Data Extraction:
On board both the CHAMP and GRACE satellites is a Spatial Triaxial Accelerometer for Research (STAR, Onera), which measures the sum of all forces on the satellites' surfaces. This measured quantity is comprised mostly of the force imparted to the satellite by atmospheric drag, with lesser constituents such as atmospheric lift, solar and Earth radiation pressure also contributing. All other non-gravitational forces on the satellite are neglected in this study. By modeling the effects of solar and Earth radiation pressure, we obtain a method of isolating the acceleration caused by atmospheric drag and lift. After these modeled forces are subtracted from the total non-gravitational forces as measured by the accelerometer, atmospheric parameters that affect drag and lift acceleration can be deduced.
Total mass density:
Satellite data is provided for our use by the CHAMP and GRACE Information System and Data Center (ISDC). Acceleration, attitude, and orbit ephemeris data files are used in the calculations pertaining to this study. The acceleration and attitude data are provided on a 1 to 10-second interval, which is processed from the high-rate raw data to remove acceleration spikes caused by spacecraft maneuvers. For CHAMP, both the along-track axis, used for density calculations, and the cross-track axis, used for wind calculations, are thought to be accurate to 3x10-9 ms-2 [Reigber et al., 2002]. The radial axis is not used in these studies because it is less sensitive by a factor of ten and has had repeated trouble since launch. For the GRACE satellites, accuracy is improved by an order of magnitude.
Eq. (1) and (2) relate accelerations of the satellite caused by drag and lift to atmospheric parameters of density and wind speed.
(1)
(2)
where
aD and aL are the
accelerations caused by drag and lift, respectively; CD
and CL are the coefficients of drag and
lift, respectively (see below for a description of the method used to estimate these parameters); A is the reference area of the
satellite; m is the mass of the satellite;
ρ is the atmospheric density; and
is the velocity of the satellite relative to the surrounding
atmosphere. In Eq. (1), the scalar acceleration acts in the direction
of
;
however, in Eq. (2), the scalar acceleration acts in a direction perpendicular to . For instance, the scalar acceleration in Eq. (2) acts in the direction of
(x
)x for a flat satellite with surface normal .
In order to obtain a more realistic relationship between acceleration and the atmospheric parameters, the satellites are broken down into a macro-model of k independent flat plates. This allows for a more accurate estimate the coefficients of drag and lift. Eq. (1) and (2) can be combined into a new vector equation relating the total acceleration to the summation of the effect of density and wind speed on each plate of the satellites:
(3)
where
is the acceleration caused by drag and lift; the subscript i
is the index for each flat plate in the macro-model, the A's
are the total areas of each plate; the
θ's are the angles of incidence of the
atmospheric particles on each plate; and the
's
are the unit normal vectors for each plate. If we make the assumption
that
is the measured satellite acceleration with the solar radiation pressure
(srp)
and earth radiation pressure (er)
removed (ie.
= star – srp – er, where star is the acceleration measured by the satellite),
Eq. (3) can readily be solved for atmospheric density, ρ,
using only the along-track axis of the accelerometer:
(4)
where
refers to the along-track unit vector.
Cross-track and radial wind speed:
It
is also possible to obtain an estimate of the neutral wind vector
component in the direction of the cross-track and radial axes of the
STAR accelerometer, using an adaptation of the method presented by
Liu et al. [2006]. The steps for this calculation are similar to
those for density calculations; however, we now have to estimate the
lift forces acting on the satellite and make the assumption that
is the measured satellite acceleration with the lift acceleration
(L),
solar radiation pressure (srp)
and earth radiation pressure (er)
removed (ie.
= star – L – srp – er).
With this assumption, the following equation relates the acceleration
caused by drag to the atmospheric parameters density and wind speed:
(5)
where
= rel – 
;
rel
is the velocity of the satellite with respect to the co-rotating
atmosphere; and
is the wind velocity with respect the the co-rotating atmosphere.
For studies involving the CHAMP satellite, only the cross-track axis
is studied for the following two reasons: neutral density and wind
speed cannot be separated from each other in calculations using the
along-track accelerometer axis, and the radial accelerometer axis
does not provide measurements of sufficient accuracy. Thus, when
solving Eq. (5) for wind speed, components of
in the along-track and radial directions are assumed to be zero. For
the GRACE satellites, winds can be estimated from both the
cross-track and radial axes. In either case, Eq. (5) is broken down
into 3 scalar equations:
(6)
where the subscripts x, y, and z refer to the along-track, cross-track, and radial axes, respectively. Assuming that the along-track wind speed is zero, and noticing the common terms in all three equations, substitutions can be made to solve for cross-track wind speed in Eq. (7) and radial wind speed in Eq. (8):
(7)
(8)
For a thorough treatment of density and wind calculations including error estimates, see Sutton et al. [2007].
Coefficients of drag and lift:
Due to the complex and elongated shape of the CHAMP satellite, the method for calculating the coefficients of drag and lift is adapted from Sentman [1961]. This method includes the effects of the incident and readmitted neutral gas particles interacting with the surface of the satellite. In addition, the neutral gas particles are assumed to have a random motion which causes an notable increase in the coefficient of drag for satellites with large areas that are nearly parallel to the direction of flow. For a flat plate, the equations for lift and drag take the following form:
(9)
(10)
where
,
,
,
, 
,
is the total area of plate i,
is the projected area of plate i in the plane perpendicular to the flow of neutral gas,
,
,
is the velocity of the incident neutral gas,
is the velocity of the readmitted neutral gas, and
is the molecular speed ratio.
Prior to December of 2007, density and wind data available on this website used the Cook [1965] method to compute coefficients of drag and lift. The advantage of the Sentman [1961] method, utilized in Version 2.0 density and wind data, is that the random motions of the neutral gas are taken into account. While this is negligible for many satellites, it must be taken into account for satellites with elongated shapes. The figure below shows the difference in the two methods versus the angle between incident flow and the satellite x-axis (nominally between 0 and 10 degrees for the CHAMP satellite, shown here between 0 and 90 degrees). These differences are reported in Sutton [2009].
References:
Cook, G. E. (1965), Satellite Drag Coefficients, Planet. Space Sci., 13, 929–946.
Liu, H., H. Luhr, S. Watanabe, W. Kohler, V. Henize, and P. Visser (2006), Zonal winds in the equatorial upper thermosphere, J. Geophys. Res. (Space Physics), 111.
Reigber, C., H. Luhr, and P. Schwintzer (2002), CHAMP mission status, Adv. Space Res., 30(2), 129–134.
Sutton E.K. (2009), Normalized Force Coefficients for Satellites with Elongated Shapes, J. Spacecraft and Rockets, 46(1), doi:10.2514/1.40940.
Sutton E.K., R.S. Nerem, and J.M. Forbes (2007), Atmospheric Density and Wind Measurements Deduced from Accelerometer Data, J. Spacecraft and Rockets, 44(6), doi:10.2514/1.28641.