Answers: (A) (B)
 (B)  (C)
  (C) (D)
 (D) 

Explanation:
<h2>(A) Gravitational force of one star on the other</h2>
According to the law of universal gravitation:
 (1)
    (1)
Where:
 is the module of the gravitational force exerted between both bodies
 is the module of the gravitational force exerted between both bodies  
 is the universal gravitation constant.
 is the universal gravitation constant.
 and
 and  are the masses of both bodies.
 are the masses of both bodies.
 is the distance between both bodies
 is the distance between both bodies
In the case of this binary system with two stars with the same mass  and separated each other by a distance
 and separated each other by a distance  , the gravitational force is:
, the gravitational force is:
 (2)
    (2)
 
 (3) This is the gravitational force between the two stars.
    (3) This is the gravitational force between the two stars.
<h2>(B) Orbital speed of each star</h2>
Taking into account both stars describe a circular orbit and the fact this is a symmetrical system, the orbital speed  of each star is the same. In addition, if we assume this system is in equilibrium, <u>gravitational force must be equal to the centripetal force</u>
 of each star is the same. In addition, if we assume this system is in equilibrium, <u>gravitational force must be equal to the centripetal force</u>   (remembering we are talking about a circular orbit):
 (remembering we are talking about a circular orbit):
So:  (4)
   (4)
Where  (5) Being
  (5) Being  the centripetal acceleration
 the centripetal acceleration 
On the other hand, we know there is a relation between  and the velocity
 and the velocity  :
:
  (6)
  (6)
Substituting (6) in (5):
  (7)
 (7)
Substituting (3) and (7) in (4):
 (8)
   (8)
Finding  :
:
  (9) This is the orbital speed of each star
 (9) This is the orbital speed of each star
<h2>(C) Period of the orbit of each star</h2><h2 />
The period  of each star is given by:
 of each star is given by:
 (10)
  (10)
Substituting (9) in (10):
 (11)
  (11)
Solving and simplifying:
 (12) This is the orbital period of each star.
  (12) This is the orbital period of each star.
<h2>(D) Energy required to separate the two stars to infinity</h2>
The gravitational potential energy  is given by:
 is given by:
 (13)
  (13)
Taking into account this energy is always negative, which means the maximum value it can take is 0 (this happens when the masses are infinitely far away); the variation in the potential energy  for this case is:
 for this case is:
 (14)
 (14)
Knowing  the total potential energy is
 the total potential energy is  and in the case of this binary system is:
 and in the case of this binary system is:
 (15)
  (15)
Now, we already have the <u>potential energy</u>, but we need to know the kinetic energy  in order to obtain the total <u>Mechanical Energy</u>
 in order to obtain the total <u>Mechanical Energy</u>  required to separate the two stars to infinity.
 required to separate the two stars to infinity.
In this sense:
 (16)
 (16)
Where the kinetic energy of both stars is:
 (17)
 (17)
Substituting the value of  found in (9):
 found in (9):
 (17)
 (17)
 (18)
 (18)
Substituting (15) and (18) in (16):
 (19)
 (19)
 (20) This is the energy required to separate the two stars to infinity.
 (20) This is the energy required to separate the two stars to infinity.