Atmospheric Drag
The atmospheric drag acceleration is given by
1
A
ad   Cd  ECF r ECF r
2 m
where, ECF r  ECI r  xr
Expres sin g these vectors in the ECI frame yields,
 ECF r 
ECI
 ECI X  Y 


  ECI Y  X 
 Z



ECI
Also,
CD = Drag coefficient
A = cross sectional area normal to the velocity vector
m = spacecraft mass
atmospheric density
ECF
r = spacecraft velocity relative to the ECF frame i.e., relative to the Earth’s
atmosphere
= rotation rate of the Earth
For example, the equation for X in the ECI frame including the effects of the
gravitational potential,  , and drag is
ECI
X
Similar equations can be written for
 1
A
 CD 
X 2
m
ECI
Y and
ECI
ECF
r [ ECI X  Y]
Z.
The primary effect of drag is to lower the apoapsis altitude of elliptical orbits until the
orbit is circularized. After that time the orbit remains circular until reentry.
The Earth’s atmospheric density is affected by a number of factors including the amount
of solar radiation striking the atmosphere and by the Earth’s magnetic field. Solar flux is
the incident radiation arriving from the sun. Extreme ultraviolet (EUV) radiation heats
the atmosphere, causing it to expand, resulting in a density increase at satellite altitudes.
EUV does not reach the Earth’s surface because the atmosphere absorbs it. Hence there
is no Earth-based means of measuring it. However, both EUV and F10.7 flux originate
from the same layer of the sun and F10.7 flux, which does reach the ground, can be used
as a proxy for EUV. F10.7 is the solar radiation with 10.7 cm wavelength. The amount
of EUV energy reaching the Earth is highly influenced by the 11-year solar cycle, which
will peak in 2001. The figure shows F10.7 flux levels for the past solar cycle as well as
predicted flux levels through 2005.
A number of density models are in use today. These include Harris – Priester, Jacchia,
Density Temperature Model (DTM), Jacchia – Roberts, Cospar, and Hedin to name the
most frequently used. A simple exponential model works to first order up to altitudes of
1000 km
 h  h0 
  0 exp  a
 H 
Where
0  reference density
h 0  reference altitude
H  scale height
H varies from 7.25 km at the Earth’s surface to 268 km at 1000 km altitude. H is the
altitude over which the density changes by the value of e = 2.72. The reference density
varies from 1.225 kg/m3 at the surface to 3.019 x 10-15 kg/m3 at 1000 km altitude
(Vallado, p 510).
Reference
Vallado, David D., Fundamentals of Astrodynamics and Applications, Space Technology
Series, McGraw Hill, 1997.
Solar Radiation Pressure
To describe solar radiation pressure forces we begin with Newton’s second law
d(mv)
 mv  mv. For thermal radiation
dt
ˆ where c is the speed of light and sˆ is a unit vector
v  cs,
F
directed from the spacecraft to the sun. Since v is constant
dm
sˆ
(1)
dt
Energy from thermal radiation is defined as (the theory of general relativity states that
even massless particles impart energy)
F  c
E  mc 2 , so
dE
dm
 c2
(2)
dt
dt
From Eq 1 and (2)
F
Where
1 dE
sˆ
c dt
dE
is the incident solar radiation on the spacecraft, i. e.,
dt
dE
 GA s / c
dt
Where G is the incident solar radiation per unit area striking the surface, As/c. G at
1 AU = 1350 watts/m2 and As/c = area of the spacecraft normal to the sun direction.
Hence,
1
F   GA s / csˆ , but
c
G 1350 watts / m 2

 4.5x106 watt sec/ m3
8
c
3x10 m / s
 4.5x106 N / m 2  PSR
Therefore,
F  PSR A s / csˆ
 3
Equation (3) assumes that all radiation is absorbed. If all radiation is reflected as specular (i. e., a
mirror effect), the force is twice that of Eq (3) since the photons are stopped by the surface and
reflected back to space at the speed of light. To accommodate this in the force equation we introduce
a reflectivity and absorptivity coefficient. As seen in the diagram below, reflected energy generates a
force, Fn , normal to the reflecting surface because energy is reflected at the same angle as the incident
energy.
In general we break the solar pressure force into the component due to absorption and the
component due to reflection, i.e.,
F  Fs  Fn
Where Fs is the force in the solar direction and Fn is the force normal to the surface
Hence,
ˆ
F  PSP As / c [sˆ  2n]
where
  absorptivity coefficient, 0    1,   1  
  reflectivity coefficient of specular reflection, 0    1
n̂ is a unit vector normal to the surface, A s / c
As / cis the area normal to the sun direction
To illustrate how solar sailing works, assume that we have a sail normal to the plane of
the paper and only specular reflected energy. In this case the resultant solar pressure
force will be normal to the sail as shown in the figure. By proper orientation of the sail,
we can pump energy into the orbit or remove energy. If we look at the equation for the
time derivative of the semimajor axis, we see that a force in the in-track direction, FI ,
increases energy (semimajor axis) while a force in the anti-velocity direction decreases
energy. The radial component of solar pressure force, FR , introduces a periodic variation
into the semimajor axis and hence the energy of the orbit. This periodic variation
averages to zero over the orbit unless the spacecraft experiences occultation
From Lagrange 's planetary equations
da
2
a(1  e 2 )

{e sin  FR 
FI }
dt n 1  e 2
r
Where FR and FI are the radial and in  track solar pressure forces.