JSR Launch Logs

Jonathan C. McDowell

Orbital Elements

Orbital Elements - Introduction

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

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:

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

Orbital derived quantities

We also consider the following auxiliary quantities:

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 .Thetar = 0 (geostationary), .Omega = .thetan (sun-synchronous), and .Omega = 0 (Molniya).

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

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 ha ~ 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-hGEO|<200 km.

The COGO defines GEO/C1 and GEO/C2 as |h-hGEO|<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.