J2 Perturbation Acceleration

The J₂ Perturbation Acceleration equation computes the three component forces in three Cartesian coordinates as they affect an Earth Satellite. This acceleration is expressed in the Earth Centered Inertial (ECI) coordinate direction `hatI, hatJ, hatK`.

[mcb]

INPUTS

• `J_2` - the coefficient representing the effect of the Earth's oblateness in the Legendre polynomial representation of the Earth's gravitational field.  This constant is provided by the vCalc constant:  J2
• `mu` - the gravitational coefficient
• `R_"Earth"` - the mean radius of the Earth.  This constant is provided by the vCalc constant: Earth - mean spherical radius (Km)
• `R` - the magnitude of the position vector (R) of a satellite relative to the ECI frame
• `X` - the x-component of the position vector (R) of the satellite relative to the ECI frame
• `Y` - the y-component of the position vector (R) of the satellite relative to the ECI frame
• `Z` - the z-component of the position vector (R) of the satellite relative to the ECI frame

THE MATH AND SCIENCE

The gravitational field around a perfectly spherical mass has an inverse square relation.  In other words, the force of a mass at a distance r from a perfectly spherical body is a function of `1/r^2`.  The force between the two objects falls off as the inverse square as distance increase between the two objects.

In the case of a satellite orbiting the Earth, this Force of Gravity would be very simple to calculate and thus the orbits easy to calculate and predict -- except the Earth is not a perfect sphere.  The density of the Earth is not uniform throughout and the shape of the Earth is really an oblate spheroid.  The Earth is slightly flatter at the poles and slightly wider at the equator.  So, the Earth's gravitational field is "perturbed", slightly different than it would be if the Earth were a perfect sphere.

The difference in the force computed for a satellite in orbit around the oblate spheroid Earth and the force computed for a satellite in orbit around a perfect spheroid Earth is a small additional force factor in the equations for predicting (or propagating) an Earth satellite's orbit.  That small additional factor is called a "perturbation".

This equation computes the acceleration factor called the `J_2` perturbation acceleration and represents it as three acceleration components in the three Cartesian coordinates of the ECI coordinate system.

Although these three force components are small, the effect on an Earth satellite is non-negligible. This J2 perturbation force will cause a satellite to vary from the orbit predicted by pure Keplarian two body computation.  To use satellites in today's complex world, we need to know the orbit of a satellite in a much more accurate way.  This J2 perturbation force, along with additional perturbations allow us to predict the actual motion of a satellite.

REFERENCE

Pak, Dennis C, "Linearized Equations for J2 Perturbed Motion Relative to an Elliptical Orbit" (2005).  Master's Thesis, San Jose State University.  See Linearized Equations for J2 Perturbed Motion Relative to an Elliptical Orbit

SEE ALSO

• J2 Perturbation Acceleration
• Non-spherical Earth Perturbation - RAAN (combined coefficients)
• J2 (constant)
• Earth Rotation Rate