Elliptical Orbit

I. Geometry of Ellipses

II. Angular Momentum

III. Energy:

Total Energy for the binary system

\[E=\frac12m_1v_1^2+\frac12m_2v_2^2-G\frac {m_1m_2}{R}\]

Relative Velocity (Definition)

If the relative velocity of the two star in a binary system:

\[v=v_1+v_2\]

The two velocities are added because the two stars are always moving in opposite directions, so their relative velocity will be the sum of the magnitudes of their individual velocities.

\[v_1=v-v_2\]

but:

\[m_1 v_1=m_2v_2\]

so:

\[v_2=\frac {m_1}{m_2} v_1\]

substuiting this:

\[v_1=v-\frac {m_1}{m_2} v_1\]

\[v_1+\frac {m_1}{m_2} v_1=v\]

\[v_1(1+\frac {m_1}{m_2})=v\]

so:

\[v_1=\frac {m_2v}{m_1+m_2}\]

and for the same for \(v_2\):

\[v_2=\frac {m_1v}{m_1+m_2}\]

so:

\[E=\frac12m_1(\frac {m_2v}{m_1+m_2})^2+\frac12m_2(\frac {m_1v}{m_1+m_2})^2-G\frac {m_1m_2}{R}\]

\[E=\frac 12m_1m_2[m_2(\frac {v}{m_1+m_2})^2+m_1(\frac {v}{m_1+m_2})^2]-G\frac {m_1m_2}{R}\]

\[E=\frac12m_1m_2(\frac {v}{m_1+m_2})^2[m_2+m_1]-G\frac {m_1m_2}{R}\]

\[E=m_1m_2\frac {v^2}{2(m_1+m_2)}-G\frac {m_1m_2}{R}\]

Total Energy, Relative Velocity:

\[E=m_1m_2[\frac {v^2}{2(m_1+m_2)}-\frac {G}{R}]\]

IV. Velocity:

Velocities at Apastron and Periastron:

At the apastron (\(v_a\)) and periastron (\(v_p\)).

Conseved Angular Momentum:

the velocities can be lelated by conservation of angular momentum. Angular momentum conservation is easy to apply at the apastron and periastron because the velocities are perpendicular to \(\vec r\)or \(\vec {F’F}\):

\[L=mvr \sin \phi\]

\(\phi =90\degree\) because \(\vec r \perp \vec v\) at apastron and periastron:

\[L=mvr=con \]

\[L_a=L_p\]

so:

\[v_ar_a=v_pr_p\]

\[v_a a(1+e)=v_p (1-e)\]

\[\frac {v_p}{v_a}=\frac {1+e}{1-e}\]

Conserved Total Energy:

\[E_a=m_1m_2[\frac {v_a^2}{2(m_1+m_2)}-\frac {G}{r_a}]\]

\[E_p=m_1m_2[\frac {v_a^2}{2(m_1+m_2)}-\frac {G}{r_p}]\]

\[E_a=E_p\]

so:

\[\frac {v_a^2}{2(m_1+m_2)}-\frac {G}{a(1+e)}=\frac {v_p^2}{2(m_1+m_2)}-\frac {G}{a(1-e)}\]

\[\frac {G}{a(1-e)} - \frac {G}{a(1+e)}=\frac {v_p^2}{2(m_1+m_2)}-\frac {v_a^2}{2(m_1+m_2)}\]

\[[\frac {G(m_1+m_2)}{a}](\frac 1{1-e}-\frac 1{1+e})=\frac 12(v_p^2-v_a^2)\]

\[[\frac {G(m_1+m_2)}{a}](\frac {2e}{1-e^2})=\frac 12v_a^2(\frac {v_p^2}{v_a^2}-1)\]

\[[\frac {G(m_1+m_2)}{a}](\frac {2e}{1-e^2})=\frac 12v_a^2[(\frac {v_p}{v_a})^2-1]\]

