#format jsmath
= Perigee Drag =
=== Approximating the drag on a decaying transfer orbit ===
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=== 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 m^3^/s^2^ 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{s}}} $ || 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 )