# Is the formula for nodal period of a near-Earth satellite given by Wikipedia correct?

Wikipedia has a formula for the nodal period of a near-Earth satellite, taking into account the oblateness of the Earth (J2), and neglecting other effects (https://en.wikipedia.org/wiki/Nodal_period). From this formula, if the eccentricity is very small (say of the order of 1e-3 or less), I have concluded that the nodal period (Tn in Wikipedia notation) would necessarily be smaller than the period of the ideal Kepler model (no aspherity for Earth), irrespective of the value of the inclination angle i in the formula. But, this seems to contradict the teachings on Celestial Mechanics (eg https://farside.ph.utexas.edu/teaching/celestial/Celestial/node93.html, cf discussion after Eq. (10.129)). Did I miss something, or Wikipedia's formula is wrong?

• There is a link to the paper by King-Hele in the Wikipedia note. The Capderou reference is not linkable because it is a textbook (Capderou Orbit and missions - Springer 2005). But this link provides more or less the same teaching:(farside.ph.utexas.edu/teaching/celestial/Celestial/node93.html). Read the discussions after Eq. (10.129). This discussion concludes that the J2 makes the satellite fly faster OR SLOWER -depending on i-, than when the satellite were orbiting a perfectly spherical (and homogeneous) planet - the Kepler case. (Thanks for pointing to the related question). May 23 '21 at 18:25
• Thanks for the edit!
– uhoh
May 23 '21 at 23:28
• Different but related: Equation for orbital period around oblate bodies, based on J2? I seem to remember that I'd actually tried a few cases to verify, but I'm not sure now. It's the equation in the answer there that you're asking about, right?
– uhoh
Nov 1 '21 at 12:31
• @uhoh, yes. According to this formula (and my calculations) $T_n(i)<T_0$, independently of the inclination i. And this independence to i is what I am trying to establish. Nov 1 '21 at 13:50
• I see, it's the "always-less-than-ness", ya that's a good question!
– uhoh
Nov 1 '21 at 14:13