#format jsmath
= Rocket Plumes =

"Once the rockets are up, who cares where they come down? That's not my department, says Wernher von Braun" --- satiric songwriter Tom Lehrer, 1962(?)

Tom Lehrer's indifferent rocket designers have different names, but are still around.  The rockets they design carry (mostly) peaceful payloads to orbit; many (sadly not all) are deorbited or launched into interplanetary space after they complete their mission.

But chemical rockets send far more propellant plume mass out their rocket nozzles than the mass of their payloads and themselves.  In missions to the exosphere and beyond (above 1000 km altitude), the propellant plume atoms may be emitted with velocities that put them in long period orbits, where they will remain until collisions or Rayleigh-scattering light pressure deflects the atoms down into the atmosphere or into interplanetary space.

Interplanetary space near the Earth's orbit accumulates propellant plume mass as well, but the solar system is vastly larger than the exosphere.  A useful upper limit for the atmosphere is the solar wind bow shock radius, approximately 100,000 kilometers radius;  the Earth's orbit around the Sun is 150,000,000 kilometers, 1500 times larger, implying a volume three billion times larger.  The propellant atom problems and mitigations described below may extend to the entire solar system in the far future; we will not discuss them here.

== Earth Orbits ==

Circular Earth orbits are divided into LEO (Low Earth Orbit), MEO (Medium ...), GEO (Geostationary ...), HEO (High ...).  LEO extends from the top of the high-drag atmosphere (approximately 300 km radius) up to 2000 km altitude, or an equatorial radius of 8378 km.  Circular Geostationary (GEO) orbits have orbital periods of one sidereal day and orbit at 42164 km radius above the equator (36786 km altitude).  MEO orbits (GPS and Glosnass, for example) are between LEO and GEO.  HEO orbits are outside of GEO and below the Moon.  While some orbits are approximately circular, many useful Earth orbits are elliptical, often with apogee far higher than perigee.

This document focuses on orbits with apogees closer to the Earth than the Moon, less than 150,000 kilometers.

The distance across the orbit, from perigee radius $ r_p $ to apogee radius $ r_a $ (on the opposite side of the Earth), is twice the "semi-major" axis $ a $, hence $ a = 0.5 * ( r_a + r_p ) $.  The orbital period T is a function of the semimajor axis $ a $,
$ T = 2 \pi \sqrt{ a^3 / \mu }$.  The standard gravitational parameter $ \mu $ is 398,600.4418 km³/s².   Here's some example orbits and periods, in seconds and sidereal days (86164.09 seconds, 366.2422 sidereal days per year):

||                       || I.S.S. || Iridium || GPS   || Comsat || 1 sday || 2 sday || 5 day  ||  Moon   || Bow Shock ||
||                       || LEO    || MEO     || MEO   || GEO    || HEEO   || HEEO   || HEEO   ||         ||
|| perigee radius km     || 6783   || 7132    || 26586 || 42164  || 8378   || 8378   || 8378   || 362600  ||
|| apogee radius km      || 6785   || 7140    || 26526 || 42164  || 75950  || 125484 || 238198 || 405400  ||
|| semi-major axis km    || 6784   || 7137    || 26558 || 42164  || 42164  || 66932  || 123288 || 384400  ||
|| perigee velocity km/s ||
|| apogee velocity km/s  ||
|| orbital period secs   || 5561.0 || 6000.0  || 43071 || 86141  || 86141  || 172282 || 430705 || 2360591 ||
|| period sidereal days  || 0.0645 || 0.0696  || 0.500 || 1.000  || 1.000  || 2.000  || 5.000  || 27.404  ||
|| orbits per solar day  || 15.54  || 14.40   || 0.499 || 0.997  || 0.997  || 0.499  || 0.199  || 0.03639 ||
|| orbits per sid. day   || 15.49  || 14.36   || 2.000 || 1.000  || 1.000  || 0.500  || 0.200  || 0.03649 ||
all numbers '''best guesses'''

== Thermal Rockets ==