Chapter 2Orbits and Launching Methods

IntroductionSatellites (spacecraft) orbiting the earth follow the same laws that govern the motion of the planets around the sun. From early times much has been learned about planetary motion through careful observations.Johannes Kepler derived empirically three laws describing planetary motion. Later Sir Isaac Newton derived Keplers laws from his own laws of mechanics and theory of gravitation. Keplers laws apply quite generally to any two bodies in space which interact through gravitation. The more massive of the two bodies is referred to as the primary, the other, the secondary or satellite.

Keplers three laws of planetary motion Keplers laws apply to any two bodies interacting in space through gravitation, the more massive of the two being the primary and the other the secondary or the satellite.KepIers First LawKeplers first law states that the path followed by a satellite around the primary will be an ellipse.

An ellipse has two focal points shown as F1 and F2 in Fig. 2.1. The center of mass of the two-body system, termed as barycenter is always centered on one of the foci.

ln our specific case, because of the enormous difference between the masses of the earth and the satellite, the center of mass coincides with the center of the earth, which is therefore always at one of the foci.

The semimajor axis of the ellipse is denoted by a, and the semiminor axis, by b. The eccentricity e is given by

Two of the orbital parameters specified for satellites (spacecraft) orbiting the earth are the eccentricity e and the semimajor axis, a . For an elliptical orbit, 0 positive => eastwardd /dt => negative => westward

It will be seen, therefore that for eastward regression, i must be greater than 90o, or the orbit must be retrograde. Cos(0 to 90 ) is positive and cos( 90 to 180) is negative

It is possible to choose values of a, e, and i such that the rate of rotation is 0.9856/day eastward. Such an orbit is said to be sun synchronous

2. Rotation of line of apsides in the orbital plane,

Line of apsides. The line joining the perigee and apogee through the center of the earth.

The other major effect produced by the equatorial bulge is a rotation of the line of apsides.

This line rotates in the orbital plane, resulting in the argument of perigee changing with time. The rate of change is given by

The units for the rate of rotation of the line of apsides will be the same as those for n

When the inclination i is equal to 63.435, the term within the parentheses is equal to zero, and hence no rotation takes place.

Denoting the epoch time by t0, the right ascension of the ascending

node by 0, and the argument of perigee by w0 at epoch gives the new values for and w at time t as

The orbit is not a physical entity, and it is the forces resulting from an oblate earth, which act on the satellite to produce the changes in the orbital parameters.

Thus, rather than follow a closed elliptical path in a fixed plane, the satellite drifts as a result of the regression of the nodes, and the latitude of the point of closest approach (the perigee) changes as a result of the rotation of the line of apsides.

With this in mind, it is permissible to visualize the satellite as following a closed elliptical orbit but with the orbit itself moving relative to the earth as a result of the changes in and w.

So, The period PA is the time required to go around the orbital path from perigee to perigee, even though the perigee has moved relative to the earth.

If the inclination is 90 ,

the regression of the nodes is zero ,

the rate of rotation of the line of apsides is d/dt = K/2

Imagine the situation where the perigee at the start of observations is exactly over the ascending node. One period later the perigee would be at an angle KPA/2 relative to the ascending node or, in other words, would be south of the equator.

The time between crossings at the ascending node would be PA (1+ K/2n), which would be the period observed from the earth.

In addition to the equatorial bulge, the earth is not perfectly circularin the equatorial plane; it has a small eccentricity of the order of 105.

This is referred to as the equatorial ellipticity.

The effect of the equatorial ellipticity is to set up a gravity gradient, which has a pronounced effect on satellites in geostationary orbit .

A satellite in geostationary orbit ideally should remain fixed relative to the earth.

The gravity gradient causes the satellites in geostationary orbit to drift to one of two stable points, which coincide with the minor axis of the equatorial ellipse.

Due to the positions of mascons and equitorial bulges there are four equilibrium points in the geostationary orbit.

Two are stable and the other two are unstable.

