﻿ Analytic Density Model

# Analytic Density Model

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## Enabling the Analytic Density Model

The default atmospheric density model used for modeling the Earth's atmosphere is the Analytic Density Model. The calculation of this model is described below. To enable the Analytic Density Model for a Spacecraft, edit the Spacecraft, proceed to the Force Model page, and select Earth from the list of celestial bodies as shown in the image below.

Then ensure that "Analytic" is selected in the Density Model drop-down menu.

To enable Analytic atmospheric density modeling via FreeFlyer script, use the following syntax:

 (Spacecraft1.Propagator AsType RK89).ForceModel.AtmosphericDensityModel = 1;

Where the AtmosphericDensityModel can be one of the following:

 •1 for Analytic density •2 for Harris-Priester density model •3 for Jacchia-Roberts density model •4 for MSIS-2000 •5 for MSIS-90 •6 for MSIS-86

Calculation of the Analytic Density

The Analytic model is based on the constant temperature hydrostatic model. In the hydrostatic model, the temperature in the atmosphere is assumed to be independent of height (isothermal assumption).

With this assumption, the density as a function of altitude, , is given by:

 (1)

where is the "scale height" and .

In general, the isothermal assumption is not in good agreement with experimental observations. However, it is more reasonable to assume that the temperature doesn’t vary quickly with height. In that case, a "segmented hydrostatic model" can be constructed with a scale height that depends on altitude. In terms of the parameter defined above, one would then have a set where the index i = 0,1,..., N indicates the appropriate altitude. The density for this model would then be given by:

 (2)

where is given by linearly interpolating between and corresponding to the tabulated altitudes bracketing the spacecraft’s altitude:

 (3)     (4)

Note that when the term:

 (5)

is identically zero and and when the same term is identically one and . The value of is and the values used for the are given below:

 Altitude (km) S = -1/h (km-1) 0 -0.160 100 -0.160 200 -0.110 300 -0.085 400 -0.067 500 -0.057 600 -0.050 700 -0.045 800 -0.040 900 -0.037 1000 -0.035

FreeFlyer uses the above model with two modifications. The first is that the height used in the computation is geodetic and second the overall density is "corrected" by a of the form

 (6)

where the default value is zero.