Quote of the Day
You must accept 1 of 2 basic premises: Either we are alone in the universe or we are not alone. Either way, the implications are staggering!
— Wernher von Braun
Yesterday, a reader asked me how to compute the totality path width for the eclipse that will cross the US on 21 Aug 2017. I wrote a post on how to perform this calculation years ago. NASA has published a path width value of 114.7 km. This width will actually vary a bit as the shadow moves across the Earth because the distance change slightly between all the bodies involved. Also, the Earth and Moon are not perfectly round, which I assume. NASA has very detailed models that even include the Moon's shape variations due to mountains and valleys.
In today's post, I will show how to compute a good approximation to NASA's result. I provide Figures 2 and 3 to show how the various parameters are defined. For the details on the analysis, please see my original post. Figure 1 shows my model for the Sun-Earth-Moon system during the eclipse.
Figure 3 shows the details on my approximation for the umbra width.
I used Equation 1 to compute the totality path width. I grabbed the data specific to 21-Aug-2017 from this web site. The rest of the information I obtained from Google searches.
- sumbra is the totality path width on the Earth
- dsun is the distance between the Earth and Sun.
- dmoon is the distance between the Earth and Moon.
- rmoon is the radius of the Moon.
- rsun is the radius of the Sun.
- α is the vertex angle of the moon's shadow cone (see Figure 1). We compute α using
- dumbra is the length of the moon's shadow cone (see Figure 1). We compute dumbra using
Figure 4 shows my calculations. I obtained 116 km, which compares favorably with NASA's 114.7 km.