Using Kepler's laws to calculate elliptical planetary motion

[kirito]

TL;DR Summary: orbital speed laws

I would appreciate a bit of explanation on how did we find e1 and e2 and if there are any useful references to learn about Kepler laws since I am lost for the most part, and would like to gain understanding and solving ability
,and if you can go into some details on how to know when there is a conservation of momentum and energy

 

[Hill]

TL;DR Summary: orbital speed laws

how did we find e1 and e2 and if there are any useful references to learn about Kepler laws

Kepler laws will not give you the equations for ##E_1## and ##E_2##. You need the Newton law of gravity for that.

 

[kirito]

Kepler laws will not give you the equations for E1 and E2. You need the Newton law of gravity for that.

so I should I think that for it to be in orbit the the gravitational force os the central force so
$$ f= m_ac = mv^2/r= G m M/r^2 $$ so I so I get $$v^2= GM/r$$ , how should I go on from there if I may ask

 

[Hill]

Please, use LaTeX.

gravitational force os the central force so f= m_ac = mv^2/r= G m M/r^2 so I so I get v^2= GM/r

You cannot assume that ##a=v^2/r## because the orbit here is not circular but rather elliptical.
You need to move from the gravitational force to the energy. Do you know the formula for gravitational potential energy?

 

[kirito]

Please, use LaTeX.

You cannot assume that ##a=v^2/r## because the orbit here is not circular but rather elliptical.
You need to move from the gravitational force to the energy. Do you know the formula for gravitational potential energy?

I don't I will try to search it up , I will also try to add latex now , thank you

 

[Janus]

Simply put, Orbital energy is the sum of kinetic energy and gravitational potential energy.
KE = 1/2mv^2
GPE = -GMm/r which is derived by applying calculus to Fg= GMm/r^2
Now, for a circular orbit, v=sqrt(GM/r)
Plugging this in for v in the KE+GPE equation(E2), and simplifying gives you -GMm/2r
It can be proven that an elliptical orbit with a semi-major axis of a has the same total orbital energy as a circular one of radius r (a circle being a special case of an ellipse where a=r)
thus E1= GMm/2a

 

[kuruman]

Your reference quotes the vis-viva equation. The link I provided shows its derivation which I think is what you want.

 

[kirito]

Simply put, Orbital energy is the sum of kinetic energy and gravitational potential energy.
KE = 1/2mv^2
GPE = -GMm/r which is derived by applying calculus to Fg= GMm/r^2
Now, for a circular orbit, v=sqrt(GM/r)
Plugging this in for v in the KE+GPE equation(E2), and simplifying gives you -GMm/2r
It can be proven that an elliptical orbit with a semi-major axis of a has the same total orbital energy as a circular one of radius r (a circle being a special case of an ellipse where a=r)
thus E1= GMm/2a

thank you for the explanation , I do admit that I have a lot of confusion in the subject that's why I asked for some resource your comment is direct and organised surely a bit more organised