Two equal-mass stars orbit their centre of mass. The distance between the stars (measured from the centre of mass of each) is 1 AU. What is the period of revolution in years? How much energy would be required to unbind these stars if they each have the same mass as the sun? The mass of the sun is 2×10^30 kg.
Can someone help please!
I got 3.5x10^39 joules? for the second part to the question.
Thanks in advance.
Knowing that you have the mass of each sun, M1=M2=M which is 2x10^30 kg, and you have the distance between the two stars a=1 AU, you can use Kepler's third law to solve for the period in seconds:
p^2=((4pi)^2(a)^3)/(Gm)
where p is measured in seconds, a is measured in metres, that is, 1 AU = 1.49x10^11 m, and the total mass m is the sum of M1 and M2, and finally G is the gravitational constant 6.67x10^-11:
p=(4pi(1.49x10^11)^3/2)/sqrt((6.67x10^-11)(2x(2x10^30)))
The final answer should be:
4.43x10^7 seconds or 1.4 years
Bill,
Would you not use:
M = 2x10^30
a = 0.5 AU
Giving:
p = sqrt(((4*pi^2)(0.5*1.49x10^11)^3)/((6.67x10^-11)(2x10^30)))
= 1.106x10^7 seconds = 0.35 years
Note: Bill's verison of Kepler's 3rd Law is incorrect.
Where Bill said: (4pi)^2, it should be: (2pi)^2 = 4*(pi^2)
To find the period of revolution in years for two equal-mass stars, you can use Kepler's Third Law, which states that the square of the period is proportional to the cube of the average distance between the stars.
1. First, convert the distance between the stars from AU (astronomical units) to meters. Since 1 AU is approximately 1.496×10^11 meters, the distance between the stars is 1.496×10^11 meters.
2. Use Newton's form of Kepler's Third Law equation to find the period of revolution in seconds.
T^2 = (4π^2 / G) * R^3
where T is the period, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), and R is the distance between the stars.
T^2 = (4π^2 / (6.67430 × 10^-11)) * (1.496×10^11)^3
3. Take the square root of both sides to find the value of T.
T = √((4π^2 / (6.67430 × 10^-11)) * (1.496×10^11)^3)
4. Convert the period from seconds to years by dividing by the number of seconds in a year. There are 3.154×10^7 seconds in a year.
T (in years) = T / (3.154×10^7)
Calculating these steps will give you the period of revolution in years for the two equal-mass stars.
Regarding the second part of the question, to find the energy required to unbind these stars, you can use the gravitational potential energy formula.
The formula for gravitational potential energy is:
U = - (G * m1 * m2) / r
where U is the gravitational potential energy, G is the gravitational constant, m1 and m2 are the masses of the stars (each equal to the mass of the sun, 2×10^30 kg), and r is the distance between the stars.
Calculating this formula with the given values will give you the energy required to unbind the stars.