Orbital perturbation

The deviation of the satellite from the predetermined path due to external disturbances is known as orbital perturbation.

The main causes of orbital perturbation are:

I. Effect of sun and moon

II. Effect of non-spherical earth

III. Atmospheric drag

IV. Solar radiation pressure

I. Effect of sun and moon

• Gravitational attraction by sun and moon causes orbital inclination of geostationary satellite to change with time if not countered by north-south station keeping, these forces would increase the orbital inclination from an initial zero degree at launch to14.67°, 26.6 years later.
• The moon and the sun both create a gravitational potential on the satellite which causes it to perturb.

II. Effect of non-spherical earth

It is known to us that earth is not perfectly spherical, their being an equatorial bulge and flattening at the poles it is called as oblate spheroid. This oblate shape of earth causes perturbation of satellite.

a.Change in orbital period

• For a spherical earth of uniform mass kepler’s third law gives the nominal mean motion n_0 as :

n0=2πTrad/sec$n_0=\frac{2\pi /}{T}rad/sec$

• The subscript ‘0’ is included as a reminder that this result applies for a perfectly spherical earth of uniform mass. However earth is described as oblate spheroid so its mean motion is given by:

n=n0[1+k1(1−1.5sin2i)a2(1−e2)1.5]$n=n_0[\frac{1+k_1(1-1.5si{n}^{2}i)}{a^2(1-e^2{)}^1.5}]$

where,k1isconstant$where,k_{1}isconstant$

aissemimajoraxis$aissemimajoraxis$

eiseccentricity$eiseccentricity$

• This orbital period taking into account Earth’s oblateness is called anomalistic period:

T=2πnsec$T=\frac{2\pi /}{n}sec$

b. Regression of nodes

• Regression of nodes is a rotation of the orbital plane in the direction opposite to the satellite motion around the axis of rotation of the earth.
• Due to regression of nodes, the nodes (ascending and descending nodes) appear to slide across the equator.
• If orbit is prograde, the nodes slide westward and if retro grade they slide eastward.
• It causes Ω$\Omega$ the right ascension of the ascending node shifts its position

dΩdt=−kcosi$\frac{d\Omega /}{dt}=-kcosi$

Where,i–inclinationangle$Where,i–inclinationangle$

• For Polar orbits regression of nodes is zero (∵i=90°)$(\because i=90°)$.

c.Rotation of line of Apsides

• The rotation of line of Apsides is the rotation of ellipse major axis around the center of the earth on a fixed orbital plane.
• Due to rotation of line of Apsides ω the argument of perigee changes with time

dωdt=k(2−2.5sin2i)$\frac{d\omega /}{dt}=k(2-2.5si{n}^{2}i)$

• At i=63.43°$i=63.43°$, no rotation takes place since the term 2−2.5sin2i=0$2-2.5si{n}^{2}i=0$

d. Drift due to Equatorial ellipticity

• The earth is not perfectly circular in the equatorial plane; it has small eccentricity of order〖 10〗^(-5).
• This leads to gravity gradient which causes the satellite in the geostationary orbit to drift to one of the two stable points called as satellite graveyards.
• These points are at 75°E$75°E$ and 105°W$105°W$. Satellites in service are prevented from drifting to these points by station keeping.

III. Atmospheric drag

• For near satellite below about 1000km, the effects of atmospheric drag are significant because the drag is greatest at the perigee.
• The drag acts to reduce the velocity at the perigee point, with the result that the does not reach the same apogee height on successive revolution.
• As a result both eccentricity (e) and semimajor axis (a) are reduced.

IV. Solar radiation pressure

• Solar radiation pressure from sun rays perturbs the satellite orbit.
• The satellite experiences change in eccentricity. It builds up for 1st sixth months and then shrinks for the next sixth months.