Answer:
(a)
(b)
Explanation:
(a) What is the mass of the star?
The Universal law of gravitation shows the interaction of gravity between two bodies:
(1)
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:
(2)
Then Newton's second law (
) will be replaced in equation (1):

By replacing (2) in equation (1) it is gotten:
(3)
Therefore, the mass of the star can be determine if M is isolated from equation (3):
(4)
But r can be known from Kepler's third law, since it gave the semi-major axis:

![a = \sqrt[3]{(7.60)^{2}}](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%5B3%5D%7B%287.60%29%5E%7B2%7D%7D)
![a = \sqrt[3]{57.76}](https://tex.z-dn.net/?f=a%20%3D%20%5Csqrt%5B3%5D%7B57.76%7D)
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):
⇒ 
⇒ 

<u>So the mass of the star is
Kg</u>
(b) What is the orbital period of the faster planet, in years?
To find the period, the equation for orbital velocity can be used:
(5)
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:
⇒ 

<u>So the period of the faster planet is 1.06 years</u>