New Blog

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So, I had a blog for a few years at artent.net, but I did not own the domain name, so I just kind of stopped blogging.

Now I've created a new blog.  The domain belongs to Google, but I think that is a step in the right direction.  This is one way to make our lives a little better---small steps in the right direction.

:)

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An Equation for Estimating how long Planets remain in Retrograde

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If you have two planets in circular orbits with semi-major axes \$a_1\$ and \$a_2\$, and periods \$T_1\$ and \$T_2\$, then the amount of time spent in retrograde is \[T_\mathrm{retro} = T_1\left|\frac{\cos ^{-1}\left(\frac{\sqrt{r}+1}{r+\frac{1}{\sqrt{r}}}\right)}{\pi\left(1-\frac{1}{r^{3/2}}\right)}\right|\] where \$r= \frac{a_2}{a_1}\$. You can apply this formula to the planets in our solar system to get approximations for their time in retrograde as seen from Earth. (These approximations are often within 10% of the correct value, but sometimes the error is larger.) \[\begin{array}{ccc} & \text{a in AU} & T_{\text{retro}} \\ \text{Mercury} & 0.387 & 23 \text{ days} \\ \text{Venus} & 0.723 & 42 \text{ days} \\ \text{Mars} & 1.524 & 73 \text{ days} \\ \text{Jupiter} & 5.204 & 121 \text{ days} \\ \text{Saturn} & 9.54 & 138 \text{ days} \\ \text{Uranus} & 19.19 & 152 \text{ days} \\ \text{Neptune} & 30.05 & 158 \t...
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