Angular momentum of orbit from orbit parameters and mass of sun

[Kaguro]

Homework Statement

A planet of mass m moves in the gravitational field of sun ( mass M). If semi major and minor axes are a and b respectively, the angular momentum of the planet is what?

Relevant Equations

Area of ellipse $\pi ab$

Kepler's laws

L = mvr = mr (dr/dt) = 2m*r*(dr/dt)/2 = 2m*(dA/dt)
So, A = (L/2m)T
so, $L = \frac{2 \pi a b m}{T}$

Now, $T^2 = \frac{4 \pi^2}{GM} a^3$
So from all these, I get
$L = \sqrt{ \frac{GM m^2 b^2}{a}}$

But answer given is
$L = \sqrt{ \frac{2GM m^2 ab}{a+b}}$
(This, they have derived from energy and angular momentum conservation.)

What is wrong? Where's this inconsistency coming from?

 

[PeroK]

Homework Statement:: A planet of mass m moves in the gravitational field of sun ( mass M). If semi major and minor axes are a and b respectively, the angular momentum of the planet is what?
Relevant Equations:: Area of ellipse $\pi ab$

Kepler's laws

L = mvr = mr (dr/dt) = 2m*r*(dr/dt)/2 = 2m*(dA/dt)
So, A = (L/2m)T
so, $L = \frac{2 \pi a b m}{T}$

Now, $T^2 = \frac{4 \pi^2}{GM} a^3$
So from all these, I get
$L = \sqrt{ \frac{GM m^2 b^2}{a}}$

But answer given is
$L = \sqrt{ \frac{2GM m^2 ab}{a+b}}$
(This, they have derived from energy and angular momentum conservation.)

What is wrong? Where's this inconsistency coming from?

I prefer your answer. So does this:

https://www.lehman.edu/faculty/anchordoqui/chapter25.pdf

See equation 25.B.20.

Why didn't you find that pdf?

 

[Delta2]

Cant find a mistake in your answer either.

Just to rewrite your first line using cross product because the way it is written as simple multiplication is wrong
$$\vec{L}=m\vec{r}\times\frac{d\vec{r}}{dt}=2m\frac{1}{2}\vec{r}\times\frac{d\vec{r}}{dt}\Rightarrow |\vec{L}|=2m\frac{dA}{dt}$$ where $A$ is the area that the radius vector $\vec{r}$ spans.

 

[Kaguro]

Ah...

I now realize that the answer given is wrong, because they assumed the orbit is circular.
When I put a=b, the two answers are consistent.

But even after assuming that a=b, they still wrote them as though they are different.

Thank you both of you.