118 lines
4.5 KiB
Markdown
118 lines
4.5 KiB
Markdown
---
|
|
title: Atmospheric drag and the density model
|
|
tags: drag, atmosphere, density, ballistic-coefficient, sun-solar-activity
|
|
summary: Why the top of the atmosphere is the hardest part of an orbit model, how density is estimated, and what the activity of the Sun does to the answer.
|
|
---
|
|
|
|
# Atmospheric drag and the density model
|
|
|
|
Drag is the only perturbation that removes energy from an orbit. Everything
|
|
else trades altitude against inclination or direction; drag trades altitude
|
|
against lifetime, and it is why most low Earth orbits end.
|
|
|
|
## Definition
|
|
|
|
**Drag** is the aerodynamic force opposite the velocity. Its magnitude is
|
|
|
|
$$
|
|
F = -\tfrac{1}{2} \rho v^2 C_d A
|
|
$$
|
|
|
|
with `rho` the atmospheric density, `v` the speed, `Cd` a drag coefficient of
|
|
order one, and `A` the projected area. What matters to the orbit is the
|
|
combination
|
|
|
|
$$
|
|
B = \frac{C_d A}{m}
|
|
$$
|
|
|
|
the inverse of the **ballistic coefficient**, and `B` is what a drag model
|
|
actually needs.
|
|
|
|
Above about 700 km the density is small enough that the force is comparable
|
|
to other perturbations, and it varies by orders of magnitude over the
|
|
altitude range of the transition region, which is the source of most of the
|
|
difficulty.
|
|
|
|
## Mathematics
|
|
|
|
### Density and altitude
|
|
|
|
The density falls by about a factor of ten for every 15 km of altitude near
|
|
500 km, and the profile is not a power law but a sum of exponential layers
|
|
separated by inversions. Around 700 to 1000 km there is a region where the
|
|
density is a factor of several higher than an exponential fit predicts, and a
|
|
satellite can gain altitude while moving through it rather than losing it.
|
|
|
|
### Temperature and the scale height
|
|
|
|
In an isothermal layer the density falls exponentially with a scale height
|
|
|
|
$$
|
|
H = \frac{k T}{m g}
|
|
$$
|
|
|
|
with `T` the temperature, which itself depends on the solar activity. A warmer
|
|
thermosphere means a larger scale height, a higher density at every altitude,
|
|
and faster decay.
|
|
|
|
### Solar activity
|
|
|
|
The thermosphere is heated by the Sun, and the heating follows the solar
|
|
ultraviolet flux. At solar maximum the density at 400 km can be a factor of
|
|
several above its minimum, and the decay rate of a satellite in the
|
|
transition region changes by the same factor. This is why an ephemeris fitted
|
|
over one solar cycle does not transfer to another.
|
|
|
|
### Effect on the orbit
|
|
|
|
The effect on the semi-major axis is a secular decay, with a rate that scales
|
|
with `rho / m` and with the inverse of the semi-major axis cubed, because the
|
|
orbit speed falls with altitude. The practical consequence is that the decay
|
|
rate is not constant over the life of the orbit; a satellite that starts in a
|
|
dense layer decays quickly and then slows as it sinks into a thinner one.
|
|
|
|
### Analytic models
|
|
|
|
A usable model expresses the density as an exponential sum in altitude, with
|
|
exponents and coefficients that depend on solar activity, latitude and day or
|
|
night. Two models are common enough to name: the exponential model used by
|
|
the SGP4 theory, which has a closed form and therefore a fitted coefficient,
|
|
and the standard atmosphere models such as US-76, which are tables and more
|
|
accurate but not invertible in closed form.
|
|
|
|
## Unit conventions
|
|
|
|
- Density in kilograms per cubic metre; grams per cubic centimetre is the
|
|
unit a data product is more likely to use.
|
|
- Area-to-mass ratio in square metres per kilogram.
|
|
- Ballistic coefficient as area-to-mass, not the reciprocal that some sources
|
|
use.
|
|
- Altitude in kilometres.
|
|
|
|
## Applicability
|
|
|
|
A drag model is the weakest part of most orbit models. It is weakest at
|
|
exactly the altitude where it matters most, in the transition region between
|
|
700 and 1000 km, where a factor of two in density is a factor of two in the
|
|
predicted lifetime and the density itself is uncertain by that much.
|
|
|
|
A model fitted to a specific satellite over a specific arc is good for that
|
|
satellite over that arc and should not be extrapolated to a different area to
|
|
mass, a different altitude, or a different solar cycle.
|
|
|
|
A mean motion derived by fitting an ephemeris over a long arc absorbs the drag
|
|
as an average and is therefore insensitive to the density model's errors,
|
|
which is why such derived values are more reliable than a propagated one.
|
|
|
|
## Standards
|
|
|
|
- US Standard Atmosphere 1976, and its updates, as the reference profile
|
|
- The density coefficients used by the SGP4 theory: Vallado et al.,
|
|
*Revisiting Spacetrack Report #3*, AIAA 2006-6753
|
|
|
|
## See also
|
|
|
|
- [[perturbations]] for the ordering of drag against `J_2` and third body
|
|
- [[ephemeris-interpolation]] for a track that stops needing a density model
|
|
- [[sgp4]] for a propagation that carries a fitted drag term |