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 |
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semimajor axis |
\large a = \sqrt[3]{ \mu / \omega^2 } |
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angular velocity |
\large \omega = \sqrt{ \mu / a^3 } |
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sidereal period |
\large T = 2\pi / \omega = 2\pi \sqrt{ a^3/\mu } |
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eccentricity |
\large e = ( r_a - r_p ) / ( r_a + r_p ) |
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velocity |
\large v_0 = \sqrt{ \mu / ( a (1 - e^2) ) } |
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true anomaly |
\Large \theta |
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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 ) |
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total velocity |
\Large v=\LARGE \sqrt{{{2\mu}\over{r}}-{{\mu}\over{a}}} |
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orbit energy parameter |
\large C_3 = \mu / a |
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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
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
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.
The eccentricities for the M orbits assume orbits mapped onto a 50km minor radius toroid.
symbol |
|
LEO 300Km |
M288 |
M360 |
GEO |
Moon |
Earth |
units |
\mu |
gravitation param. |
3.98600448E14 |
1.3271244E20 |
m3/s2 |
||||
|
relative to |
earth |
earth |
earth |
earth |
earth |
Sun |
|
a_g |
gravity |
8.938095 |
2.437062 |
1.911369E |
0.02242078 |
0.002697573 |
0.005930053 |
s |
T |
sidereal period |
5431.00996 |
14393.4323 |
17270.5433 |
86164.100 |
2360591.577 |
31558149.76 |
s |
T_s |
synodic period |
5431.944772 |
14400 |
17820 |
86400 |
2551442.9 |
31558149.76 |
s |
T/2\pi |
Sidereal / 2pi |
864.37208 |
2290.78585 |
2748.69228 |
13713.44093 |
375698.7212 |
5022635.5297 |
s |
\omega |
Angular velocity |
1.15690E-03 |
4.36531E-04 |
3.63809E-04 |
7.29212E-05 |
2.66171E-06 |
1.99099E-07 |
rad/s |
|
orbits/year |
5810.7332 |
2192.5382 |
1827.2819 |
366.25640 |
13.36879 |
1.00000 |
|
a |
semimajor axis |
6678000 |
12788970.60 |
14440980.32 |
42164169.86 |
384399000 |
1.4959826E11 |
m |
R_a |
apogee radius |
6678000 |
12838970.60 |
14490980.32 |
42164169.86 |
405696000 |
1.5209823E11 |
m |
R_p |
perigee radius |
6678000 |
12738970.60 |
14440980.32 |
42164169.86 |
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 |