Differences between revisions 22 and 54 (spanning 32 versions)
⇤ ← Revision 22 as of 2011-04-01 22:33:21 →
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Editor: KeithLofstrom
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← Revision 54 as of 2011-04-03 20:21:47 → ⇥
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Editor: KeithLofstrom
Comment:
Deletions are marked like this. Additions are marked like this.
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----
MORE LATER add pointers to orbit discussions
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|| gravitational parameter || $ \large \mu = a^3 \omega^2 = G M $ ||<|7-3>{{attachment:Anomaly.png|Anomaly|width=400}}|| || gravitational parameter || $ \large \mu = a^3 \omega^2 = G M $ ||<|8-3>{{attachment:Anomaly.png|Drawing from Wikipedia|width=400}}||
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|| $J_2$ Speedup factor || $ \large \omega'/\omega = 1+1.5 |J_2| (R_{eq}/a)^2$ ||
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|| eccentric anomaly<<BR>>ellipse center angle||$\large E = \Large { { e+\cos(\theta) } \over { 1+e\cos(\theta)}}$|| || periapsis || apoapsis ||
|| mean anomaly, || $ \large M = E - e \sin( E ) $||<)> earth || perigee         || apogee ||
|| time from perigee || $ \large t = M / \omega $||<)> sun || perihelion           || apohelion ||		 || eccentric anomaly<<BR>>ellipse center angle||$\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right)$|| || periapsis || apoapsis ||
|| mean anomaly, || $ \large M = E - e \sin( E )         $||<)> earth || perigee || apogee ||
|| time from perigee || $ \large t = M / \omega ? ? ? $||<)> sun || perihelion || apohelion ||
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|| radial velocity        || $ \large v_r = e v_0 \sin( \theta )   $||
|| total velocity<<BR>>tangent to orbit    || $ \Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}$||
|| orbit energy parameter     || $ \large C_3 = \mu / a                            $||
 || radial velocity || $ \large v_r = e v_0 \sin( \theta )$||<:>$\huge\rightarrow$||$ 0 $||$ 0 $||
|| total velocity<<BR>>tangent to orbit|| $\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}$||<:>$\huge\rightarrow$||$\large v=v_p,~ r=r_p$||$ \large v=v_a,~ r=r_a$||
|| orbit energy parameter || $ \large C_3 = \mu / a = v_p v_a $||<-3:> $\large \mu = a v_p v_a ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 a = r_p + r_a $||
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In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are  In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are
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The rotation is expressed as $ \vec \Omega = \omega \hat k $, where $ \omega $ is the angular velocity of the orbit .   The rotation is expressed as $ \vec \Omega = \omega \hat k $, where $ \omega $ is the angular velocity of the orbit .
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Coriolis acceleration: $ \Large \vec a_{Coriolis} = -2 \vec \Omega \times \dot { \vec r } $ Coriolis acceleration: $ \LARGE \ddot{\vec r}_{Coriolis} = -2 \vec \Omega \times \dot { \vec r } $
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Centrifugal acceleration: $ \Large \vec a_{Centrifugal} = - \vec \Omega \times \vec \Omega \times \vec r $ Centrifugal acceleration: $ \LARGE \ddot{\vec r}_{Centrifugal} = - \vec \Omega \times \vec \Omega \times \vec r $
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Centripedal (gravity) acceleration: $ \Large \vec a_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $
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In scalar equations: FROM HERE DOWN, WORK IN PROGRESS, NOT VERIFIED:
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$ \ddot x = -2 \omega \dot y - \omega^2 x $ Centripedal (gravity) acceleration: $ \LARGE \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j ) $
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$ \ddot y = 2 \omega \dot x + 3 \omega^2 y $ Scalar equations:
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$ \ddot z = -\omega^2 z $ $ \LARGE \ddot x = -2 \omega \dot y - \omega^2 x $

$ \LARGE \ddot y = 2 \omega \dot x + 3 \omega^2 y $

$ \LARGE \ddot z = -\omega^2 z $
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If $e$ is very small, we can assume $e^2 \approx 0 $.
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$\Large {1\over{1+e\cos(\theta)}}={{1-e\cos(\theta)}\over{\left(1+e\cos(\theta)\right)\left(1-e\cos(\theta)\right)}}$