From the conservation of angular momentum:

\[\frac {v_p}{v_a}=\frac {1+e}{1-e}\]

so:

\[[\frac {G(m_1+m_2)}{a}](\frac {2e}{1-e^2})=\frac 12v_a^2[(\frac {1-e}{1+e})^2-1]\]

but:

\[(\frac {1+e}{1-e})^2-1=\frac {(1+e)^2-(1-e)^2}{(1-e)^2}=\frac {2.2e}{(1-e)^2}\]

so:

\[[\frac {G(m_1+m_2)}{a}](\frac {2e}{1-e^2})=\frac 12v_a^2\frac {2.2e}{(1-e)^2}\]

\[[\frac {G(m_1+m_2)}{a}](\frac {1}{1-e^2})=v_a^2\frac {1}{(1-e)^2}\]

\[v_a^2=[\frac {G(m_1+m_2)}{a}]\frac {(1-e)^2}{1-e^2}\]

so:

\[v_a^2=[\frac {G(m_1+m_2)}{a}]\frac {1-e}{1+e}\]

substituting this in the ratio \(v_p/v_a\):

\[v_p^2=(\frac {1+e}{1-e})^2.v_a^2 \]

\[v_p^2=(\frac {1+e}{1-e})^2[\frac {G(m_1+m_2)}{a}]\frac {1-e}{1+e}\]

\[v_p^2=[\frac {G(m_1+m_2)}{a}]\frac {1+e}{1-e}\]

Velocities at Apastron and Periastron

\[v_a^2=[\frac {G(m_1+m_2)}{a}]\frac {1-e}{1+e}\]

\[v_p^2=[\frac {G(m_1+m_2)}{a}]\frac {1+e}{1-e}\]

\(E=\frac12m_1[\frac {G(m_1+m_2)}{a}]\frac {1-e}{1+e}+\frac12m_2[\frac {G(m_1+m_2)}{a}]\frac {1+e}{1-e}-G\frac {m_1m_2}{R}\); R=2a

\[E=\frac12m_1[\frac {G(m_1+m_2)}{a}]\frac {1-e}{1+e}+\frac12m_2[\frac {G(m_1+m_2)}{a}]\frac {1+e}{1-e}-G\frac {m_1m_2}{2a}\]

\[E=G\frac {m_1m_2}{2a}[(\frac {m_1+m_2}{m_2})\frac {1-e}{1+e}+(\frac {m_1+m_2}{m_1})\frac {1+e}{1-e}-1]\]

\[[(\frac {m_1+m_2}{m_2})\frac {1-e}{1+e}+(\frac {m_1+m_2}{m_1})\frac {1+e}{1-e}-1]\]

common numinator:

\[m_1m_2(1+e)(1-e)\]

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\[E=-G\frac {m_1m_2}{2a}\]

but:

\[E=m_1m_2[\frac {v^2}{2(m_1+m_2)}-\frac {G}{r}]\]

so

\[-G\frac {m_1m_2}{2a}=m_1m_2[\frac {v^2}{2(m_1+m_2)}-\frac {G}{r}]\]

\[\frac {-G}{2a}=\frac {v^2}{2(m_1+m_2)}-\frac {G}{r}\]

\[\frac {v^2}{2(m_1+m_2)}=(\frac {G}{r}-\frac {G}{2a})\]

\[v^2=G(m_1+m_2)(\frac {2}{r}-\frac {1}{a})\]

V. Observing elliptical orbits

In studying the Doppler shifts of elliptical orbits as compared with circular orbits, there are three important differences:

(1) The velocity is not constant.

(2) The velocity is not perpendicular to the line from the center of mass to the orbiting object.

(3) Even when situated in the orbital plane, the radial velocity curve is influenced by the observer's position relative to the major axis of the ellipse.

Correcting Doppler Shifts

To correct Doppler shifts due to orbit inclination, the observed ellipse must be adjusted. If an elliptical orbit is tilted, its projection on the sky will still form an ellipse, but with a different eccentricity than the actual ellipse. Detecting tilt involves examining the apparent position of the foci; if they differ from the center of mass, the orbit is inclined. Determining the inclination requires adjusting the position of the center of mass iteratively until the projected orbit aligns well with observations.