Photogravitational force might be deduced as follows.

$ F = \phi_1 (T_{s1} T_{s2}/r_{12})^4 $

where phi is photogravitaional constant and Ts is surface temperature of astronomic object.

The difference between surface temperature and core temperature is also important for near field gravity. with lattitude, it increases resulting in value between 9.78 and 9.82 m/s^2 To take care of differntial effect, It can be modified as follows.

$ F = \phi_2 (\nabla T_{s1} \nabla T_{s2}/r_{12})^4 $

$ F = \phi_2 m_{p1} m_{p2} /r_{12}^4 $

where $ m_{p1}= m_{i1} \nabla T_{s1}/\nabla T_0 $ and $ m_{p2}= m_{i2} \nabla T_{s2}/\nabla T_0 $

We want to match centrifugal and centripetal force,

$ F = m_{i2} r_{12} \omega^2= \phi_2 m_{p1} m_{p2} /r_{12}^4 $

We have derived the inverse biquadrate counterpart of Kepler 3rd law, $ \omega^2= \phi_2 m_{p1} (\nabla T_{s2}/\nabla T_0) /r_{12}^5 $

See alsoEdit