Why is Nodal precession affected by the rotational period of the planet?

The $J_2$ term in Nodal Precession equation has the rotational period of the planet. I understand that the oblateness of the sphere with an equatorial bulge induces a torque on the orbit of a satellite, which causes it to precess. But why does the planet's rotation affect this process?

• Check this section - en.wikipedia.org/wiki/Nodal_precession#Equation, where they discuss the J2 term. It contains the "central body's rotational rate". – karthikeyan Apr 16 '18 at 12:44
• From the links in the (currently unanswered) question For the mathematical relationship between J2 (km^5/s^2) and dimensionless J2 - which one is derived from the other? you can see that at least there, $J_2$ is coming from zonal harmonic expansion of a measured potential, not derived from a simple equation. I suspect the Wikipedia article is using a mathematical sleight of hand here, or is just plain lazy. – uhoh Apr 16 '18 at 13:28
• Thanks. It could be the latter, of wikipedia being lazy. I hope some more digging and time will help me! – karthikeyan Apr 16 '18 at 13:40
• Looks like J_2 here is in ECEF frame. Generally gravity field is defined in two ways. One with no centrifugal accounting. And with centrifugal force accounting. It seems the latter is mentioned is wiki. – Prakhar Apr 16 '18 at 14:22
• @karthikeyan if you come to a conclusion and no one else has posted an answer, it's perfectly fine to answer your own question and accept it. – uhoh Apr 16 '18 at 14:45

Why is Nodal precession affected by the rotational period of the planet?

It doesn't, at least not directly.

The first equation there for the rate of precession $$\omega_p$$:

$$\omega_p = -\frac{3}{2} \frac{R_E^2}{(a(1-\epsilon^2))^2} J_2 \omega \cos(i)$$

depends on the parameters of the orbit ($$a, \epsilon, \omega$$, i) and the Earth's equatorial radius $$R_E$$ and its "second dynamic form factor ($$-\sqrt{5}C_{20}$$)" or $$J_2$$ term. There is no explicit dependence on the rotational rate of the earth nor should one be expected, as $$J_2$$ is an expression related to an axisymmetric term.

But the Wikipedia article goes on to use a different equation that tries to predict the value $$J_2$$ term based on some equilibrium model of a rotating body. The article says:

This last quantity is related to the oblateness as follows:

$$J_2 = \frac{2 \epsilon_E}{3} - \frac{R_E^3 \omega_E^2}{3 GM_E}$$

without really giving a source for this or an explanation.

In my opinion the Wikipedia article could be improved by explaining that this equation isn't necessarily fundamental, nor should it be used to generate a value for $$J_2$$ that would then be used to propagate a spacecraft's orbit.

The oblateness or $$\epsilon_E$$ in the equation comes from the surface of the Earth, whereas an accurate measurement of $$J_2$$ will only come from careful, experimental measurements of Earth's gravitational field, which results from the real 3D distribution of mass within the entire volume of the Earth.

A small rocky body's $$J_2$$ would be unrelated to it's rotation rate since centrifugal forces would not define it's shape. The larger the body and the more that it could flow in order to reach an equilibrium mass distribution however, the closer the link would be between its rotation rate and the mass distribution and therefore the link between it's actual $$J_2$$ and the one predicted by the second equation.

• – ProfRob Jan 18 '20 at 10:15