$\Large ~ ~ ~ ~ ~ ~ ~ = {{1-e\cos(\theta)}\over{\left(1-e^2\cos^2(\theta)\right)}}$

$\large ~ ~ ~ ~ ~ ~ ~ \approx 1-e\cos(\theta) ~ ~ ~ $ ... since $ \large e^2 \approx 0 $

$\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right)$

$\large \cos( E ) = { \Large {{e+\cos(\theta)}\over{1+e\cos(\theta)}}} \large { \approx \left(e+\cos(\theta)\right)\left( 1-e\cos(\theta) \right) } $

$ \large ~ ~ ~ ~ ~ ~ ~ \approx e + \cos(\theta) - e^2 \cos(\theta) - e \cos^2( \theta ) $

$ \large ~ ~ ~ ~ ~ ~ ~ \approx \cos(\theta) + e \left( 1 - \cos^2( \theta ) \right) $

$ \Large \cos( E ) \approx \cos(\theta) + e\sin^2( \theta ) $

$ \large \sin( E ) = \sqrt{ 1 - \cos^2( E ) } \approx \sqrt{ 1 - \left( \cos(\theta) + e \sin^2( \theta ) \right)^2 } $

$ \large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ 1 - \left( \cos^2(\theta) + 2e \cos(\theta)\sin^2( \theta ) + e^2 \sin^4( \theta ) \right) } $

$ \large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ 1 - \cos^2(\theta) - 2e \cos(\theta)\sin^2( \theta ) } $

$ \large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ \sin^2(\theta) \left( 1 - 2e \cos(\theta) \right) } $

$ \Large \sin( E ) \approx \sin(\theta ) \left( 1 - e \cos(\theta) \right)$

$ \large M = E - e \sin( E ) \approx E - e \left( \sin(\theta ) \left( 1 - e \cos(\theta) \right) \right) $

$ \large M \approx E - e \sin(\theta) + e^2 \cos(\theta) $

$ \Large M \approx E - e \sin(\theta) $

$ \large \cos( M ) \approx \cos( E - e \sin(\theta) ) = \cos( E ) \cos( e \sin(\theta) ) + \sin( E ) \sin( e \sin(\theta) ) $

$ \large \sin( e X ) \approx e X - (e X)^3 / 6 + (e X)^5 / 120 + ... \approx e X ~ ~ ~ $ higher order terms are < e^2^

$ \large \cos( e X ) \approx 1 - (e X)^2 / 2 + (e X)^4 / 24 + ... \approx 1 ~ ~ ~ $ higher order terms are < $ e^2 $

$ \large \cos( M ) \approx \cos( E ) + e \sin( E ) \sin(\theta) ) $

$ \large \cos( M ) \approx \cos(\theta) + e\sin^2( \theta ) + e \left( \sin(\theta ) \left( 1 - e \cos(\theta) \right) \right) $

$ \Large \cos( M ) \approx \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) ) $

$ \large \sin( M ) \approx \sin( E - e \sin(\theta) ) = \sin( E ) \cos( e \sin(\theta) ) - \cos( E ) \sin ( e \sin(\theta) ) $

$ \large \sin( M ) \approx \sin( E ) - e \cos( E ) \sin(\theta) $

$ \large \sin( M ) \approx \sin(\theta ) \left( 1 - e \cos(\theta) \right) - e \left( \cos(\theta) + e\sin^2( \theta )\right)\sin(\theta) $

$ \large \sin( M ) \approx \sin(\theta) - e \sin(\theta)\cos(\theta) - e\sin(\theta)\cos(\theta) + e^2\sin^2(\theta) $

$ \Large \sin( M ) \approx \sin(\theta)( 1 - 2 e \cos(\theta)) $

We want to find the locus of x and y, rotated to vertical by angle -M, and subtracting [ 0, a ] .

