What Are the Orbital Periods and Star Mass for These Hypothetical Planets?

[ltung]

Planet A: Mass - 8 × 10^24 kg, Radius 8,200 km, Temperature - 500 K, Semi-major axis - 0.52 AU
Planet B: Mass - 3 × 10^27 kg, Radius - 57000 km, Temperature - 150 K, Semi-major axis - 6.4 AU
Planet C: Mass - 6 × 10^23 kg, Radius - 3500 km, Temperature - 105 K, Semi-major axis -37.8 AU

1. If the orbital period of planet A is 190 days, what are the periods of the other two planets? What is the mass of the star around which these planets orbit?
2. Which of these planets could retain an atmosphere of oxygen molecules?

Based on the 190 days orbital period, p^2 = k a^3, which plugging in the semimajor axis, I get k for 1.93 and used it to find the orbital periods of hte other planets.
Then I don't know how to do the other part of question 1 and question 2. There is a similar question on this forum, but it didn't help me with answering this question.

Thanks

 

[ideasrule]

There's an expression for k that involves the star's mass. Do you know what it is, or can you derive it if you don't?

 

[qraal]

Planet A: Mass - 8 × 10^24 kg, Radius 8,200 km, Temperature - 500 K, Semi-major axis - 0.52 AU
Planet B: Mass - 3 × 10^27 kg, Radius - 57000 km, Temperature - 150 K, Semi-major axis - 6.4 AU
Planet C: Mass - 6 × 10^23 kg, Radius - 3500 km, Temperature - 105 K, Semi-major axis -37.8 AU

1. If the orbital period of planet A is 190 days, what are the periods of the other two planets? What is the mass of the star around which these planets orbit?
2. Which of these planets could retain an atmosphere of oxygen molecules?

Based on the 190 days orbital period, p^2 = k a^3, which plugging in the semimajor axis, I get k for 1.93 and used it to find the orbital periods of hte other planets.
Then I don't know how to do the other part of question 1 and question 2. There is a similar question on this forum, but it didn't help me with answering this question.

Thanks

The mass of the star can be worked out relative to our Sun's mass by the fact that the orbital period is proportional to the inverse square root of the mass of the primary. In this case the planet takes 190 days to orbit at a distance of 0.52 AU. Earth takes 365.25636 days to orbit our Sun at 1 AU. Thus the mass of the other star relative to the Sun can be worked out via the relation I mentioned. The full equation is: p² = 4.π².a³/[G.(M_s+M_p)] ...where M_p & M_s are the masses of the Primary and Secondary respectively, in this case the star and planet. In most cases the planetary mass is so small that the mass of the star is a near enough approximation.

As for retaining oxygen molecules what do your notes tell you about atmospheric loss times? The short answer is all those planets will retain oxygen, but I'll let you work out why yourself. You do have notes on the equations? Or are you asking because you don't?