Two newly discovered planets follow circular orbits around a star in a distant part of the galaxy. The orbital speeds of the planets are determined to be 44.7 km/s and 56.2 km/s. The slower planet's orbital period is 8.61 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?
I got M= 5.57 E32 and it is wrong, making my time wrong. What equation do I use for (a)?
To find the mass of the star, you can utilize Newton's version of Kepler's Third Law of Planetary Motion, which relates the orbital period of a planet, the radius of its orbit, and the mass of the star it revolves around.
The equation for Kepler's Third Law is:
T^2 = (4π^2 / GM) * r^3
Where:
T is the orbital period of the planet
G is the gravitational constant (approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2)
M is the mass of the star
r is the radius of the planet's orbit
First, we need to convert the given orbital period of the slower planet from years to seconds. Since there are 365.25 days in a year (taking into account leap years), the conversion factor is:
1 year = 365.25 days * 24 hours * 60 minutes * 60 seconds
Now we have all the necessary information to solve for the mass of the star:
T = 8.61 years = 8.61 * 365.25 * 24 * 60 * 60 seconds
r (radius) needs to be found from the given orbital speeds.
Since the orbital speed of a planet can be expressed as the square root of (G * M / r), we can solve for r:
For the slower planet:
v = 44.7 km/s (convert to m/s by multiplying by 1000)
v = sqrt(G * M / r)
r = G * M / v^2
Now substitute the values into the equation for the slower planet:
r = (6.67430 x 10^-11) * M / (44.7 * 1000)^2
Now that you have the value of r for the slower planet, substitute it into the equation for Kepler's Third Law:
(8.61 * 365.25 * 24 * 60 * 60)^2 = (4π^2 / (6.67430 x 10^-11 * M)) * r^3
Rearrange the equation and solve for M:
M = (4π^2 * r^3) / (G * T^2)
Now put in the known values and calculate the mass of the star.
I hope this helps! Let me know if you have any more questions.