$ \large r= a \Large{{1-e^2}\over{1+e\cos(\theta)}} \large \approx a ( 1 - e \cos(\theta) ) $

$ \large x = r \cos( \theta ) \approx a ( 1 - e \cos(\theta) ) \cos( \theta ) $

$ \large y = r \sin( \theta ) \approx a ( 1 - e \cos(\theta) ) \sin( \theta ) $

Rotate the vector [ x , y ] by -M to :

$ \large [ x', y' ] = \left( \begin{array}{cc} \cos( M ) & \sin( M ) \\ -\sin( M ) & \cos( M ) \end{array} \right) [ x, y ] $

$ \large x' = \cos( M ) x + \sin( M ) y $

$ \large x' \approx ( \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) ) ) ~ a ( 1 - e \cos(\theta) ) \cos( \theta ) ~ + ~ \sin(\theta)( 1 - 2 e \cos(\theta)) ~ a ( 1 - e \cos(\theta) ) \sin(\theta) $

$ \large x' / a \approx ( \cos(\theta) + 2e\sin^2( \theta ) - 2e\sin^2\cos(\theta) ) ( \cos( \theta ) - e \cos^2^(\theta) ) ~ + ~ \sin^2(\theta)( 1 - 3 e \cos(\theta)) + e \cos^2(\theta) ) $




$ \large y' = \cos( M ) y - \sin( M ) x $

$ \large y' \approx ( \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) ) ) ~ a ( 1 - e \cos(\theta) ) \sin( \theta ) ~ - ~ \sin(\theta)( 1 - 2 e \cos(\theta)) ~ a ( 1 - e \cos(\theta) ) \cos(\theta) $






