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Close Range Gravitational Acceleration of Ellipsoidal Body
Category : Blog Marketing
Saturday, January 31st 2009

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sys1cor1

sys1cor2

sys1cor3

sys1cor1²

sys1cor2²

sys1cor3²

radius1²

radius1²

radius2²

Because of rotation, the shape of a planet resemble an ellipsoid with pole-to-pole distance being smaller than its equatorial radius (oblate spheroid).

Assuming that sys1cor3-axis is colinear with its rotational axis, while both sys1cor1-axis and sys1cor2-axis is coplanar with its equatorial plane, such ellipsoid could be represented in Cartesian coordinate with the following equation

sys1cor1

sys1cor2

sys2cor3

sys2cor1

sys2cor2

Since the radius of said ellipsoid on both sys1cor1-direction and sys1cor2-direction is equal, we could easily transform the Cartesian Ellipsoid equation above into a Cyclindrical coordinate with the following intervariable relationship :

sys1cor1 = sys2cor1 cos(sys2cor2)
sys1cor2 = sys2cor1 sin(sys2cor2)
sys1cor3 = sys2cor3

Thus the Equation representing the Ellipsoid in Cylindrical Coordinate is :

sys2cor1²/radius1² + sys2cor3²/radius2² = 1

sys2cor1(sys2cor3)

radius1

sys2cor3²

radius2²

sys2cor3=-radius2..radius2

sys2cor3

sys2cor2=-π..π

radius2

-radius2

radius1

sys2cor3²

radius2²

GF(sys2cor1,sys2cor2,sys2cor3) dsys2cor2 dsys2cor1 dsys2cor3

sys2cor1

By integrating the gravity function GF(sys2cor1,sys2cor2,sys2cor3) within the limit mentioned above, we get the formula for calculating the gravity acceleration caused by an ellipsoid body.

This GF(sys2cor1,sys2cor2,sys2cor3) could (will have to) be substituted with gravity function of your choice. 
For Newtonian Physics :

GF(sys2cor1,sys2cor2,sys2cor3) =

sys2cor1 cos(sys2cor2) - sys1cor10
sys2cor1 sin(sys2cor2) - sys1cor20
sys2cor3 - sys1cor30

G ρ(sys2cor1,sys2cor2,sys2cor3)

((sys2cor1 cos(sys2cor2) - sys1cor10)²+(sys2cor1 sin(sys2cor2) - sys1cor20)²+(sys2cor3 - sys1cor30)²)³

Where :
   G = Gravity Constant
   ρ(sys2cor1,sys2cor2,sys2cor3) = density of the planet in said position
   [ sys1cor10 , sys1cor20 , sys1cor30 ] = Position Vector of the accelerated point-object

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Close Range Gravitational Acceleration of Ellipsoidal Body
Category : Blog Marketing
Saturday, January 31st 2009

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Summary
This comic explains the computation required to calculate close range gravitational acceleration of an Ellipsoidal body. Explanation include Cartesian-Cylindrical coordinate transformation and how to fit a Gravity function into the ellipsoid integration.