JSR Launch Logs

Jonathan C. McDowell

Orbital Elements

Orbital Elements - Introduction

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:

Earth-related properties

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.

Orbital derived quantities

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_r, the longitude drift rate, or difference between the orbital angular velocity and the Earth's rotational angular velocity:
^.Theta_r = 2Pi 1/T - 1/T_GEO

Some categories

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_r = 0 (geostationary), ^.Omega = ^.theta_n (sun-synchronous), and ^.Omega = 0 (Molniya).

GEO/S, the geostationary earth orbit with T = 23:56 satisfies ^.Theta_r = 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; (however, SPACECOM defines deep space to begin at 3hr45 min, corresponding approximately to a limitation on particular sensors).
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. During spacecraft reentries, breakup is usually within 10 km of an altitude of 78 km, according to an Aerospace Corp. study, which also lends support to defining the boundary of space near to 80 km.

In contrast, the X-Prize adopted the "Karman line" at 100 km.

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.

Ballistic missiles fly on a variety of trajectories with perigees between -6300 and ~ -2000 km. For short range missiles, often we only know the range but wish to know the apogee. The apogee which gives the minimum energy requirement for a given range Rho depends on the altitude of burnout h and in the limit of short ranges is h_a ~ h + 1/ 4 Rho ignoring atmospheric drag (which admittedly is most especially not to be ignored for short range, low altitude flights).

The IADC guidelines define LEO as 2000 km altitude or less, and GEO as i<15.0 and |h-h_GEO|<200 km.

The COGO defines GEO/C1 and GEO/C2 as |h-h_GEO|<10? km. and i< 3.0 or i≥ 3.0 respectively. In 2016 I redefined GEO/S and GEO/I to match these. Their broader GEO range of 1226 to 1656 minutes period seems too wide to me; I use 1380 to 1500 minutes.

Hyperbolic orbits

Hyperbolic orbits are sometimes labelled with a negative value of the semimajor axis, and I adopt this convention. I further extend it by retaining the definitions for apogee and perigee height even though the apogee doesn't have direct physical significance. In astronautical contexts the value C3, the square of the asymptotic velocity, is often used; it is related to the semimajor axis by

a = Mu/C3

where for the Earth, Mu=398603.2 in (km, s) units. The orbital categories I adopt are tabulated in Orbital Categories.