MORE LATER
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Sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds M288 sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds
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1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds.
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1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds === A Table of orbits ===
|| || || LEO 300Km || M288 || M360 || GEO || Moon || Earth || units ||
|| $ \mu $ || gravitation param. ||<:-5> 3.98600448e14 || 1.3271244e20 || m^3^/s^2^||
|| ||<)> relative to ||<:-5> earth || Sun || ||
|| $ J_2 $ || Oblate-ness ||<:-5> -1.082626683e-3 earth radius = 6378000m || -6e-7 || ||
||$\omega'/\omega$||J2 speedup fraction|| 1.4813e-3|| 4.0389e-4|| 3.1677e-4 || 3.7158e-5 || 4.4707e-7 || 2e-17 || ||
|| $ a_g $ || gravity || 8.938095 || 2.437062 || 1.9113693 || 0.02242078 || 0.002697573|| 0.005930053 || s ||
|| $ T $ || sidereal period || 5431.010 || 14393.4323 || 17270.5433 || 86164.100 || 2360591.577|| 31558149.76 || s ||
|| $ T_s $ || synodic period || 5431.945 || 14400 || 17820 || 86400 || 2551442.9 || 31558149.76 || s ||
|| $ T/2\pi$|| Sidereal / 2pi || 864.3721 || 2290.78585 || 2748.69228 || 13713.44093|| 375698.7212|| 5022635.5297|| s ||
|| $ \omega$|| Angular velocity || 1.1569e-03|| 4.36531e-04|| 3.63809e-04|| 7.29212e-05|| 2.66171e-06|| 1.99099e-07 || rad/s ||
|| || orbits/year || 5810.733 || 2192.5382 || 1827.2819 || 366.25640 || 13.36879 || 1.00000 || ||
|| $ a $ || semimajor axis || 6678000 || 12788971 || 14440980 || 42164170 || 384399000 || 1.4959826e11 || m ||
|| $ R_a $ || apogee radius || 6678000 || 12838976 || 14490980 || 42164170 || 405696000 || 1.5209823e11 || m ||
|| $ R_p $ || perigee radius || 6678000 || 12738976 || 14440980 || 42164170 || 363104000 || 1.4709829e11 || m ||
|| $ e $ || eccentricity || 0.000000 || 0.001951 || 0.001728 || 0.000000 || 0.055401 || 0.016711 || ||
|| $ V_0 $ || mean velocity || 7725.84 || 5582.79 || 5253.76 || 3074.66 || 1023.16 || 29784.81 || m/s ||
|| $ V_a $ || apogee velocity || 7725.84 || 5571.90 || 5244.68 || 3074.66 || 966.47 || 29287.07 || m/s ||
|| $ V_p $ || perigee velocity || 7725.84 || 5593.68 || 5262.84 || 3074.66 || 1079.84 || 30282.55 || m/s ||
|| $ C_3 $ || orb. specific energy|| -59688597 || -31167516 || -27602035 || -9453535 || -1036945 || -887125553 || J/kg ||
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----
Note: The following table uses the classic formula $ \omega^2 a^3 = \mu $ from the books, and does not take into account the oblate spheroid shape of the Earth, which adds centripedal force for low orbits and thus should increase $ \omega $ and $ | C_3 | $. So please check these numbers.		 Note: This table uses the classic formula $ \omega^2 a^3 = \mu $, and does not take into account the oblate spheroid shape of the Earth, and many other perturbations. However, with the perturbations included, and with good data from ground stations and GPS to establish positions and velocities, we really can compute these numbers to this many decimal places. So while the numbers above are actually far less accurate, they represent the precision of the measurements we will someday compute.
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|| symbol || || LEO 300Km || M288 || M360 || GEO || Moon || Earth || units ||
|| $ \mu $ || gravitation param. ||<:-5> 3.98600448E+14 || 1.3271244E+20 || m^3^/s^2^ ||
|| ||<)> relative to || earth || earth || earth || earth || earth || Sun || ||
|| $ a_g $ || gravity || 8.938095E+00 || 2.437062E+00 || 1.911369E+00 || 2.242078E-01 || 2.697573E-03 || 5.930053E-03 || seconds ||
|| $ T $ || sidereal period || 5431.009959 || 14393.432269 || 17270.543331 || 86164.099662 || 2360591.57743 || 31558149.7635 || seconds ||
|| $ T_s $ || synodic period || 5431.944772 || 14400.000000 || 17820.000000 || 86400.000000 || 2551442.90000 || 31558149.7635 || seconds ||
|| $ T/2\pi $ || Sidereal / 2pi || 864.372081 || 2290.785853 || 2748.692283 || 13713.440926 || 375698.721205 || 5022635.5297 || seconds ||
|| $ \omega $ || Angular velocity || 1.1569092E-03 || 4.3653142E-04 || 3.6380937E-04 || 7.2921159E-05 || 2.6617072E-06 || 1.9909866E-07 || rad/sec ||
|| || orbits/year || 5810.73318 || 2192.53822 || 1827.28185 || 366.25640 || 13.36879 || 1.00000 || ||
|| $ a $ || semimajor axis || 6678000.00 || 12788970.60 || 14440980.32 || 42164169.86 || 384399000.00 || 149598261000 || meters ||
|| $ R_a $ || apogee radius || 6678000.00 || 12838970.60 || 14490980.32 || 42164169.86 || 405696000.00 || 152098232000 || meters ||
|| $ R_p $ || perigee radius || 6678000.00 || 12738970.60 || 14440980.32 || 42164169.86 || 363104000.00 || 147098290000 || meters ||
|| $ e $ || eccentricity || 0.000000 || 0.001951 || 0.001728 || 0.000000 || 0.055401 || 0.016711 || ||
|| $ V_0 $ || mean velocity || 7725.84 || 5582.79 || 5253.76 || 3074.66 || 1023.16 || 29784.81 || m/s ||
|| $ V_a $ || apogee velocity || 7725.84 || 5571.90 || 5244.68 || 3074.66 || 966.47 || 29287.07 || m/s ||
|| $ V_p $ || perigee velocity || 7725.84 || 5593.68 || 5262.84 || 3074.66 || 1079.84 || 30282.55 || m/s ||
|| $ C_3 $ || orb. specific energy || -59688596.59 || -31167516.16 || -27602035.26 || -9453534.82 || -1036944.55 || -887125553 || J/kg ||		 REFS:

 . http://articles.adsabs.harvard.edu/full/1981AJ.....86..912G
 . http://en.wikipedia.org/wiki/Kepler_orbit
 . http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion
 . http://en.wikipedia.org/wiki/Orbital_period
 . http://en.wikipedia.org/wiki/Mean_anomaly
 . http://en.wikipedia.org/wiki/Eccentric_anomaly ... where I got the picture, then modified it

