = 108 km = 108*10³ m , T(dactyl)= 1.54 days = 1.54 *24*60*60 sec,

We have ,

Using these values in equation (1),    , M (243 lda) can be calculated from the equation below.

.........(2)   

After calculations, M (243 lda) = 1.5 * 10¹⁷ kg (Answer of part B)

Comparing this with mass of Hektor 624 (7.9*10¹⁸ kg) ,​ mass of 243 lda is less than that of 624 Hektor. (Answer of part A).