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| || The origin elliptical Kepler orbit has four parameters we need to compute two $ \Delta V $ burns for our transfer orbit: <<BR>><<BR>> * Apogee $ r_a $ <<BR>><<BR>> * Perigee $ r_p $ <<BR>><<BR>> * Inclination $ i $ <<BR>><<BR>> * argument of perigee $ \omega $ <<BR>><<BR>>We can compute the semimajor axis $ a $ and the eccentricity $ e $ from the apogee and perigee <<BR>><<BR>>The angle of the ascending and descending nodes do not affect the $ \Delta V $ burns, though they do affect when we make our burns so we insert into the desired destination region of the M288 orbit.<<BR>><<BR>>|| {{http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/2000px-Orbit1.svg.png||width=300}} || | || [[http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/2000px-Orbit1.svg.png|{{http://upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Orbit1.svg/2000px-Orbit1.svg.png||width=300}}]] || The origin elliptical Kepler orbit has four parameters we need to compute two $ \Delta V $ burns for our transfer orbit: <<BR>><<BR>> * Apogee $ r_a $ <<BR>><<BR>> * Perigee $ r_p $ <<BR>><<BR>> * Inclination $ i $ <<BR>><<BR>> * argument of perigee $ \omega $ <<BR>><<BR>>We can compute the semimajor axis $ a $ and the eccentricity $ e $ from the apogee and perigee <<BR>><<BR>>The angle of the ascending and descending nodes do not affect the $ \Delta V $ burns, though they do affect when we make our burns so we insert into the desired destination region of the M288 orbit.<<BR>><<BR>>|| |
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| || Blue curve:<<BR>><<BR>>An orbit in the horizontal X,Y plane<<BR>><<BR>>apogee = apoapsis = 6<<BR>>perigee= periapsis = 2<<BR>>semimajor axis = 4<<BR>>eccentricity = 0.5 || {{ attachment:orb01.png | |width=256 }} || || Red curve:<<BR>><<BR>>A similar orbit with a 45 degree '''argument of perigee''' <<BR>><<BR>>(rotate around vertical Z axis): || {{ attachment:orb02.png | |width=256 }} || || Green curve:<<BR>><<BR>>A similar orbit with a 30 degree '''inclination''' <<BR>><<BR>>(rotate around horizontal X axis):<<BR>><<BR>><<BR>>note that the ascending and descending nodes, where the orbit crosses the equatorial plane, do not have the same radius from the origin|| {{ attachment:orb04.png | |width=256 }} || |
|| [[ attachment:orb01.png |{{ attachment:orb01.png | |width=256 }}]] || Blue curve:<<BR>><<BR>>An orbit in the horizontal X,Y plane<<BR>><<BR>>apogee = apoapsis = 6<<BR>>perigee= periapsis = 2<<BR>>semimajor axis = 4<<BR>>eccentricity = 0.5 || || [[attachment:orb02.png |{{ attachment:orb02.png | |width=256 }}]] || Red curve:<<BR>><<BR>>A similar orbit with a 45 degree '''argument of perigee''' <<BR>><<BR>>(rotate by $ \omega $ around vertical Z axis): || || [[attachment:orb04.png |{{ attachment:orb04.png | |width=256 }}]] || Green curve:<<BR>><<BR>>A similar orbit with a 30 degree '''inclination''' <<BR>><<BR>>(rotate by $ i $ around horizontal X axis):<<BR>><<BR>><<BR>>note that the ascending and descending nodes, where the orbit crosses the equatorial plane, do not have the same radius from the origin|| |
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| || $ \mu $ || earth gravitational constant || 3.986004418e14 m^3^/s^2^ || || $ T $ || sidereal orbital period (s) || $ \Large { { 2 \pi } \over \omega } $ || || $ a $ || semimajor axis (m) || $ 0.5 ( r_a + r_p ) $ || $ \large \left( \mu \left( T \over { 2 \pi } \right)^2 \right)^{1/3} $ || || $ e $ || eccentricity (unitless) || $ \Large { { r_a } \over { a } } - 1 $ || $ \Large { { r_a - r_p } \over { r_a + r_p } } $ || || $ r_p $ || perigee, periapsis (m) || $ a ( 1 - e ) $ || || $ r_a $ || apogee, apoapsis (m) || $ a ( 1 + e ) $ || || $ v_o $ || orbit velocity (m/s) || $ \large \sqrt{ \mu ( 1 - e^2 ) / a } $ || || $ v_p $ || perigee velocity (m/s) || $ ( 1 + e ) v_0 $ || || $ v_a $ || apogee velocity (m/s) || $ ( 1 - e ) v_0 $ || || $ \theta $|| true anomaly, orbit angle from perigee || radians or degrees || || $ r $|| radius (m) || $ \large a ( 1 - e^2 ) / ( 1 + e \cos( \theta ) ) $ || ||$v_{\perp}$|| velocity perpendicular to radius (m/s) || $ \large v_0 ( 1 + e \cos( \theta ) ) $ || || $ v_r $ || radial velocity (m/s) || $ \large e v_0 \sin( \theta ) $ || |
Changing Orbits to M288
How do we get from any arbitrary orbit to M288? The following techniques may not be optimal for minimal \Delta V , but they set an upper bound. We will assume Kepler orbits and a circular M288 destination orbit, and heavy objects unaffected by light pressure. Actual thinsat orbits are elliptical and precess over a year with J2 and light pressure.
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The origin elliptical Kepler orbit has four parameters we need to compute two \Delta V burns for our transfer orbit: |
Maneuvers use the least \Delta V farther out, so we will use different strategies depending on whether the apogee of the origin orbit is higher or lower than the M288 orbit r_288 .
Describing an arbitrary Kepler Orbit
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Blue curve: |
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Red curve: |
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Green curve: |
gnuplot source and you will need to crop and convert and resize using gimp to reproduce the above images.
\mu |
earth gravitational constant |
3.986004418e14 m3/s2 |
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T |
sidereal orbital period (s) |
\Large { { 2 \pi } \over \omega } |
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a |
semimajor axis (m) |
0.5 ( r_a + r_p ) |
\large \left( \mu \left( T \over { 2 \pi } \right)^2 \right)^{1/3} |
e |
eccentricity (unitless) |
\Large { { r_a } \over { a } } - 1 |
\Large { { r_a - r_p } \over { r_a + r_p } } |
r_p |
perigee, periapsis (m) |
a ( 1 - e ) |
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r_a |
apogee, apoapsis (m) |
a ( 1 + e ) |
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v_o |
orbit velocity (m/s) |
\large \sqrt{ \mu ( 1 - e^2 ) / a } |
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v_p |
perigee velocity (m/s) |
( 1 + e ) v_0 |
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v_a |
apogee velocity (m/s) |
( 1 - e ) v_0 |
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\theta |
true anomaly, orbit angle from perigee |
radians or degrees |
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r |
radius (m) |
\large a ( 1 - e^2 ) / ( 1 + e \cos( \theta ) ) |
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v_{\perp} |
velocity perpendicular to radius (m/s) |
\large v_0 ( 1 + e \cos( \theta ) ) |
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v_r |
radial velocity (m/s) |
\large e v_0 \sin( \theta ) |
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