Near Circular Orbits

General Elliptical Orbits

In the orbital plane, neglecting the J_2 spherical oblateness parameter :

gravitational parameter

\large \mu = a^3 \omega^2 = G M

Drawing from Wikipedia

semimajor axis

\large a = \sqrt[3]{ \mu / \omega^2 }

angular velocity

\large \omega = \sqrt{ \mu / a^3 }

J_2 Speedup factor

\large \omega'/\omega = 1+1.5 |J_2| (R_{eq}/a)^2

sidereal period

\large T = 2\pi / \omega = 2\pi \sqrt{ a^3/\mu }

eccentricity

\large e = ( r_a - r_p ) / ( r_a + r_p )

velocity

\large v_0 = \sqrt{ \mu / ( a (1 - e^2) ) }

true anomaly
orbit angle from focus

\Large \theta

eccentric anomaly
ellipse center angle

\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right)

periapsis

apoapsis

mean anomaly,

\large M = E - e \sin( E )

earth

perigee

apogee

time from perigee

\large t = M / \omega ? ? ?

sun

perihelion

apohelion

radius

\large r=a\Large{{1-e^2}\over{1+e\cos(\theta)}}

\huge\rightarrow

\large r_p =( 1-e )a

\large r_a =( 1+e )a

perpendicular velocity

\large v_{\perp}= v_0 ( 1+e \cos( \theta ))

\huge\rightarrow

\large v_p =(1+e)v_0

\large v_a =(1-e)v_0

radial velocity

\large v_r = e v_0 \sin( \theta )

\huge\rightarrow

0

0

total velocity
tangent to orbit

\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}}

\huge\rightarrow

\large v=v_p,~ r=r_p

\large v=v_a,~ r=r_a

orbit energy parameter

\large C_3 = \mu / a = v_p v_a

\large \mu = a v_p v_a ~ ~ ~ ~ ~ ~ ~ ~ ~ 2 a = r_p + r_a

Fictional Forces in Orbit

In the rotating frame of a circular orbit, counterclockwise viewed from above the orbital plane, the directions are

direction

unit vector

description

x

\vec i

Tangential to (along the line of) the orbit, in the orbital plane, pointing clockwise or westward

y

\vec j

Radially outwards from the center of rotation, in the orbital plane

z

\vec k

Perpendicular to the orbital plane, northwards

The rotation is expressed as \vec \Omega = \omega \hat k , where \omega is the angular velocity of the orbit . The radial vector \vec r is composed of \vec r = x \vec i + y \vec j + z \vec k .

Coriolis acceleration: \LARGE \ddot{\vec r}_{Coriolis} = -2 \vec \Omega \times \dot { \vec r }

Centrifugal acceleration: \LARGE \ddot{\vec r}_{Centrifugal} = - \vec \Omega \times \vec \Omega \times \vec r

FROM HERE DOWN, WORK IN PROGRESS, NOT VERIFIED:

Centripedal (gravity) acceleration: \LARGE \ddot{\vec r}_{Centripedal} = \omega^2 ( 2 y \vec k - x \vec i - z \vec j )

Scalar equations:

\LARGE \ddot x = -2 \omega \dot y - \omega^2 x

\LARGE \ddot y = 2 \omega \dot x + 3 \omega^2 y

\LARGE \ddot z = -\omega^2 z

Locus of Elliptical Orbit Position in Rotating Frame

If e is very small, we can assume e^2 \approx 0 .

