Calculating Orbital Periods for Spherical Planets in Orbit

[NasuSama]

Homework Statement

Spherical Planet X (mass MX = 7.81x10^24 kg, radius RX = 2.09x10^6 m) travels in a circular orbit of radius ro = 4.39x10^11 m around the star Gort. Its period of orbit is Τ = 390 Earth days.

b) Planet Q is also in circular orbit around the star Gort, at radius 24.76x10^11 m. Find the period of orbit of this planet.

Homework Equations

T²/r² = 4π²/(GM)

The Attempt at a Solution

I found the mass of the Gort to be 4.41 * 10^(31), using the same form I indicated under "equations". Then...

T = √((4.39 * 10^(11) + 24.76 * 10^(11)) * 4π²/(6.67 * 10^(-11) * 4.41 * 10^(31)))
≈ 5.765 * 10^8 seconds

Since 1 day = 24 * 3600 seconds, we have...

5.765 * 10^8 seconds * days / (24 * 3600) seconds → 6673 days, and the answer is marked incorrectly.

I don't get why this happens. I reported this to my professor, and he said that he got the correct answer that is different from the answer I have.

 

[gneill]

You don't need to muck around with the mass of the star if you just assume that it's much, much greater than that of the two planets. Nor do you need to worry about unit conversions so long as you maintain the same units throughout. Just apply the statement of Kepler's third law directly and form the appropriate ratios.

As for units, might as well let the time unit TU be "days", and since both radii in terms of 10¹¹m, use that as the distance unit; DU = 10¹¹m. So planet X has T = 390 TU; r = 4.39 DU.

 

[voko]

The formula you have for Kepler's third law is incorrect. It must relate the square of the period with the cube of the radius.

Secondly, you don't need to compute the mass of Gort (unless you also have the mass of Q, in which case you could use a more accurate formula). The right hand side is the same for both planets, so you could equate their left hand sides directly. You don't need to convert time units to seconds in this case, you can just use days.

 

[NasuSama]

You don't need to muck around with the mass of the star if you just assume that it's much, much greater than that of the two planets. Nor do you need to worry about unit conversions so long as you maintain the same units throughout. Just apply the statement of Kepler's third law directly and form the appropriate ratios.

As for units, might as well let the time unit TU be "days", and since both radii in terms of 10¹¹m, use that as the distance unit; DU = 10¹¹m. So planet X has T = 390 TU; r = 4.39 DU.

Thank you very much! Then, it's just...

390²/4.39³ = T²/24.76³
T = √(390²/4.39³ * 24.76³)

 

[gneill]

Thank you very much! Then, it's just...

390²/4.39³ = T²/24.76³
T = √(390²/4.39³ * 24.76³)

Yup. What's your result?

 

[NasuSama]

Yup. What's your result?

That is approximately 5220.

 

[gneill]

That is approximately 5220.

Looks good