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 35.1 km/s and 63.3 km/s. The slower planet's orbital period is 6.74 years. (a) What is the mass of the star? (b) What is the orbital period of the faster planet, in years?
To find the mass of the star, we can use the formula for the orbital period of a planet in a circular orbit:
T = (2πr) / v
Where T is the orbital period, r is the radius of the orbit, and v is the orbital speed.
(a) To find the mass of the star, we need the radius of the orbit. We don't have the value directly, but we can use the fact that the planets have circular orbits to find it. In a circular orbit, the centripetal force is balanced by the gravitational force:
Fc = Fg
mv^2 / r = GmM / r^2
Where Fc is the centripetal force, Fg is the gravitational force, m is the mass of the planet, M is the mass of the star, r is the radius of the orbit, v is the orbital speed, and G is the gravitational constant.
We can cancel out the mass of the planet, m, from both sides and rewrite the equation as:
v^2 = G*M / r
To find the radius of the orbit, we'll use the given orbital speed of the slower planet, v = 35.1 km/s, and its period, T = 6.74 years. First, we need to convert the orbital velocity to m/s:
v = 35.1 km/s = 35.1 * 1000 m/s = 35,100 m/s
Now we can rearrange the equation to solve for r:
r = G*M / v^2
Plugging in the values for G and v, we get:
r = (6.67430 * 10^-11 N(m/kg)^2) * M / (35,100 m/s)^2
Simplifying the equation and substituting the known values:
r = (6.67430 * 10^-11 N(m/kg)^2) * M / (35,100 m/s)^2
Now, we can solve for r:
r = 2.121172 * 10^15 M
We'll use this expression for r in the next step.
To find the mass of the star, M, we can use the given orbital period of the slower planet, T = 6.74 years. First, we need to convert the period to seconds:
T = 6.74 years * 365.25 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute
Simplifying the equation, we get:
T = 2.126636 * 10^8 s
Now we can rearrange the formula for orbital period to solve for the mass of the star:
T = (2πr) / v
M = (T * v^2) / (4π^2 * r)
Plugging in the known values, we get:
M = (2.126636 * 10^8 s * (35,100 m/s)^2) / (4π^2 * 2.121172 * 10^15 M)
Simplifying the equation, we get:
M = 1.449249 * 10^30 kg
Therefore, the mass of the star is approximately 1.449249 * 10^30 kg.
(b) Now that we have the mass of the star and the orbital speed of the faster planet, we can use the same formula for orbital period to find the orbital period of the faster planet.
T = (2πr) / v
Plugging in the known values for the faster planet's orbital velocity, v = 63.3 km/s, and the radius of the orbit, r, which we found earlier, we get:
T = (2π * 2.121172 * 10^15 M) / (63,300 m/s)
Now, we can calculate the orbital period:
T = (2π * 2.121172 * 10^15 M) / (63,300 m/s)
T = 7.74 years
Therefore, the orbital period of the faster planet is approximately 7.74 years.