Calculating the Mass of a Star from a Planet's Orbital Period

[A_I_]

Suppose there is a planetary system in which a planet with an average distance of 6 AU from the star has an orbital period of 3 years. What is the mass of the star?

The answer should be in SOLAR MASSES.


I tried to use the formula: P^2 = (4pi^2*a^3)/G(m+M)
but it didnt work:(

any hint?
thanks

 

[Tide]

This is a simple variation problem and you should recognize that

$$\frac {M P^2}{R^3}$$

is a constant so you can easily setup the appropriate proportion.

 

[Blueshift5]

I would first like to commend you for attempting to use the formula to calculate the mass of the star from the given information. However, it seems that there may have been an error in your calculations. The formula you used is actually the correct one for calculating the mass of a star from a planet's orbital period and distance. I would suggest checking your calculations and making sure all the units are consistent (e.g. using astronomical units for distance and years for orbital period).

To help you get started, I can provide some guidance on how to use the formula. P is the orbital period of the planet in years, a is the average distance between the planet and the star in AU, and G is the gravitational constant. The variables m and M represent the masses of the planet and the star, respectively. Since we are trying to find the mass of the star, we can rearrange the formula to solve for M:

M = (4pi^2*a^3)/(G*P^2)

Plugging in the values given in the problem, we get:

M = (4pi^2*(6 AU)^3)/(G*(3 years)^2)

Now, it's important to note that the units for AU and years are not in a form that can be used with the gravitational constant, which has units of (m^3)/(kg*s^2). Therefore, we will need to convert the units to a more appropriate form. For example, you could convert AU to meters by using the conversion factor 1 AU = 1.496 x 10^11 meters. Similarly, years can be converted to seconds using the conversion factor 1 year = 3.154 x 10^7 seconds.

Once you have converted the units, you can plug in the values into the formula and solve for the mass of the star in kilograms. Then, to convert to solar masses, you can divide the mass by the mass of the Sun, which is approximately 1.989 x 10^30 kg.

I hope this helps and good luck with your calculations! Remember to always double-check your units and calculations to ensure accuracy.