- #1

- 613

- 2

Starting with the Lagrangian

[tex]\mathcal{L} = \left(\frac{1}{2}m (r')^2 + \frac{1}{2}m r^2 (\theta ')^2\right)+\frac{G m M}{r}[/tex]

Applying the Euler Lagrange eqn to prove conservation of momentum conjugate to angle

[tex]0 = m r^2\theta '=L[/tex]

And solving for angular velocity

[tex]\theta '=\frac{L}{m r^2}[/tex]

Then applying the eqn to the radial component

[tex]m r (\theta ')^2-\frac{G m M}{r^2}= m a[/tex]

Assuming on a stable orbit net force on r should be zero and substituting in what I found for angular velocity

[tex]\frac{L^2}{m r^3}-\frac{G m M}{r^2}=0[/tex]

Then solving for L

[tex]L = \sqrt{G m^2M r}[/tex]

Plugging in values from wiki

[tex]L = \sqrt{\left(6.67\times 10^{-11}\right)\left(6\times 10^{24}\right)^2\left(2\times 10^{30}\right)\left(15\times 10^6\right)}[/tex]

[tex]L = 2.68395*10^38[/tex]

But I think I'm two orders of magnitude off, I remember it being to the power 40 not 38

Have I done something wrong here or made any incorrect assumptions?