# In the calculation of sunrise, where do the magic numbers come from?

In this question about how to calculate sunrise, there is a link to a page that describes a algorithm to calculate sunrise sunset.

But when I start to follow the instructions I find that there is a lot of magical constants in that instruction and it is hard to understand what is going on.

For example this part:

1. Calculate the Sun's mean anomaly $$M = 0.9856 t - 3.289$$

Where did he find $0.9856$ and $3.289$?

And in the next part

1. Calculate the Sun's true longitude $$L = M + 1.916 \sin(M) + 0.020 \sin(2 M) + 282.634$$ Note: $L$ potentially needs to be adjusted into the range $[0,360)$ by adding/subtracting $360$.

Where did he find $1.916$, $0.020$ and $282.634$?

## 1 Answer

The mean anomaly relates position and time of an orbiting body. It is zero at the perihelion and increases with time. Its formula is: $M=M_0+nt$.

So, in this case $M_0=-3.289°$ is the mean anomaly when the measurement was made. $n=0.9856$ is the mean motion, which is $2\pi$ divided by the duration of the full orbit.

More information here.