# Deflexion massive particle near the sun

I am trying to solve an exercise similar to the 25.21 of the MTW: deflexion of a massive particle near a star. But i face some trouble solving it.

I start from the equation $$u'^2 + u^2 = \frac{B^2}{J^2}[\frac{E^2}{c^4 A^2} - m^2]c^2$$ where $$A = B^{-1} = (1-r_0/r)$$, $$J$$ is the angular momentum, m the mass of the particle, $$u = 1/r$$ and $$E$$ is the energy. I know how to solve this for a massless photon putting m = 0 and defining an impact parameter b. But here I have to define a second impact parameter since there are two terms now whereas the solution of the MTW depends only on $$\beta = v/c$$ and b. I don't have any idea where does the speed plays a role here ?

For the photon, we had expanded both A and B ( in the weak field approximation with the eddington parametrisation ) to find $$u" + u = (\alpha + \gamma )GM/(c^2b^2)$$ and we deduce easily the deflection from this equation.

But here must I expand E to introduce the speed ? If someone could make the begin of the calculation or give me some clues, it would be very helpful to me.

Thanks a lot !

• I can't help you because I don't fully understand your equations. (BTW if you put an equation between $\$\$\dots\$\ it gets written in a row by itself and is much more readable). What does $u'$ mean? Differentiation wrt what? A sketch of your solution preceding your first equation would give me some hint. – Elio Fabri Jan 4 at 20:24
• I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. – Ben Crowell Jan 5 at 15:59