Kepler's Third Law and Motion of Two Point Masses

In summary, the conversation discusses the attempt to solve equations in the paper 'Gravitational Radiation and the Motion of Two Point Masses' by Peters (1964). The equations are used to calculate results for a binary star system with each mass being equal to 1 solar mass. The results show that the separation between the stars is approximately 10 solar radii, the period is 4.5 days, and the lifetime for decay is 3x1012 years. However, when attempting to solve the equations using the given values for solar mass, solar radius, G, and c, the resulting values for B and T do not match the expected results. The conversation also mentions difficulties with applying Kepler's third law to the problem.
  • #1
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I'm trying to work through some equations in the paper 'Gravitational Radiation and the Motion of Two Point Masses' (Peters, 1964) but I can't get out the right values

1. Homework Statement
For a binary star system with each mass = 1 solar mass, the equations give the results:
Separation ~ 10 solar radii
Period ~ 4.5 days
Lifetime for decay ~ 3x1012 years

Homework Equations


T = a4/4B
B = (64/5)G3m1m2(m1+m2)/c5

The Attempt at a Solution


Using solar mass = 1.989 x 1030kg,
solar radius = 695700 km
G = 6.67408 x 10-11 m3kg-1s-2
c = 3 x 108 ms-1

I get B = (1/(3 x 108)5)(64/5)(6.67408 x 10-11)3 x 2 x 1.989 x1030 x 103 = 6.229 x 10-42

T = (10 x (695700 x 1000))4/4B = ~1.12 x 1051 s, which isn't at all near 3 x 1012 years :( Help?
 
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  • #2
Oh, also regarding Kepler's third law. My lecture notes give it as

(G/4pi2)(m1 + m2) t2=a3

where t is orbital period in years, masses are in solar units, and a is in au. This formula also isn't working for me yet?
 

FAQ: Kepler's Third Law and Motion of Two Point Masses

1. How does Kepler's Third Law explain the motion of two point masses?

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This means that the farther a planet is from its star, the longer it takes to complete one orbit. This law also applies to the motion of two point masses orbiting each other, where the orbital period is determined by the distance between the two masses and their combined mass.

2. What is the significance of the semi-major axis in Kepler's Third Law?

The semi-major axis is the longest radius of an elliptical orbit and is used to determine the size and shape of the orbit. In Kepler's Third Law, the square of the orbital period is directly proportional to the cube of the semi-major axis. This means that the semi-major axis plays a crucial role in determining the orbital period of a planet or two point masses.

3. How does Kepler's Third Law relate to Newton's Law of Universal Gravitation?

Kepler's Third Law and Newton's Law of Universal Gravitation are closely related. Newton's law states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This means that the orbital period of two point masses is determined by their combined mass and the distance between them, which is consistent with Kepler's Third Law.

4. Can Kepler's Third Law be used to determine the masses of two point masses?

Yes, Kepler's Third Law can be used to determine the masses of two point masses. By measuring the orbital period and the semi-major axis of the orbit, the combined mass of the two point masses can be calculated using the equation M = 4π²a³/GT², where M is the combined mass, a is the semi-major axis, G is the gravitational constant, and T is the orbital period.

5. How does the distance between two point masses affect their orbital period according to Kepler's Third Law?

The distance between two point masses is directly proportional to their orbital period according to Kepler's Third Law. This means that the farther apart the two masses are, the longer it takes for them to complete one orbit around each other. This relationship is represented by the equation T² ∝ a³, where T is the orbital period and a is the semi-major axis.

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