James asked:
> Can someone explain how the Perigee & Apogee Height of a sat
> is calculated from a TLE?
The method required depends on the required accuracy. Assuming that approximate values at the epoch (date and time) of the elements
will suffice, here is a quick method:
1. Compute semi-major axis
a = (8681663.653 / n0) ^ (2/3)
Where a = semi-major axis, km
n0 = mean motion at epoch, rev/d
2. Compute perigee height
hp = a(1 - e0) - 6371
where hp = perigee height, km
e0 = eccentricity at epoch
6371 = Earth's mean radius
3. Compute apogee height
ha = a(1 + e0) - 6371
where ha = apogee height, km
e0 = eccentricity at epoch
6371 = Earth's mean radius
Applying this method to the following satellite's orbit:
USA 129 15.0 3.0 0.0 5.3 v
1 24680U 96072A 02233.10391782 .00026000 00000-0 30679-3 0 04
2 24680 97.8820 295.4481 0500000 269.2694 90.7304 14.81194991 02
where e0 = 0.0500000
n0 = 14.81194991 rev/d
results in hp = 283 km and ha = 983 km, both rounded to the nearest km.
To get the most accurate result, requires taking full account of Earth's oblateness, in which case the results nearest the epoch of
the above elements are hp = 304 km and ha = 989 km, both rounded to the nearest km.
Here is a brief explanation of the role of Earth's oblateness in determining perigee and apogee height.
Due to its oblateness, Earth's equatorial radius is 6378 km and its polar radius is 6357 km (both rounded to the nearest km), so a
given perigee distance relative to the centre of the Earth, will result in different perigee heights relative Earth's surface,
depending on the latitude over which perigee is located.
Further complicating matters, Earth's oblateness causes the long axis of a satellite's orbit to rotate (or precess) about the Earth.
For the above orbit, one complete rotation takes 110 days.
Still further complicating matters, Earth's oblateness causes a satellite's orbital eccentricity to oscillate as its long axis
rotates. Eccentricity is greatest when the perigee is at its most northerly point, and least when the perigee is at its most
southerly point.
For the above orbit, the perigee and apogee distance relative to the centre of the Earth, varies over a range of about 15 km, over a
period of 55 days.
To fully account for Earth's oblateness, requires the use of the SGP4 and SDP4 orbital models, which are the basis of the tles, and
an algorithm to compute height relative to Earth's oblate surface. The calculations require the use of a computer.
Ted Molczan
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This archive was generated by hypermail 2b29 : Sat Dec 14 2002 - 14:21:04 EST