# Apsidal Precession: What Am I Getting Wrong?

In Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft (https://ui.adsabs.harvard.edu/abs/2017AJ....153..121P/abstract), one finds the precession of Mercury’s perihelion due to the oblateness (quadrupole) of the Sun to be:

$$\displaystyle \dot \varpi_{J_2} = \frac {3}{2} \frac {nJ_2}{(1-e^2)^2} \left( \frac {R_\odot}{a} \right)^2 \left(1 - \frac {3}{2} sin^2 i \right)$$

(equation 3, p. 2). A few lines down, we read that it amounts to about 0.03″ per century—later refined to 0.0286″ per century.

However, when I do the calculation, with $$a$$ = 57.90905 Gm, $$e$$ = 0.20563, $$i$$ = 3.38°, $$R_\odot$$ = 696,342 km (all four from Wikipedia), $$n$$ = 4.09°/d (calculated from formula 33.6 in Meeus 1998), and $$J_2$$ = 2.25 × 10⁻⁷ (paper’s abstract), I get 0.01855″ per century.

What am I doing wrong?

• Plugging in those numbers, I got 0.02844" per century, which is roughly 1.53 times your value and fairly close to theirs, so I think the parameter values you chose are definitely right. Apr 29, 2021 at 23:38
• Can you please detail your calculation? Once you calculate $\dot \varpi$, you need to convert to arcseconds per century; that might be where I screw up… Apr 30, 2021 at 2:34
• Here's a Wolfram Alpha link confirming the answer, although it's given in milliarcseconds per year. Apr 30, 2021 at 2:43
• Ha! That’s what I had wrong. The value obtained is in degrees per day (which I didn’t know); this is simply multiplied by 3600 to convert to seconds per day, then by 36525 to convert to seconds per century. Excellent. Thanks! :) Apr 30, 2021 at 2:50

• The reason your answer came out in degrees per day is that you plugged in $n$ with those units. Assuming you use the same units for both parts of the ratio Rsun/a, the only thing left that has units is $n$, so that will set the units of your answer. Apr 30, 2021 at 16:36