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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?

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    $\begingroup$ 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. $\endgroup$
    – HDE 226868 ♦
    Apr 29, 2021 at 23:38
  • $\begingroup$ 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… $\endgroup$
    – Pierre Paquette
    Apr 30, 2021 at 2:34
  • $\begingroup$ Here's a Wolfram Alpha link confirming the answer, although it's given in milliarcseconds per year. $\endgroup$
    – HDE 226868 ♦
    Apr 30, 2021 at 2:43
  • $\begingroup$ 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! :) $\endgroup$
    – Pierre Paquette
    Apr 30, 2021 at 2:50

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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.

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    $\begingroup$ 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. $\endgroup$
    – Eric Jensen
    Apr 30, 2021 at 16:36

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