Kepler's Law: constant, mean distance, ratio

[jlaw]

Hi E-Instructors,

Would someone kindly tell me if my answers is anywhere close to being right (and/or how to get there) and fly me in the right direction for the last question?

Problem:
A result of Kepler's Harmonic Law is that the mass of a planet (m)
with a satellite is directly proportional to the cube of the mean distance (d)
from the satellite to the planet and inversely proportional to the square of the period of revolution (p).
Early astronomers estimated the mass of the earth to be 5.976 x 10^24 kg
and observed that the moon orbited the earth with a period of 27.322 days
at a mean distance of 384.4 x 10^3 km.

1. Write a formula for the mass of a planet according to Kepler.
I think it's
m = k*d^3........I used the letter k for the constant.
p^2

2. Find the proportionally constant using the observations and estimates for the earth and moon.

5.976 * 10^24 = k * (384.4 * 10^3)^3
........................ 27.322^2

5.976 * 10^24 = k * (56800235.584 * 10^9)
........................ 746.492

5.976 * 10^24 = k * (5.68 * 10^16)
........................ 746.492

5.976 * 10^24 = ____k_________
....................131.424 * 10^16

5.976 * 10^24 = ______k_____
....................1.31 * 10^18

7.829 * 10^42 = k ??? I'm I close?

3. Find the mass of Mars based on observations that Phobos orbits Mars in 7.65 hours at a mean distance of 9330 km.
If the above is right then would I just write the formula like this:

m = k * (d)^3
........p^2

m = 7.829 * 10^42 * (9330)
.............7.65^2

Then would it be solved the same way as question # 2? :idea:

4. What is the approximate ratio of the mass of the earth to the mass of Mars?
I do not know how to start this one... :?:

Thank you for your always appreciated help.
Sincerely,
Julia

 

[Deleted member 4993]

A result of Kepler's Harmonic Law is that the mass of a planet (m)
with a satellite is directly proportional to the cube of the mean distance (d)
from the satellite to the planet and inversely proportional to the square of the period of revolution (p).
Early astronomers estimated the mass of the earth to be 5.976 x 10^24 kg
and observed that the moon orbited the earth with a period of 27.322 days
at a mean distance of 384.4 x 10^3 km.

1. Write a formula for the mass of a planet according to Kepler.
I think it's
m = k*d^3........I used the letter k for the constant.
p^2

2. Find the proportionally constant using the observations and estimates for the earth and moon.

5.976 * 10^24 = k * (384.4 * 10^3)^3
........................ 27.322^2

5.976 * 10^24 = k * (56800235.584 * 10^9)
........................ 746.492

5.976 * 10^24 = k * (5.68 * 10^16)
........................ 746.492

5.976 * 10^24 = ____k_________
....................131.424 * 10^16 <<<<This step is incorrect -- how did 10^16 get to denominator?

 

[jlaw]

5.976 * 10^24 = ____k_________
....................131.424 * 10^16 <<<<This step is incorrect -- how did 10^16 get to denominator?

Thank you Subhotosh. Would I need to divided 5.976 * 10^24 and 131.424 *10^16 instead? to
get = 0.0454711 * 10^8
= 4.547 * 10^6 :?:

 

[Deleted member 4993]

Suppose you had to solve for 'k' from the following equation:

\(\displaystyle A \, = \, \frac{k \cdot B}{C}\)

you would get:

\(\displaystyle A \cdot C = \, k \cdot B\)

\(\displaystyle k \, = \, \frac{A \cdot C}{B}\)

follow the exact same steps