\Large {1\over{1+e\cos(\theta)}}={{1-e\cos(\theta)}\over{\left(1+e\cos(\theta)\right)\left(1-e\cos(\theta)\right)}}

\Large ~ ~ ~ ~ ~ ~ ~ = {{1-e\cos(\theta)}\over{\left(1-e^2\cos^2(\theta)\right)}}

\large ~ ~ ~ ~ ~ ~ ~ \approx 1-e\cos(\theta) ~ ~ ~ ... since \large e^2 \approx 0

\large E=\arccos\Large\left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}\right)

\large \cos( E ) = { \Large {{e+\cos(\theta)}\over{1+e\cos(\theta)}}} \large { \approx \left(e+\cos(\theta)\right)\left( 1-e\cos(\theta) \right) }

\large ~ ~ ~ ~ ~ ~ ~ \approx e + \cos(\theta) - e^2 \cos(\theta) - e \cos^2( \theta )

\large ~ ~ ~ ~ ~ ~ ~ \approx \cos(\theta) + e \left( 1 - \cos^2( \theta ) \right)

\Large \cos( E ) \approx \cos(\theta) + e\sin^2( \theta )

\large \sin( E ) = \sqrt{ 1 - \cos^2( E ) } \approx \sqrt{ 1 - \left( \cos(\theta) + e \sin^2( \theta ) \right)^2 }

\large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ 1 - \left( \cos^2(\theta) + 2e \cos(\theta)\sin^2( \theta ) + e^2 \sin^4( \theta ) \right) }

\large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ 1 - \cos^2(\theta) - 2e \cos(\theta)\sin^2( \theta ) }

\large ~ ~ ~ ~ ~ ~ ~ \approx \sqrt{ \sin^2(\theta) \left( 1 - 2e \cos(\theta) \right) }

\Large \sin( E ) \approx \sin(\theta ) \left( 1 - e \cos(\theta) \right)

\large M = E - e \sin( E ) \approx E - e \left( \sin(\theta ) \left( 1 - e \cos(\theta) \right) \right)

\large M \approx E - e \sin(\theta) + e^2 \cos(\theta)

\Large M \approx E - e \sin(\theta)

\large \cos( M ) \approx \cos( E - e \sin(\theta) ) = \cos( E ) \cos( e \sin(\theta) ) + \sin( E ) \sin( e \sin(\theta) )

\large \sin( e X ) \approx e X - (e X)^3 / 6 + (e X)^5 / 120 + ... \approx e X ~ ~ ~ higher order terms are < e2

\large \cos( e X ) \approx 1 - (e X)^2 / 2 + (e X)^4 / 24 + ... \approx 1 ~ ~ ~ higher order terms are < e^2

\large \cos( M ) \approx \cos( E ) + e \sin( E ) \sin(\theta) )

\large \cos( M ) \approx \cos(\theta) + e\sin^2( \theta ) + e \left( \sin(\theta ) \left( 1 - e \cos(\theta) \right) \right)

\Large \cos( M ) \approx \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) )

\large \sin( M ) \approx \sin( E - e \sin(\theta) ) = \sin( E ) \cos( e \sin(\theta) ) - \cos( E ) \sin ( e \sin(\theta) )

\large \sin( M ) \approx \sin( E ) - e \cos( E ) \sin(\theta)

\large \sin( M ) \approx \sin(\theta ) \left( 1 - e \cos(\theta) \right) - e \left( \cos(\theta) + e\sin^2( \theta )\right)\sin(\theta)

\large \sin( M ) \approx \sin(\theta) - e \sin(\theta)\cos(\theta) - e\sin(\theta)\cos(\theta) + e^2\sin^2(\theta)

\Large \sin( M ) \approx \sin(\theta)( 1 - 2 e \cos(\theta))

We want to find the locus of x and y, rotated to vertical by angle -M, and subtracting [ 0, a ] .

\large r= a \Large{{1-e^2}\over{1+e\cos(\theta)}} \large \approx a ( 1 - e \cos(\theta) )

\large x = r \cos( \theta ) \approx a ( 1 - e \cos(\theta) ) \cos( \theta )

\large y = r \sin( \theta ) \approx a ( 1 - e \cos(\theta) ) \sin( \theta )

Rotate the vector [ x , y ] by -M to :

\large [ x', y' ] = \left( \begin{array}{cc} \cos( M ) & \sin( M ) \\ -\sin( M ) & \cos( M ) \end{array} \right) [ x, y ]

\large x' = \cos( M ) x + \sin( M ) y

\large x' \approx ( \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) ) ) ~ a ( 1 - e \cos(\theta) ) \cos( \theta ) ~ + ~ \sin(\theta)( 1 - 2 e \cos(\theta)) ~ a ( 1 - e \cos(\theta) ) \sin(\theta)

