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| Given apogee $ r_a $ and perigee $ r_p $ and $ \mu_e = $ 3.986e14 m^3 /s^2 and angle from perigee, true anomaly: $ \Large \theta $ | Given apogee $ r_a $ and perigee $ r_p $ and $ \mu_e = $ 3.986e14 m^3^/s^2^ and angle from perigee, true anomaly: $ \Large \theta $ |
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| ( note: $a$ is more commonly used for the semimajor axis. I use $s$ to avoid confusion with apogee ) |
Perigee Drag
Approximating the drag on a decaying transfer orbit
What is the velocity and radius versus time for an elliptical orbit?
Given apogee r_a and perigee r_p and \mu_e = 3.986e14 m3/s2 and angle from perigee, true anomaly: \Large \theta
eccentricity: \large e = ( r_a - r_p ) / ( r_a + r_p ) |
semimajor axis: \large s = ( r_a + r_p ) / 2 |
characteristic velocity: \large v_0 = \LARGE\sqrt{{\mu \over 2}\left({1 \over r_a}+{1 \over r_p}\right)} |
eccentric anomaly (ellipse center angle): \large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right) |
mean anomaly: \large M = E - e \sin( E ) |
time from perigee: \large t = M / \omega |
radius: \large r=s\Large{{1-e^2}\over{1+e\cos(\theta)}} |
perigee radius: \large r_p =( 1-e ) s |
perpendicular velocity: \large v_{\perp}= v_0 ( 1+e \cos( \theta )) |
perigee velocity: \large v_p =(1+e)v_0 |
total velocity, tangent to orbit: \Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}} |
angular momentum: \large L =\LARGE \sqrt{ { 2 \mu ~ r_a ~ r_p } \over { r_a + r_p } } |
( note: a is more commonly used for the semimajor axis. I use s to avoid confusion with apogee )