I am working on calculating axial precession of a planet orbiting a star, starting by attempting to follow along with Wikipedia's article on the topic. I have gotten through most of it without much trouble, but I can't figure out what's going on in the final step. Here is the equation Wikipedia gives for axial precession caused by the gravity of the sun acting on a tilted Earth:
$$\frac{dψ}{dt} = \left(\frac{3}{2}\right) \left(\frac{GM}{a^3 (1 - e^2)^{3/2}}\right) \left(\frac{C - A}{C}\right) \left(\frac{\cos(ϵ)}{ω}\right)$$
Where
- $GM = 1.3271244×10^{20} \frac{m^3}{s^2}$
- $a = 1.4959802×10^{11} m$
- $e = 0.016708634$
- $\frac{C − A}{C} = 0.003273763$
- $ϵ = 23.43928°$
- $ω = 7.292115×10^{-5} rad/s$
Plugging that all in and working it out gives me $\frac{dψ}{dt} = 2.45 × 10^{−12} rad^{-1} s^{-1}$, which mostly agrees with what Wikipedia gives. However, Wikipedia omits the radian unit. The next thing Wikipedia does is convert that value to arcseconds per year.
I can do the math to know that the conversion factors given by Wikipedia are correct - $1 rad = \frac{360 × 60 × 60}{2π}″ = \frac{1.296 × 10^6}{2π}″$ and $1 y = 365.25 × 24 × 60 × 60 s = 3.15576 × 10^7 s$ - but I can't figure out why Wikipedia does what it does to get to its stated final value of 15.948788 arcseconds per year, and how the units work out.
I can convert from per-second to per-year by multiplying $2.45 × 10^{−12} rad^{-1} s^{-1}$ by $3.15576 × 10^7 s/y$ to get $7.731612 × 10^{-5} rad^{-1} y^{-1}$. But the radian unit still has a negative exponent, which means converting it to arcseconds should mean multiplying by $\frac{2π \: rad}{1.296 × 10^6″}$ to get a value of $3.748391 × 10^{-10} ″^{-1} y^{-1}$. To get Wikipedia's number I have to instead multiply by $\frac{1.296 × 10^6″}{2π \: rad}$, which gives the number Wikipedia lists but has the units $\frac{″}{rad^2 y}$.
How does Wikipedia go from the negative exponent on the angle unit to a positive exponent? Am I making a mistake in my calculations leading up to this step?