\large x' / a \approx ( \cos(\theta) + 2e\sin^2( \theta ) - 2e\sin^2\cos(\theta) ) ( \cos( \theta ) - e \cos^2^(\theta) ) ~ + ~ \sin^2(\theta)( 1 - 3 e \cos(\theta)) + e \cos^2(\theta) )

\large y' = \cos( M ) y - \sin( M ) x

\large y' \approx ( \cos(\theta) + e\sin^2( \theta ) ( 2 - \cos(\theta) ) ) ~ a ( 1 - e \cos(\theta) ) \sin( \theta ) ~ - ~ \sin(\theta)( 1 - 2 e \cos(\theta)) ~ a ( 1 - e \cos(\theta) ) \cos(\theta)

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Periods of M orbits

M orbits describe the number of minutes an orbit takes travel once around the earth and return to the same position overhead. For server sky, these are integer fractions of a 1440 minute synodic day; this makes it easier to calculate the sky position given the orbital parameters and the time of day. the M288 orbit returns to the same apparent position 5 times per day, or 365.256...*5 times per year. That position moves around the earth 365.256...+1 times per year. So the total number of orbits per year, relative to the stars, is 365.256...*6+1 orbits per year. That is divided into the year length in seconds to yield the

M288 sidereal orbit time = ( 365.256... * 86400 ) / ( 365.256... * 6 + 1 ) = 86400 / ( 6 + 1/365.256... ) = 14393.43227 seconds

1 year = 365.256363004 days of 86,400 seconds, or 31558149.7635456 seconds.

A Table of orbits

LEO 300Km

M288

M360

GEO

Moon

Earth

units

\mu

gravitation param.

3.98600448e14

1.3271244e20

m3/s2

relative to

earth

Sun

J_2

Oblate-ness

-1.082626683e-3 earth radius = 6378000m

-6e-7

\omega'/\omega

J2 speedup fraction

1.4813e-3

4.0389e-4

3.1677e-4

3.7158e-5

4.4707e-7

2e-17

a_g

gravity

8.938095

2.437062

1.9113693

0.02242078

0.002697573

0.005930053

s

T

sidereal period

5431.010

14393.4323

17270.5433

86164.100

2360591.577

31558149.76

s

T_s

synodic period

5431.945

14400

17820

86400

2551442.9

31558149.76

s

T/2\pi

Sidereal / 2pi

864.3721

2290.78585

2748.69228

13713.44093

375698.7212

5022635.5297

s

\omega

Angular velocity

1.1569e-03

4.36531e-04

3.63809e-04

7.29212e-05

2.66171e-06

1.99099e-07

rad/s

orbits/year

5810.733

2192.5382

1827.2819

366.25640

13.36879

1.00000

a

semimajor axis

6678000

12788971

14440980

42164170

384399000

1.4959826e11

m

R_a

apogee radius

6678000

12838976

14490980

42164170

405696000

1.5209823e11

m

R_p

perigee radius

6678000

12738976

14440980

42164170

363104000

1.4709829e11

m

e

eccentricity

0.000000

0.001951

0.001728

0.000000

0.055401

0.016711

V_0

mean velocity

7725.84

5582.79

5253.76

3074.66

1023.16

29784.81

m/s

V_a

apogee velocity

7725.84

5571.90

5244.68

3074.66

966.47

29287.07

m/s

V_p

perigee velocity

7725.84

5593.68

5262.84

3074.66

1079.84

30282.55

m/s

C_3

orb. specific energy

-59688597

-31167516

-27602035

-9453535

-1036945

-887125553

J/kg

Note: This table uses the classic formula \omega^2 a^3 = \mu , and does not take into account the oblate spheroid shape of the Earth, and many other perturbations. However, with the perturbations included, and with good data from ground stations and GPS to establish positions and velocities, we really can compute these numbers to this many decimal places. So while the numbers above are actually far less accurate, they represent the precision of the measurements we will someday compute.

The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.

REFS:

NearCircularOrbits (last edited 2022-09-14 00:16:43 by KeithLofstrom)