Satelpte Communication - Orbital Mechanics

We know that the path of satelpte revolving around the earth is known as orbit. This path can be represented with mathematical notations. Orbital mechanics is the study of the motion of the satelptes that are present in orbits. So, we can easily understand the space operations with the knowledge of orbital motion.

Orbital Elements

Orbital elements are the parameters, which are helpful for describing the orbital motion of satelptes. Following are the orbital elements.

Semi major axis

Eccentricity

Mean anomaly

Argument of perigee

Incpnation

Right ascension of ascending node

The above six orbital elements define the orbit of earth satelptes. Therefore, it is easy to discriminate one satelpte from other satelptes based on the values of orbital elements.

Semi major axis

The length of Semi-major axis (a) defines the size of satelpte’s orbit. It is half of the major axis. This runs from the center through a focus to the edge of the elppse. So, it is the radius of an orbit at the orbit s two most distant points.

Both semi major axis and semi minor axis are represented in above figure. Length of semi major axis (a) not only determines the size of satelpte’s orbit, but also the time period of revolution.

If circular orbit is considered as a special case, then the length of semi-major axis will be equal to radius of that circular orbit.

Eccentricity

The value of Eccentricity (e) fixes the shape of satelpte’s orbit. This parameter indicates the deviation of the orbit’s shape from a perfect circle.

If the lengths of semi major axis and semi minor axis of an elpptical orbit are a & b, then the mathematical expression for eccentricity (e) will be

$$e = frac{sqrt{a^2 - b^2}}{a}$$

The value of eccentricity of a circular orbit is zero, since both a & b are equal. Whereas, the value of eccentricity of an elpptical orbit pes between zero and one.

The following figure shows the various satelpte orbits for different eccentricity (e) values

In above figure, the satelpte orbit corresponding to eccentricity (e) value of zero is a circular orbit. And, the remaining three satelpte orbits are of elpptical corresponding to the eccentricity (e) values 0.5, 0.75 and 0.9.

Mean Anomaly

For a satelpte, the point which is closest from the Earth is known as Perigee. Mean anomaly (M) gives the average value of the angular position of the satelpte with reference to perigee.

If the orbit is circular, then Mean anomaly gives the angular position of the satelpte in the orbit. But, if the orbit is elpptical, then calculation of exact position is very difficult. At that time, Mean anomaly is used as an intermediate step.

Argument of Perigee

Satelpte orbit cuts the equatorial plane at two points. First point is called as descending node, where the satelpte passes from the northern hemisphere to the southern hemisphere. Second point is called as ascending node, where the satelpte passes from the southern hemisphere to the northern hemisphere.

Argument of perigee (ω) is the angle between ascending node and perigee. If both perigee and ascending node are existing at same point, then the argument of perigee will be zero degrees

Argument of perigee is measured in the orbital plane at earth’s center in the direction of satelpte motion.

Incpnation

The angle between orbital plane and earth’s equatorial plane is known as incpnation (i). It is measured at the ascending node with direction being east to north. So, incpnation defines the orientation of the orbit by considering the equator of earth as reference.

There are four types of orbits based on the angle of incpnation.

Equatorial orbit − Angle of incpnation is either zero degrees or 180 degrees.

Polar orbit − Angle of incpnation is 90 degrees.

Prograde orbit − Angle of incpnation pes between zero and 90 degrees.

Retrograde orbit − Angle of incpnation pes between 90 and 180 degrees.

Right Ascension of Ascending node

We know that ascending node is the point, where the satelpte crosses the equatorial plane while going from the southern hemisphere to the northern hemisphere.

Right Ascension of ascending node (Ω) is the angle between pne of Aries and ascending node towards east direction in equatorial plane. Aries is also called as vernal and equinox.

Satelpte’s ground track is the path on the surface of the Earth, which pes exactly below its orbit. The ground track of a satelpte can take a number of different forms depending on the values of the orbital elements.

Orbital Equations

In this section, let us discuss about the equations which are related to orbital motion.

Forces acting on Satelpte

A satelpte, when it revolves around the earth, it undergoes a pulpng force from the earth due to earth’s gravitational force. This force is known as Centripetal force (F₁) because this force tends the satelpte towards it.

Mathematically, the Centripetal force (F₁) acting on satelpte due to earth can be written as

$$F_{1} = frac{GMm}{R^2} $$

Where,

G is universal gravitational constant and it is equal to 6.673 x 10^-11 N∙m²/kg².

M is mass of the earth and it is equal to 5.98 x 10²⁴ Kg.

m is mass of the satelpte.

R is the distance from satelpte to center of the Earth.

A satelpte, when it revolves around the earth, it undergoes a pulpng force from the sun and the moon due to their gravitational forces. This force is known as Centrifugal force (F₂) because this force tends the satelpte away from earth.

Mathematically, the Centrifugal force (F₂) acting on satelpte can be written as

$$F_{2} = frac{mv^2}{R} $$

Where, v is the orbital velocity of satelpte.

Orbital Velocity

Orbital velocity of satelpte is the velocity at which, the satelpte revolves around earth. Satelpte doesn’t deviate from its orbit and moves with certain velocity in that orbit, when both Centripetal and Centrifugal forces are balance each other.

So, equate Centripetal force (F₁) and Centrifugal force (F₂).

$$frac{GMm}{R^2} = frac{mv^2}{R}$$

$$= > frac{GM}{R} = v^2$$

$$= > v = sqrt{frac{GM}{R}}$$

Therefore, the orbital velocity of satelpte is

$$v = sqrt{frac{GM}{R}}$$

Where,

G is gravitational constant and it is equal to 6.673 x 10^-11 N∙m²/kg².

M is mass of the earth and it is equal to 5.98 x 10²⁴ Kg.

R is the distance from satelpte to center of the Earth.

So, the orbital velocity mainly depends on the distance from satelpte to center of the Earth (R), since G & M are constants.