How to calculate the orbital period and the density of a planetary body?

Question

I'm a student studying in 9th grade, I am making this fictional habitable planet named "Darwin B" for a planet making competition. It orbits a sun-like star at a distance of 1.15 AU or 172 million kilometres in a nearly circular orbit. Its rotational period is 19 hours, 38 minutes. Its mass is $6.15×10^{24}kg$ and its radius is about 6,743 kilometres. I have to calculate its orbital period and density but I'm weak in maths and don't know how to. Please help.

Answer 1

The formula for orbital period is given on Wikipedia:

$$T=2\pi \sqrt\frac{a^3}\mu$$

where:

• $T$ is the orbital period in seconds
• $a$ is the orbit's semi-major axis in meters
• $\mu = GM$ is the standard gravitational parameter
  • $G$ is the gravitational constant
  • $M$ is the mass of the more massive body in kilograms

So $T = 2 \pi \sqrt \frac { (172 \cdot 10^9) ^ 3 } { 6.674 \cdot 10^{-11} \cdot 1.9884 \cdot 10^{30} }$. Can you take it from here?

As for density, the volume of a sphere is given by the formula $V = \frac43\pi r^3$; density ($\rho = \frac MV$, with $M$ the mass) is usually given in grams per cubic centimeter, so it makes sense to convert to those units. That gives the following calculation:

$$\frac { 6.15 \cdot 10^{27} } { \frac43\pi (6.743 \cdot 10^8)^3 }$$

Answer 2

Orbital Period

According to Kepler's Third Law, the orbital period $T$ is defined as

$$T=2\pi\sqrt \frac{a^3}{\mu}$$

$T$ is, as said before, the orbital period (i.e. the time for an object - in this case, the planet - to complete an orbit around the massive, central object - in this case, the star) measured in seconds.
$a$ is the object's semi-major axis (the longest diameter of an ellipse - in this case, the greatest distance between star and planet).
$\mu=GM$ with $G$ being the Gravitational Constant and $M$ the mass of the massive object (the star).
(from Wikipedia - Orbital Period)

By inserting the values, we get $$T=2\pi\sqrt \frac{(1,15AU)^3}{GM}$$

Note that in this case, the mass of the planet is not relevant. What we need instead is the mass of the star, which you have not given. Since you assumed a sun-like star, we can insert the sun's standard gravitational parameter for $G\times M$:

$$T=2\pi\sqrt \frac{(1.72\times10^{11}m)^3}{1,33\times10^{20} \frac {m^3}{s^2}}\approx38863930\,s$$

which is around 449.81 days.

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Density

For the density, we know that $$\rho=\frac{m}{V}$$

Approximating the planet's shape to a sphere with a Volume $V=\frac{4}{3}\pi r^3$, we get $V\approx1.284\times10^{21}m^3$

Thus, the density is $$\rho\approx 4789\frac{kg}{m^3}$$

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Notes and Disclaimer

I am no physics expert myself, I am only a student. I do not take any responsibility regarding the correctness of my calulations.

I encourage you to only take the formulas (either from this post or just look them up) and do the math yourself. Since you participate in a competition (I do not know the rules of the competition, i.e. whether it is allowed to ask others for help), it is anyway better to actually do the work yourself. Please only consider this post as a reference to check whether your calculations seem to be correct (assuming that mine are, which I hope) and don't simply copy it.

I do not take any responsibility if you use this post other than described above, which includes potentially being ruled out from the competition.

I hope this helps.