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| Deletions are marked like this. | Additions are marked like this. |
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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 $ ||<|7-3>{{attachment:Anomaly.png|Drawing from Wikipedia|width=400}}|| |
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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=\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 || |
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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}}}$||$\large v=v_p, r=r_p $||$ \large v=v_a, r=r_a $|| || orbit energy parameter || $ \large C_3 = \mu / 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 z \vec k - x \vec i - y \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 $ | In 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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| 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 || || || $ a_g $ || gravity || 8.938095 || 2.437062 || 1.911369E || 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 precise, they represent the accuracy 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 || |
Near Circular Orbits
MORE LATER add pointers to orbit discussions
General Elliptical Orbits
In the orbital plane, neglecting the J_2 spherical oblateness parameter :
gravitational parameter |
\large \mu = a^3 \omega^2 = G M |
|
||
semimajor axis |
\large a = \sqrt[3]{ \mu / \omega^2 } |
|||
angular velocity |
\large \omega = \sqrt{ \mu / a^3 } |
|||
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 |
\Large \theta |
|||
eccentric anomaly |
\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 |
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 |
\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}} |
\large v=v_p, r=r_p |
\large v=v_a, r=r_a |
|
|| orbit energy parameter || \large C_3 = \mu / 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 )
In 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
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 |
|
||||
a_g |
gravity |
8.938095 |
2.437062 |
1.911369E |
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 precise, they represent the accuracy of the measurements we will someday compute.
The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.