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the satellite remains above the Northern Hemisphere near apogee for approximately 11 hours/orbit. Mission planners choose perigee altitude to meet the satellite's mission constraints. Typical perigee altitudes vary from 200 to 1,000 km. We can calculate the eccentricity and apogee altitude using the semimajor axis, perigee altitude, and equations from Table 6-2.

In a Sun-synchronous orbit, the satellite orbital plane remains approximately fixed with respect to the Sun. We do this by matching the secular variation in the right ascension of the ascending node (Eq. 6-19) to the Earth's rotation rate around the Sun. A nodal precession rate of0.9856 deg/day will match the Earth's average rotation rate about the Sun. Because this rotation is positive, Sun-synchronous orbits must be retrograde. For a given semimajor axis, a, and eccentricity, we can use Eq. (6-19) to find the inclination for the orbit to be Sun-synchronous.

### 63.3 Perturbations From Atmospheric Drag

The principal nongravitational force acting on satellites in low-Earth orbit is atmospheric drag. Drag acts in the opposite direction of the velocity vector and removes energy from the orbit This energy reduction causes the orbit to get smaller, leading to further increases in drag. Eventually, the altitude of the orbit becomes so small that the satellite reenters the atmosphere.

The equation for acceleration due to drag on a satellite is:

where p is atmospheric density,^ is the satellite's cross-sectional area, m is the satellite's mass, V is the satellite's velocity with respect to the atmosphere, and CD is the drag coefficient »2.2 (See Table 8-3 in Sec. 8.1.3 for an extended discussion of CD).

We can approximate the changes in semimajor axis and eccentricity per revolution, and the lifetime of a satellite, using the following equations:

Aam = -2* (CDA/m)a> pp exp (-c) [/0 + 2e/, ] (6-22)

= -2k (CDA/m)a Pp exp (^c) [/, + e (/0 +/2)/2] (6-23)

where pp is atmospheric density at perigee, c = ae / H,H is density scale height (see column 25, Inside Rear Cover), and /, are Modified Bessel Functions* of order i and argument c. We model the term m / (CDA), or ballistic coefficient, as a constant for most satellites, although it can vary by a factor of 10 depending on the satellite's orientation (see Table 8-3).

For near circular orbits, we can use the above equations to derive the much simpler expressions:

where P is orbital period and Kis satellite velocity.

A rough estimate of the satellite's lifetime, L, due to drag is

where, as above, H is atmospheric density scale height given in column 25 of the Earth Satellite Parameter tables in the back of this book. We can obtain a substantially more accurate estimate (although still very approximate) by integrating Eq. (6-24), taking into account the changes in atmospheric density with both altitude and solar activity level. We did this for representative values of the ballistic coefficient in Fig. 8-4 in Sec. 8.1.

### 6.2.4 Perturbations from Solar Radiation

Solar radiation pressure causes periodic variations in all of the orbital elements. Its effect is strongest for satellites with low ballistic coefficients, that is, light vehicles with large frontal areas such as Echo. The magnitude of the acceleration, aR, in m/s2 arising from solar radiation pressure is aR » -4.5 x + r)Alm (6-29)

where A is the satellite cross-sectional area exposed to the Sun in m2, m is the satellite mass in kg, and r is a reflection factor, (r = 0 for absorption; r = 1 for specular

* Values for /, can be found in many standard mathematical tables.

reflection at normal incidence; and r » 0.4 for diffuse reflection.) Below 800 km altitude, acceleration from drag is greater than that from solar radiation pressure; above 800 km, acceleration from solar radiation pressure is greater.

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