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

Anomaly

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
orbit angle from focus

\Large \theta

eccentric anomaly
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

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 )

total velocity
tangent to orbit

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

orbit energy parameter

\large C_3 = \mu / a



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.

3.98600448E+14

1.3271244E+20

m3/s2

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