Differences between revisions 39 and 50 (spanning 11 versions)

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
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 ||<-3:> \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 )

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
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:

-  ⇤ ← Revision 39 as of 2011-04-02 08:25:08 → 
  Size: 7653
  Editor: KeithLofstrom
  Comment:
+   ← Revision 50 as of 2011-04-03 01:05:32 → ⇥
  Size: 8368
  Editor: KeithLofstrom
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 3:
-----
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|Drawing from Wikipedia|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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+Line 11:
+|| $J_2$ Speedup factor                    || $ \large \omega'/\omega = 1+1.5 |J_2| (R_{eq}/a)^2$ ||
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+Line 16:
-|| eccentric anomaly<<BR>>ellipse center angle||$\large E=\Large{{e+\cos(\theta)}\over{1+e\cos(\theta)}}$|| || periapsis    || apoapsis               ||
+|| eccentric anomaly<<BR>>ellipse center angle||$\large E=\arccos\Large left({{e+\cos(\theta)}\over{1+e\cos(\theta)}}right)$|| || periapsis || apoapsis ||
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+Line 23:
-|| orbit energy parameter              || $ \large C_3 = \mu / a = v_p v_a                        $||<-3:>  $\large \mu = a v_p v_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        $||=
-Line 74:
+Line 73:
-|| $ a_g $  || gravity             || 8.938095  || 2.437062   || 1.911369E  || 0.02242078 || 0.002697573|| 0.005930053  || s        ||
+|| $ 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        ||
-Line 89:
+Line 90:
-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.
+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.
-Line 92:
+Line 93:
+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