# How J3 affect orbit's parameters

I'm trying to calculate the effect of J3 on my orbit's parameter (a, e, I, $$\Omega$$, ...)

I start with the equation of the gravitational potential of the Earth :

$$U = \frac{\mu}{r}\left(1 - \sum^\infty_{l=1} \left(\frac{R}{r}\right)^lJ_lP_l(sin \phi)\right)$$

With

$$P_l$$: the l-th Legendre's polynomials

(r, $$\lambda$$, $$\phi$$) is my spherical referential

So for the J3 perturbation, I have (I keep only the term where $$J_3$$ appears) :

$$U_3 = \frac{\mu}{r} \left(\frac{R}{r}\right)^3J_3P_3(sin \phi)$$

To determine the effect of this, I have to use Lagrange equations :

And now, I have a problem: How can I rewrite $$U_3$$ in function of the parameter of my orbit: a, e, I, $$\Omega$$, $$\omega$$ and M. Does anyone know how to transform my equation in term os osculating elements? Thanks !

• What do you mean by $R_3$? Do you need to propagate or study this phenomena with the Lagrange equations? They may go singular in some orbits. To calculate the spherical harmonics with Cartesian coordinates, you can use the Pines Equations. These, and other methods, are discussed in this PhD dissertation: "Efficient Models for the Evaluation and Estimation of the Gravity Field Public Deposited", Brandon A. Jones, scholar.colorado.edu/concern/graduate_thesis_or_dissertations/… Apr 25 '20 at 17:50
• $R_3$ is a typo in my message, I have edited to replace $R_3$ by $U_3$. My final goal is to express $U_3$ in function of a, e, i ... the osculating elements of the orbit. But I don't know how I can do that. I've heard of Hansen theory but I didn't understand it Apr 25 '20 at 18:12