- #1
E92M3
- 68
- 0
Why is the total energy of an elliptical orbit given by:
[tex]E_{tot}=\frac{-GMm}{2a}[/tex]
Where a=semi major axis.
I agree for a circular orbit I can do the following:
[tex]F_c=F_g[/tex]
[tex]ma_c=\frac{GMm}{r^2}[/tex]
[tex]\frac{v^2}{r}=\frac{GM}{r^2}[/tex]
[tex]v^2=\frac{GM}{r}[/tex]
Since the total energy also equal to the kinetic plus potential energy we have:
[tex]E_{tot}=\frac{1}{2}mv^2-\frac{GMm}{r}=\frac{1}{2}m\frac{GM}{r}-\frac{GMm}{r}=\frac{-GMm}{2r}[/tex]
Ok this is a similar form for circular orbit. But how can we just put a in instead of r for elliptical orbit? What is the justification?
[tex]E_{tot}=\frac{-GMm}{2a}[/tex]
Where a=semi major axis.
I agree for a circular orbit I can do the following:
[tex]F_c=F_g[/tex]
[tex]ma_c=\frac{GMm}{r^2}[/tex]
[tex]\frac{v^2}{r}=\frac{GM}{r^2}[/tex]
[tex]v^2=\frac{GM}{r}[/tex]
Since the total energy also equal to the kinetic plus potential energy we have:
[tex]E_{tot}=\frac{1}{2}mv^2-\frac{GMm}{r}=\frac{1}{2}m\frac{GM}{r}-\frac{GMm}{r}=\frac{-GMm}{2r}[/tex]
Ok this is a similar form for circular orbit. But how can we just put a in instead of r for elliptical orbit? What is the justification?