Astrophysics: Deriving Newton's Gravitational Formula from Kepler's

In summary, Dervie Newton's form of Kepler's third law, which describes the orbital motion of two stars in circular orbits with masses M1 and M2, separation a, and period P, can be obtained by combining the equations for the centre of mass and centripetal force. This results in the formula F = GMm/r^2, where F is the gravitational force, G is the gravitational constant, M is the total mass of the two stars, m is the mass of one of the stars, and r is the distance between them.
  • #1
knowlewj01
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0

Homework Statement



Dervie Newton's form of Kepler's third law.

decrribing the orbital motion of two stars in circular orbits with masses M1 and M2, separation a, and period P

ie.

Obtain

F=[tex]\frac{ G M1 M2 }{ a^2 }[/tex]

From

M1+M2=[tex]\frac{4 \pi^2 a^3 }{GP^2}[/tex]

Homework Equations



Centre of mass:

M1r1 = M2r2

a = r1 + r2

P = [tex]\frac{2\pi r}{v}[/tex]

The Attempt at a Solution



[not to good at this LaTeX thing so i'll wing it]

1: switch the (M1 + M2) For P^2

P^2 = (4π^2 a^3)/(G(M1 + M2))

switch the P for the term above:

(4π^2 r^2)/v^2 = (4π^2 a^3)/(G(M1 + M2))

π's cancel:

r^2/v^2 = a^3 / G(M1 + M2)

problem is here that i don't know what the r is, do i have to work out this for r1 and r2 seperatly?

anyone done this before that could point me in the right direction?
 
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  • #3
Thanks, I think i got it, it looks a bit eggy but i got there, there may be something wrong.

M = [tex]\frac{4\pi^2r^3}{GP^2}[/tex]

P2 = [tex]\frac{4\pi^2r^3}{GM}[/tex]

[tex]\frac{4\pi^2r^2}{v^2}[/tex] = [tex]\frac{4\pi^2r^3}{GM}[/tex]

[tex]\frac{r^2}{v^2}[/tex] = [tex]\frac{r^3}{GM}[/tex]

[tex]\Rightarrow[/tex] acceleration = [tex]\frac{v^2}{r}[/tex]

a = centrepetal acceleration

[tex]\frac{r}{a}[/tex] = [tex]\frac{r^3}{GM}[/tex]

a = [tex]\frac{GM}{r^2}[/tex]

[tex]\Rightarrow[/tex] F= ma

F = [tex]\frac{GMm}{r^2}[/tex]
 

Related to Astrophysics: Deriving Newton's Gravitational Formula from Kepler's

1. How does Kepler's third law relate to Newton's gravitational formula?

Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of its semi-major axis. This law can be used to derive Newton's gravitational formula, which 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.

2. What is the significance of Newton's gravitational formula in astrophysics?

Newton's gravitational formula is a fundamental law in astrophysics that explains how objects with mass interact with each other through the force of gravity. It is essential in understanding the motion of planets, stars, and galaxies, and is also used in the study of celestial mechanics and gravitational waves.

3. What are the steps to derive Newton's gravitational formula from Kepler's third law?

To derive Newton's gravitational formula from Kepler's third law, one must first express the orbital period and semi-major axis in terms of the gravitational constant, the masses of the objects, and the distance between them. Secondly, the formula for the centripetal force must be substituted into Kepler's third law equation. Finally, by equating this equation to the formula for the force of gravity, Newton's gravitational formula can be derived.

4. Can Newton's gravitational formula be applied to objects other than planets?

Yes, Newton's gravitational formula can be applied to any two objects with mass, regardless of their size or composition. It is commonly used to calculate the force of gravity between celestial bodies, but it also applies to smaller objects, such as apples falling to the ground due to Earth's gravitational pull.

5. How does the inverse square law in Newton's gravitational formula impact the strength of gravitational force?

The inverse square law in Newton's gravitational formula means that the strength of the gravitational force decreases as the distance between two objects increases. This means that the closer two objects are to each other, the stronger the force of gravity between them will be. Conversely, the farther apart they are, the weaker the force of gravity will be.

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