Kepler, Newton, Gravity, error

[NCStarGazer]

Looking at the gravitational equation

F=G*(M*m)/r^2

and centripetal force

F = ((V^2)/r)*m

If you set the two equal and solve for G you get:

G = ((V^2)*r)/M

Substituting (4*pi^2*r^2)/T^2 for V^2 you now have

G = (4*pi^2*r^3)/(M*T^2)

With solution for G, look at Kepler's law with Newton's update,

(M+m)*P^2 = (4*pi^2*a^3)/G

Substituting the G solved for into Kepler's equation and consider working with a perfect circle, r will equal a and T will equal P.

Now, once you substitute in the solved G, simplify...

You Get

(M+m)*P^2 = P^2*m

This is obviously not true!

Please let me know where the error is in my observation.

Thanks!

 

[jmb]

look at Kepler's law with Newton's update,

(M+m)*P^2 = (4*pi^2*a^3)/G

No! The law is $MP^2 = 4\pi^2a^3/G$ (see here). In which case there is no contradiction (hardly surprising given the above form of Kepler's Law is derived from Newton's law of gravity).

 

[NCStarGazer]

Well... I am confused, as the link you reference has 4π2a3 = P2G(M + m) under "Conversions for Unknowns", which is the same as what I referenced (M+m)*P^2 = (4*pi^2*a^3)/G.

 

[D H]

A bit out of order. jmb's response first.

(M+m)*P^2 = (4*pi^2*a^3)/G

No! The law is $MP^2 = 4\pi^2a^3/G$

Sorry, jmb, but NCStarGazer has the right expression.

$$\tau^2 = \frac{4\pi^2}{G(M+m)} \,a^3$$
The source of NCStarGazer's error:

Looking at the gravitational equation

F=G*(M*m)/r^2

and centripetal force

F = ((V^2)/r)*m

You are assuming that the r in Newton's law of gravity and the r in the centripetal force equation are one and the same. They aren't. Both objects are orbiting their center of mass. Given that the distance between a pair of objects of mass m and M is r, the distance from the object with mass m to the center of mass is

$$r_{cm} = \frac M{M+m}\,r$$

The centripetal force needed to sustain a circular orbit about the center of mass is thus

$$F=\omega^2 r_{cm} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{M}{M+m}\,r$$

Combining this with the gravitational force yields

$$\frac {GMm}{r^2} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{M}{M+m}\,r$$

Solving for the period,

$$\tau^2 = \frac{4\pi^2}{G(M+m)}\,r^3$$

 

[NCStarGazer]

D H

Thanks, I see how the center of mass can bring clarity. In working with your post I am still having trouble getting

$$\frac{GMm}{r^2}= \left(\frac{2\pi}{\tau}\right)^2\frac{M}{M+m}\,r$$

To yield your answer, I placed it in Maple and it always returns a version that has Gm(M+m); not G(M+m) in denominator ( I cannot find how to reduce the Gm to be only G), I did it by hand and I came up with the same. Did I miss something or was there a typo or such?

Appreciate the help!

 

[D H]

A typo. ω²r has units of acceleration, not force. Restarting with the equation for the center of mass,

$$r_{cm} = \frac M{M+m}\,r$$

The centripetal force needed to sustain a circular orbit about the center of mass is thus

$$F=m\omega^2 r_{cm} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{Mm}{M+m}\,r$$

Combining this with the gravitational force yields

$$\frac {GMm}{r^2} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{Mm}{M+m}\,r$$

Solving for the period,

$$\tau^2 = \frac{4\pi^2}{G(M+m)}\,r^3$$

 

[jmb]

Sorry, jmb, but NCStarGazer has the right expression.

Ughh, my bad. I read this when I was very tired and tried to answer the question too quicky (something I really shouldn't do!).

I saw NCStarGar's use of $r$ in the centripetal and gravitational force equations and assumed he was making the assumption that $M>>m$ and ignoring centre of mass issues. Of course had I taken on the significance of what he/she meant by "Newton's update" I would have realized that wasn't the intent...

Sorry NCStarGazer and thanks to D H for spotting this. Very embarrassed now!