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 43.3 km/s and 58.6 km/s. the slower planet’s orbital period is 7.60 years. (a) what is the mass of the star? (b) what is the orbital period of the faster planet, in years?
(a) What is the mass of the star?
The Universal law of gravitation shows the interaction of gravity between two bodies:
Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.
For this particular case M is the mass of the star and m is the mass of the planet. Since it is a circular motion the centripetal acceleration will be:
Then Newton's second law () will be replaced in equation (1):
By replacing (2) in equation (1) it is gotten:
Therefore, the mass of the star can be determine if M is isolated from equation (3):
But r can be known from Kepler's third law, since it gave the semi-major axis:
However, a can be expressed in astronomical units:
One astronomical unit is defined as the distance between the Earth and the Sun ():
r and v will be expressed in meters before being replaced in equation (4):
So the mass of the star is Kg
(b) What is the orbital period of the faster planet, in years?
To find the period, the equation for orbital velocity can be used:
Notice that the distance of the faster planet from the Star (r) is needed, that can be found using equation (4) in terms of r and the mass of the star:
It is necessary to express the velocity of the faster planet in meters.
Equation (5) can be rewritten in terms of T:
There are 31536000 seconds in 1 year:
So the period of the faster planet is 1.06 years
found in the nucleus
has mass of one amu