These two stable points are separated by 180 on the equator and are at approximately 75 E longitude and 105 W longitude.

Satellites in service are prevented from drifting to these points through station-keeping maneuvers.

Because old, out-of-service satellites eventually do drift to these points, they are referred to as satellite graveyards.

The effect of equatorial ellipticity is negligible on most other satellite orbits.

d) Atmospheric dragFor near-earth satellites, below about 1000 km, the effects of atmospheric drag are significant.

Because the drag is greatest at the perigee, the drag acts to reduce the velocity at this point, with the result that the satellite does not reach the same apogee height on successive revolutions.

The result is that the semimajor axis and the eccentricity are both reduced.

Drag does not noticeably change the other orbital parameters, including perigee height.

In the program used for generating the orbital elements given in the NASA bulletins, a pseudo-drag term is generated, which is equal to one-half the rate of change of mean motion.

An approximate expression for the change of major axis can be derived as follows

The change in mean motion,n is

where the 0 subscripts denote values at the reference time t0, and

is the first derivative of the mean motion.

The mean anomaly is also changed, an approximate value for the change being:

The changes resulting from the drag term will be significant only for long time intervals, and for present purposes it will be ignored.

Inclined OrbitsThe orbital elements are defined with reference to the plane of the orbit.

The position of the plane of the orbit is fixed (or slowly varying) in space.

The location of the earth station is usually given in terms of the local geographic coordinates which rotate with the earth. And the earth station quantities may be the azimuth and elevation angles and range.

In calculations of satellite position and velocity in space, rectangular coordinate systems are generally used .

So transformations between coordinate systems are therefore required.

Calculation for elliptical inclined orbits:- the first step is to find the earth station look angles and range

The look angles for the ground station antenna are the azimuth and elevation angles required at the antenna so that it points directly at the satellite.

Elevation is measured upward from local horizontal plane

Azimuth is measured from north eastward to the projection of the satellite path onto the local horizontal plane

Determination of the look angles and range:The quantities used are 1. The orbital elements, 2. Various measures of time3. The perifocal coordinate system, which is based on the orbital plane4. The geocentric-equatorial coordinate system, which is based on the earths equatorial plane5. The topocentric-horizon coordinate system, which is based on the observers horizon plane.

The satellites with inclined orbits are not geostationary, and therefore, the required look angles and range will change with time.

The two major coordinate transformations needed are:

The satellite position measured in the perifocal system is transformed to the geocentric-horizon system in which the earths rotation is measured, thus enabling the satellite position and the earth station location to be coordinated.

The satellite-to-earth station position vector is transformed to the topocentric-horizon system, which enables the look angles and range to be calculated.

CalendarsA calendar is a time-keeping device in which the year is divided into months, weeks, and days.

Calendar days are units of time based on the earths motion relative to the sun.

It is more convenient to think of the sun moving relative to the earth.

But this motion is not uniform, and so a fictitious sun, termed the mean sun, is introduced.

The mean sun does move at a uniform speed but requires the same time as the real sun to complete one orbit of the earth.

Ie period of mean sun = period of real sun.

This time being the tropical year.

A day measured relative to this mean sun is termed a mean solar day.

Calendar days are mean solar days.

A tropical year contains 365.2422 days.

In order to make the calendar year, also referred to as the civil year, it is normally divided into 365 days.

The extra 0.2422 of a day is significant, and for example, after 100 years, there would be a discrepancy of 24 days between the calendar year and the tropical year.

Julius Caesar made the first attempt to correct the discrepancy by introducing the leap year, in which an extra day is added to February whenever the year number is divisible by 4.

This gave the Julian calendar, in which the civil year was 365.25 days on average, a reasonable approximation to the tropical year.

Again a difference of days per year exists.

ie On every 400 years we are actually adding an extra 3 days.

