Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

**General Description**

A polar orbit for a satellite of the Earth is one where the plane of the elliptic orbit lies on or is very close to the axis of the Earth's rotation. Such a satellite is usually placed in an orbit low enough that it circles the Earth many times a day. Since the Earth rotates on its axis while the satellite follows its orbit, the satellite gets a detailed view of the Earth with fairly extensive coverage. The satellite takes images of the Earth below and sends them back to receiving stations where the images can be analyzed for many diverse purposes such as crop monitoring and management, search and rescue, monitoring territorial waters, and so forth.

**Orbit Parameters**

The macro developed for three-dimensional analysis of orbits in this chapter is well suited for exploring the characteristics or polar orbits. To demonstrate this, we will look at the polar orbit for a typical monitoring satellite called Radarsat 2.

The parameters for this satellite are available and are seen, in very small font, in the image shown next:

The Radarsat 2 orbit parameters are summarized in the terminology of this web site as follows:

Altitude: 798 km

Inclination: 98.574 degrees

Period: 100.7 min, 14 orbit/day

Repeat Cycle: 24 days

Semi-Major Axis (**a**): 7176.07 km = 7176070 m

*
Note: The Semi-Major axis is shown on a drawing of an ellipse in the first topic of
Chapter 7.*

Eccentricity (e): 0.001251477

A derivation of how these data can be used to find other orbital parameters is beyond
the scope of this chapter and topic. We shall state them and the reader can find
derivations
here and
there.
For a semi-major axis **a** and eccentricity e, the perigee
(that is, closest approach) of the orbit is:

perigee = **a *** (1 - e) = 7176070 * (1 - 0.001251477)
= 7167089.313 m

The quoted altitude is an average value, not useful for our purposes. The altitude of the perigee is

altitude = perigee - earthRadius = 7167089.313 - 6371010.0 = 796079.313 m

The velocity at perigee is related to the standard gravitational parameter G*M,
the semi-major axis **a**, and the eccentricity by the following formula

Vperigee = (G*M * (1 + e) / (**a** * ( 1 - e)))**0.5**

= (398600441800000
* (1 + 0.001251477) / (7176070 * ( 1 - 0.001251477)))**0.5**

= 7462.237 m/s

For our purposes, we will assume the orbit is such that at t = 0, the Latitude and Longitude of the satellite are both 0 and that this also corresponds to the perigee of the orbit. There is enough information to fill in the parameters for the Elliptic worksheet.

Latitude, Longitude, Altitude = 0, 0, 796079.313, respectively.

Since we are assuming the initial conditions apply to perigee, the Elevation must be 0.0 degrees to specify a strictly tangential velocity (no radial component). The inclination of the orbit is the angle of the plane of the orbit measured from the equator and in the direction where the satellite crosses the equator in a northerly direction. Since Azimuth on the worksheet is measured from north, we have

Elevation, Azimuth = 0.0, -8.574 degrees, respectively.

The orbit period, T, is provided by the analytic calculation and it is based on the formula

T**2**
= [4*π^{2}/G*M]**a ^{3}**

This yields T = 6049.79837274296 s = 100.8 min which closely matches the published number (100.7 min). By using a step size of 0.604979837274296 and suitable calculation steps per output (100) and number of outputs (1400) we can generate output for 14 orbits each with 100 points of plotting to produce a representation of the orbit coverage.

**Orbit Coverage**

**
**

For each orbit of 100.8 minutes, the Earth below rotates by an amount of 360 * 100.8 / (24 * 3600) = 25.2 degrees. Looking at the plot of orbit sub-points, we can see that each track is separated by this amount. After 14 orbits the Earth will turn 14*T/(24*3600) *360 = 352.905 degrees and there is an offset of about 7.1 degrees from one day to the next for the same set of 14 tracks.

During the orbit, the altitude varies because the orbit is elliptic, not circular. The farthest point of the orbit is the apogee and its value is given as

apogee = **a *** (1 + e) = 7176070 * (1 + 0.001251477) = 7185050.687 m
which is an altitude of 814040.687 m compared with 796079.313 m for
the perigee.

**Sun Synchronous Orbits**

This subject is mentioned for the sake of completeness. Sun synchronous orbits are those that are inclined as is the polar orbit studied above. The spherical Earth model produces regular elliptic orbits whose characteristics do not change with time. But the Earth is not a perfect sphere. It is closely shaped as an oblate spheroid with the equatorial and polar radii of 6378.160 and 6356.775 km, respectively. This shape induces the plane of the satellite orbit to slowly rotate around the polar axis with the rate of rotation varying with the amount of inclination from the polar axis. By selecting the proper amount of inclination, the rate at which the satellite orbit's plane rotates can be made to match the rate at which the Earth revolves around the Sun, about 0.986 degrees per day (360 degrees per 365.2422 days). The satellite in such an orbit will have the same angular relation to the Sun while this orbit is maintained. This is of interest for such satellites since they can always obtain sunlight for power generation and see the Earth with the same constant illumination.

**Next**

Newton proposed a thought experiment in which a projectile was launched by cannon into Earth orbit from a mountaintop. Gerald Bull had the dream of launching a satellite into orbit from a desert site in Iraq. These ideas are examined in the next topic.

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