1.5: Orbital categories

A pure Keplerian orbit is described by six classical orbital elements:

• a, the semi-major axis (or mean orbital radius)

• e, the eccentricity,

• i, the inclination to the equatorial plane,

• Omega, the longitude of the ascending node at which the orbit crosses the equator northbound.

• Omega, the argument of perigee, the angle between the ascending node and the perigee.

• T₀, the time at which the satellite was at perigee.

At time T₀, the satellite is at perigee, the closest point to the Earth, with distance a(1-e) from the Earth's center.

The following terrestrial quantities are of interest:

• R_E = 6378.0 km, the nominal Earth equatorial radius.

• T_GEO = 23:56, the Earth sidereal rotation period. (Note: in this section times are expressed as hours:minutes).

• T_s = 1:24, the orbital period for a satellite skimming the Earth surface. It may be found more accurately by

  T_s = 4 Pi² / GM R_E³

  where M is the Earth mass.

• Theta_n = -0.986 deg/day, the rate of advance of the sidereal longitude of noon.

The Earth is not a point mass; field harmonics and atmospheric drag affect the orbit. We therefore consider instantaneous Keplerian elements valid at time T_e, the orbital epoch. We also consider the following auxiliary quantities:

• h_p, the perigee height

  h_p = a(1-e)-R_E

• a_p, the apogee height

  a_p = a(1+e)-R_E

• Phi, the true anomaly, or angle between perigee and the current position.

• T, the orbital period:

  T = T_s ( a / R_E ) ³

  by Kepler's third law.

• t, the time.

• M, the mean anomaly,

  M = 2Pi ( t - T₀ ) / T

• T_E, the time at equator crossing.

• ^.Omega, the precession of the orbital plane; for Earth,

  ^.Omega = (9.943 deg/d)( R_E / a )^3.5 cos i

• ^.Omega, the advance of perigee.

  ^.Omega = (9.943 deg/d)( R_E / a )^3.5 ( 2 - 2.5 sin² i)

• ^.Theta_l, the longitude drift rate, or difference between the orbital angular velocity and the Earth's rotational angular velocity:

  ^.Theta_l = 2Pi 1/T - 1/T_GEO

Earth satellites use a wide variety of orbital parameters. We can, however, group satellite orbits in some broad categories.

There are three mathematically special orbits, corresponding to ^.Theta_l = 0 (geostationary), ^.Omega = ^.theta_n (sun-synchronous), and ^.Omega = 0 (Molniya).

• GEO/S, the geostationary earth orbit with T = 23:56 satisfies ^.Theta_l = 0 and in addition, the actual geocentric longitude stays constant if i = 0.0 and e = 0.0 (so that the angular velocity vectors of Earth and the orbit coincide).

• SSO, sun-synchronous orbit, with

  T = 3:47 ( - cos i ) 3/7.

  This orbit has inclinations between 97 and 103 degrees.

• Molniya-type orbit, with i = 63.43.

• We can also note the natural special cases of i = 0 (equatorial eastbound) and i = 90 (polar). Since launches usually take advantage of the direction of Earth spin to gain velocity, the special case i = 180 (equatorial westbound) is not of interest.

• Orbits with apogees beyond the lunar sphere of influence are strongly affected by lunar gravity, and are lumped with deep space probes. The lunar orbital radius is 384400 km. The L2 point is at 326380 km. The lunar sphere of influence begins at 318200 km. The Earth-Sun L1 point is at 1497600 km in the solar direction.

We define orbital regimes in broad boxes around these special orbits, and adopt the following extra boundaries:

• a lower boundary of space, at 80 km (the mean mesopause);

• a transition boundary at 150 km (the F layer), below which satellite orbits are unstable to decay on timescales of hours.

• The LEO/MEO boundary at 2 hr orbital period, corresponding approximately to the lower edge of the intense part of the equatorial radiation belts;

• a special, arbitrary value e = 0.5 to divide circular and mildly elliptical orbits from highly elliptical orbits.

The mesopause is actually typically at 85-90km; I adopt 80 km to be generous and to match the 1960s USAF definition of space. The D layer of the ionosphere is 75-95 km; E-layer is 95-150 km; F layer is 150 and above. F1 at 170 km and F2 reflecting layer at about 250-450 km; topside ionosphere is above F2's max (at 300-400 km) up to the (O/H-He) transition layer at 500-1000 km.

The highest flying non-rocket plane is Helios, which reached 29 km on 2001 Aug 14. Balloons reach up to 50 km.

Code	Name	T	i	e	hp	ha
ATM	Atmospheric	N/A	0.0 - 180.0	0.0 - 1.0	<80	0 - 80
TAO	Trans-Atm.	N/A	0.0 - 180.0	0.0 - 1.0	0-80	>80
SO	Suborbital	N/A	0.0 - 180.0	0.0 - 1.0	<0	>80
LEO/E	Equatorial	1:26 - 2:00	0.0 - 20.0	0.0 - 0.21	80 - 1682	80 - 3284
LEO/I	Intermediate	1:26 - 2:00	20.0 - 85.0	0.0 - 0.21	80 - 1682	80 - 3284
LEO/P	Polar	1:26 - 2:00	85.0 - 95.0	0.0 - 0.21	80 - 1682	80 - 3284
LEO/S	Sun-Synch	1:26 - 2:00	95.0 - 104.0	0.0 - 0.21	80 - 1682	80 - 3284
LEO/R	Retrograde	1:26 - 2:00	104.0 - 180.0	0.0 - 0.21	80 - 1682	80 - 3284
MEO	Medium	2:00 - 23:00	0.0 - 180.0	0.0 - 0.50	80 - 34680	1682 - 55209
HEO	Highly Ellip	4:03 - 23:00	0.0 - 180.0	0.50 - 0.92	80 - 14331	13000 - 69280
HEO/M	Molniya	11:30 - 12:30	62.0 - 64.0	0.50 - 0.77	80 - 7294	19489 - 41854
GTO	GEO Transfer	10:00 - 12:30	0.0 - 85.0	0.50 - 0.77
GEO/S	Stationary	23:55.5-23:56.5	0.0 - 2.0	0.00 - 0.01	35353 - 35795	35775 - 36217
GEO/I	Inclined GEO	23:55.5-23:56.5	0.0 - 20.0	0.00 - 0.05	33667 - 35795	35775 - 37903
GEO/T	Synchronous	23:55.5-23:56.5	0.0 - 180.0	0.00 - 0.85	80 - 35795	35775 - 71510
GEO/D	Drift GEO	23:00 - 25:00	0.0 - 2.0	0.00 - 0.05	32628 - 37028	34681 - 39198
GEO/ID	Inclined Drift	23:00 - 25:00	0.0 - 20.0	0.00 - 0.05	32628 - 37028	34681 - 39198
GEO/NS	Near-synch	23:00 - 25:00	0.0 - 180.0	0.0 - 0.85	80 - 37028	34681 - 73976
DSO	Deep Space	> 25:00	0.0 - 180.0	0.0 - 0.50	>15325	>37028
DHEO	Deep Eccentric	>25:00	0.0 - 180.0	0.50 - 1.00	>80	>58731
CLO	Cislunar		0.0 - 180.0	0.0 - 1.00		>318200
EEO	Earth Escape
HCO	Heliocentric
PCO	Planetocentric
PEO	Planetary escape trajectory
SSE	Solar System Escape