By the year 1582, an appreciable discrepancy once again existed between the civil and tropical years. Pope Gregory XIII took matters in hand by abolishing the days October 5 through October 14, 1582, to bring the civil and tropical years into line and by placing an additional constraint on the leap year in that years ending in two zeros must be divisible by 400 without remainder to be reckoned as leap years.

This dodge was used to miss out 3 days every 400 years.

To see this, let the year be written as X00 where X stands for the hundreds. For example, for 1900, X = 19. For X00 to be divisible by 400, X must be divisible by 4. Now a succession of 400 years can be written as X+ n, X+ (n+ 1), X + (n + 2), X + (n + 3),X + (n + 4),where n is any integer from 0 to 9. If X+ n is evenly divisible by 4, then the adjoining three numbers are not, so these three years would have to be omitted.

For example let us take X= 20 and n=0; then in the years 2000, 2100,2200, 2300, 2400, 0nly 2000 and 2400 area leap year and the remaining are not even though it is divisible by 4.

The resulting calendar is the Gregorian calendar, which is the one in use today.

Universal timeUniversal time coordinated (UTC) is the time used for all civil timekeeping purposes.

And it is the time reference which is broadcast by the National Bureau of Standards as a standard for setting clocks.

It is based on an atomic time-frequency standard.

The fundamental unit for UTC is the mean solar day.

In terms of clock time, the mean solar day is divided into 24 h, an hour into 60 min, and a minute into 60 s. Thus there are 86,400 clock seconds in a mean solar day.

Satellite-orbit epoch time is given in terms of UTC.

Universal time is equivalent to Greenwich mean time (GMT), as well as Zulu (Z) time.

For computations, UT will be required in two forms: as a fraction of a day and in degrees.

Given UT in the normal form of hours, minutes, and seconds, it is converted to fractional days as or

In turn, this may be converted to degrees as

Sidereal timeSidereal time is time measured relative to the fixed stars.

One complete rotation of the earth relative to the fixed stars is not a complete rotation relative to the sun.

This is because the earth moves in its orbit around the sun.

The sidereal day is defined as one complete rotation of the earth relative to the fixed stars.

One sidereal day has 24 sidereal hours, 1 sidereal hour has 60 sidereal minutes, and 1 sidereal minute has 60 sidereal seconds.

But sidereal times and mean solar times, are different even though both use the same basic subdivisions.

The relationships between the two systems,

1 mean solar day= 1.0027379093 mean sidereal days = 24h and (0.0027379093 x 24 )hrs = 24 hrs 0.065709823 hrs= 24 hrs (0. 065709823 x 60) min = 24 hrs 3.942589392 min = 24 hrs 3 m (0.942589392 x 60) sec= 24 h 3 m 56.55536 s sidereal time

= 56.55536 + 3 x 60 + 24 x 60 x 60 = 86,636.55536 mean sidereal seconds

1 mean sidereal day = (1/1.0027379093 ) mean solar days= 0.9972695664 mean solar days = 23 h 56 m 04.09054 s mean solar time = 86,164.09054 mean solar seconds

The orbital plane or perifocal coordinate systemIn the orbital plane, the position vector r and the velocity vector v specify the motion of the satellite.

Determination of position vector r : From the geometry of the ellipse and

r can also be calculated using

Determination of the true anomaly , v can be done in 2 stages Stage 1 : Find the mean anomaly M .Stage 2 : Solve Keplers equation.

Stage 1: The mean anomaly M at time t can be found as

Here, n is the mean motion, and Tp is the time of perigee passage.

For the NASA elements,

Therefore,

Substitute for Tp gives

Stage 2 : Solve Keplers equation.

Keplers equation is formulated as follows

[.........

In satellite orbital calculations, time is often measured from the instant of perigee passage. Denote the time of perigee passage as T and any instant of time after perigee passage as t. Then the time interval of significance is t - T. Let A be the area swept out in this time interval, and let Tp be the periodic time. Then, from Keplers second law,The mean motion is n = 2/Tp and the mean anomaly is M = n (t - T). Combining these

Combining c= ae is the perigee, position of earth. So area is swept for the time t- T is AEarlier we got the area as

combining these equations This is the Keplers equation....................................]

