themightyweeaboo wrote:Also, does anyone know a formula I could use to solve this?
The ratio of orbital velocities for a star to its planet is 0.0083. The system has an inclination derived of
75 degrees, and the mass of the star is 0.66 solar masses. What is the mass of the planet in Jupiter
masses?
A bit late on this (astronomy isn't really my thing, but astrodynamics is), but if you're still stuck on this:
Two bodies exerting gravitational influence on each other orbit about their common center of mass. The center of mass is the point at which all of the mass balances. In this case, if the masses of the star and planet are
and
, respectively, and their distances from the center of mass are
and
, then the center of mass can be found by setting
We know the mass of the star, so if we can determine the distances from the center of mass, we can solve for the mass of the planet.
Now, given only the ratio of orbital velocities, we will need to assume that the objects are in circular orbits in order to do this. If the planets were in eccentric orbit, the ratio of velocities would be meaningless unless it was stated at which points in the orbit this occurs - as in an eccentric orbit, the velocity (including its magnitude) is always changing. However, in a circular orbit the magnitude of the velocity (also known as speed) is constant (although the direction of the velocity is changing). For the purposes of this problem I am going to proceed with this assumption, however this question is poorly written as it should state the orbits are circular. This cannot be assumed in general - for example, Mercury's orbit has an eccentricity of ~0.2, which means it is ~50% closer to the sun at perihelion than at aphelion. Saturn's orbit has an eccentricity of ~0.055, which puts it about 11% closer at perihelion than aphelion. This is non-negligible. Anyway, if we continue with the circular assumption, we can find the velocity of an orbit as a function of its radius and orbital period.
We know that velocity is distance traveled over time, and if the orbit is circular then the magnitude of velocity will be constant. The circumference of a circle is given by
, where
is the radius. So if we denote the orbital period - or time it takes to complete one orbit - as
, then the velocity is given by:
.
Now, let's write this for both the star and the planet:
Now, let's appeal to Kepler's 3rd law: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. Here, since the orbit is circular, the square of the period,
, is proportional to the the cube of the radius,
. So we know that
is a constant.
Now, since we're only concerned about ratios, not absolute values, we can say that velocity is proportional to the radius over the period. If
denotes "is proportional to", then
Now, since we know that the square of the period is proportional to the cube of velocity, we can take the square root of this and write
. If we sub this in, we find that
Now, since we know
, we can divide these out and sub in. Note that since we are actually taking the ratio of
to
when we do this, this becomes an equivalency, not a proportion.
or
But wait! This doesn't seem right at all! How can the radius of the planet's orbit be much smaller than the star's? If you continued solving, you'd see that the mass of the planet would be much larger than that of the star, which isn't realistic. In reality, the star would be traveling much faster about the center of mass (which it will be much closer to). So let's assume they mean the planet's orbital velocity magnitude is 0.0083 of that of the star, and
Now, we can rewrite the first equation (center of mass) as
and get
If
kg, then
kg
which is 0.0447 Jupiter masses.
All in all - this is a poor question due to its lack of stating that the orbits are circular, and its (likely inadvertent) assertion that the planet is heavier than the star.
