## Solar Eclipse August 21st 2017 – Calculations 1 – Greatest Eclipse

If you haven’t been living in a cave then you should know that in August this year, 2017, something awesome is planned for North America and if you have been living in a cave then you will be very surprised when the world goes dark for the total solar eclipse of August 21st.  Seriously if you are not aware of the magnitude of this event then get googling, go to www.eclipse2017.org or maybe www.space.com or eclipse.gsfc.nasa.gov and get yourself informed and start making your travel plans!

The solar eclipse takes place when the moon passes between the earth and the sun and  due to some very favorable geometry blocks out all (total) sunlight during the transit.  To observe the effect you need to be in the right place at the right time and in the case of the August 2017 eclipse this is along a line that passes across North America.  Only on a very narrow part of the line, the umbral shadow, will the effect be total (and awesome) although in the penumbral  wider zone there will a partial obscuration.  To get the best effect you need to be in the narrow strip of totality.

Predicting the eclipse requires knowledge of the motions of the sun, moon and earth but fortunately the hard work has been done by eclipse experts and is publicly available from sources such as Fred Espenak and it is Fred’s calculations that form the basis of the rest of this post.  The calculated eclipse elements are published here https://eclipse.gsfc.nasa.gov/SEbeselm/SEbeselm2001/SE2017Aug21Tbeselm.html are referenced here and acknowledged “Eclipse Predictions by Fred Espenak, NASA’s GSFC”.  Thanks!

An eclipse can be described in many ways but thanks to the work of Bessel and later  Chauvenet, a scheme was devised to characterize an eclipse by describing the shadow axis between sun, moon and earth. The Besselian elements, themselves described as a series of time dependent polynomials, can then be evaluated during the period of the eclipse to determine local circumstances of an observer at any point with respect to the shadow axis.

Being the constantly curious type I thought it might be useful to attempt to evaluate the position of the eclipse central line from the published elements.  Now the maths behind employing the elements is published in a variety of sources which are probably not on your bookshelf and in my case I embarked on a shopping spree, “The Explanatory Supplement to the Astronomical Almanac“, Astronomical Algorithms and a few others!

Firstly, let us calculate the “Greatest Eclipse” which is defined according to www.astropixels.com as “‘the’instant’when’the’axis’of’the’Moon’s’shadow’passes’closest’to’the’center’of’ Earth.'”

Now using Fred’s elements… for the position (x,y) of the shadow cone on the fundamental plane…

$x=-0.129576+0.5406409t-0.0000293t^2-0.0000081t^3\\y=0.485417-0.1416394t-0.0000905t^2+0.0000021t^3$

These two polynomials can be evaluated for any value of t during the period of the eclipse to give the intersection of the shadow cone.  t is a time in hours relative to the reference time, which is 18:00 for this eclipse.

$t = t_1 - t_0$

$t_0=18.00\quad (TDT)$

Additionally there is a conversion required to convert between Terrestrial Dynamical Time (TDT) and Universal Time (UT).  The time difference between these two standards is “deltaT” which is estimated and published as part of the elements as 68.4 seconds. DeltaT varies and estimates of the actual difference between TDT and UT will become more accurate as we near August 21st but this difference will be a fraction of a second and not significant for my purposes at least!

So… to calculate the “Greatest Eclipse” we vary t and minimise x^2+y^2

t=0.44451491…

t=18:26:40.3 TDT

t=18:25:31.9 UTC

Now the actual eclipse location (which we will calculate next) will end up at a location in Kentucky which observes Central Daylight Time, which currently has a 5 hour offset to UTC so that gives a local time of 13:25:31.9 CDT.  So something awesome happens around 13:25 somewhere in Kentucky…

Calculating the location of the intersection of the shadow cone with the earth surface is much trickier and for this step I refer to the work “Prediction and Analysis of Solar Eclipe Circumstances” by Wentworth Williams JR….

Returning to the Besselian elements….

$d=11.86697-0.013622t-0.000002t^2\\\mu=89.24545+15.003937t$

and then employing the following:

$\rho_1 = \sqrt{1-e^2\cos^2{d}}$

$\cos\,{ d }_{ 1 } =\frac { \sqrt { 1-e^{ 2 } } \quad\cos\,{ d } }{ \rho _{ 1 } }$

$\sin{{d}_{1}}=\sin{d}/{\rho}_{1}$

${y}_{1} = y/{\rho}_{1}$

${\zeta}_{1} = \sqrt{ {\kappa}^{2}-{x}^{2}-{{y}_{1}}^{2}}$

where

$\kappa = 1 + 0.15678503\times{10}^{-3} \quad{h}_{1}$

$\tan { \textcircled{H}}=\frac { x }{ { \zeta }_{ 1 }\cos ^{ }{ { d }_{ 1 }-{ y }_{ 1 }\sin ^{ }{ { d }_{ 1 } } } }$

$\sin { { \phi }_{ 1 }=\frac { { \zeta }_{ 1 }\sin ^{ }{ { d }_{ 1 }+{ y }_{ 1 }\cos ^{ }{ { d }_{ 1 } } } }{ \kappa } }$

$\tan{\phi} = 1.003364 \tan{{\phi}_{1}}$

${\lambda}_{e}=\mu-\textcircled{H}$

${\lambda}={\lambda}_{e}-1.002738 \Delta{T}$

And then using the values for t=0.4445149 we obtain…

$\rho_1 = 0.996789058$

$\cos\,{ d }_{ 1 } = 0.9785101$

$\sin{{d}_{1}} = 0.2061987$

${y}_{1} = 0.4237993$

${\zeta}_{1} = 0.8989609$

Assuming we calculate for sea level so h=0

$\kappa = 1$

$\tan { \textcircled{H}} = 0.1397787$

$\sin { { \phi }_{ 1 }} = 0.6000565$

$\phi = 0.6451845 radians$

$\phi = 36.9663509 degrees$

${\lambda}_{e}= 1.53515234$

${\lambda}=87.6719701 degrees$

i.e. We calculate the location of the point of Greatest Eclipse to be Latitude 36.9663509 N and Longitude 87.6719701 W which is here just outside Cerulean Kentucky.  I’m pleased to note that these co-ordinates also are consistent with Fred’s page of elements

 Lat = 36°58.0'N      Long = 087°40.3'W

Using this method and varying t it is possible to plot the center line… now all I need is a map and a marker pen…

So where will you be on August 21st 2017?  (The people of Cerulean are in for a good afternoon, weather permitting).

(Special acknowledgement to Robert Nufer who helped me remove some errors due to degree/radian conversion, without Robert I would not have got this far.  In particular the published polynomials for d and mu are in degrees, these need to be converted to radians for the trig functions.  Additionally an extra final acknowledgement to Fred “Mr. Eclipse” Espenak for providing the elements without whom these calculations would have not been possible).