Keplers equation is

E is the eccentric anomaly.Once E is found, v can be found from an equation known as Gauss equation, which is

For near-circular orbits where the eccentricity is small, an approximation for v directly in terms of M is

r may be expressed in vector form in the perifocal coordinate system (0r PQW frame) as r = (r cos v )P + (r sin v)Q

Local Mean Solar Time andSun-Synchronous OrbitsThe celestial sphere is an imaginary sphere of infinite radius, where the points on the surface of the sphere represent stars or other celestial objects.

The celestial equatorial plane coincides with the earths equatorial plane, and the direction of the north celestial pole coincides with the earths polar axis.

The the suns meridian is shown in figure .

line of Aries is

The angular distance along the celestial equator, measured eastward from the point of Aries to the suns meridian is the right ascension of the sun, denoted by S.

In general, the right ascension of a point P, is the angle, measured eastward along the celestial equator from the point of Aries to the meridian passing through P. This is shown as P.

The hour angle of a star is the angle measured westward along the celestial equator from the meridian to meridian of the star. Thus for point P the hour angle of the sun is (P S) measured westward

In astronomy, an object's hour angle (HA) is defined as the difference between the current local sidereal time (LST) and the right ascension () of the object.hour angle = local sidereal time (LST) - right ascension

1 HA = 15 degrees

ie right ascension of the sun Sright ascension of a point P Pfor point P the hour angle of the sun is (P S)

The apparent solar time of point P is the local hour angle of the sun, expressed in hours, plus 12 h. Ie apparent solar time of point P =LHA of the sun+
12 h

The 12 h is added because zero hour angle corresponds to midday, when the P meridian coincides with the suns meridian.

Because the earths path around the sun is elliptical rather than circular, and also because the plane containing the path of the earths orbit around the sun (the ecliptic plane) is inclined at an angle of approximately 23.44, the apparent solar time does not measure out uniform intervals along the celestial equator, in other words, the length of a solar day depends on the position of the earth relative to the sun.

To overcome this difficulty a fictitious mean sun is introduced, which travels in uniform circular motion around the sun.

The time determined in this way is the mean solar time.

Figure shows the trace of a satellite orbit on the celestial sphere.

Point A corresponds to the ascending node.

The hour angle of the sun from the ascending node of the satellite is - S measured westward.

The hour angle of the sun from the satellite (projected to S on the celestial sphere) is - S + .

thus the local mean (solar) time is since 1 degree = 1/15 HA (hour angle)

To find :- Consider the spherical triangle defined by the points ASB.

This is a right spherical triangle because the angle between the meridian plane through S ( projection of satellite) and the equatorial plane is a right angle.

The triangle contains the inclination i and latitude

Inclination i is the angle between the orbital plane and the equatorial plane.

Latitude is the angle measured at the center of the sphere going north along the meridian through S.

The solution of the right spherical triangle gives

The local mean (solar) time for the satellite is therefore (by substituting the value of )

As the inclination i approaches 90 angle approaches zero.

The right ascension of the sun S can be calculated as follows,Consider the earth completes a 360 orbit around the sun in 365.24 days.

where d is the time in days from the vernal equinox (or line of Aries , .

For a sun-synchronous orbit the local mean time must remain constant.

The advantage of a sun-synchronous orbit is that the each time the satellite passes over a given latitude, the lighting conditions will be approximately the same.

So it can be used for placing weather satellites and environmental satellites.

Eq. for tSAT shows that for a given latitude and fixed inclination i , the only variables are S and .

In effect, the angle (- S ) must be constant for a constant local mean time.

Let 0 represent the right ascension of the ascending node at the vernal equinox and ' the time rate of change of .

For this to be constant the coefficient of d must be zero, or

The orbital elements a, e, and i can be selected to give the required regression of 0.9856 east per day.

These satellites follow near-circular, near-polar orbits.