Programming and Simulation

Project #1B (20%)

Can you predict the next solar eclipse?

On August 21, 2017, millions of people gathered in a narrow band in the United States to witness a total eclipse of the sun. A solar eclipse occurs when the moons orbit coincides with the suns orbit such that the moon casts a shadow on the earth. Persons within that shadow observe the moon completely eclipsing the sun.

The path of a solar eclipse can be predicted in advance using the theory of Besselian Elements. The basic equations to predict the path of a solar eclipse are given below.

You are required to implement these equations in MATLAB, determine the path of the solar eclipse of April 20, 2023, and plot it on a map.

NOTE: you are not required to study Besselian theory, only to implement the equations in MATLAB.

Values Provided

The following table of Besselian elements is provided by NASA.

SOURCE: https://eclipse.gsfc.nasa.gov/SEsearch/SEdata.php?Ecl=20230420

How to use the table:

x, y rectangular coordinates of the moon with respect to the fundamental plane measured as a fraction of the equatorial radius d right declination angle of the point on the celestial sphere towards which the axis of the moons shadow is directed ephemeris hour angle of the point above e 0.00669454, ellipticity of the earth due to its non-spherical shape

A reference time is given which corresponds to t = 0. The values of x,y,d and at this time are given in the line corresponding to n = 0. Values at other times can be obtained from the polynomial equation using the coefficients in the table. Angles are given in degrees, and time is measured in hours.

Equations to determine the eclipse path

At any given time, the following equations can be used to determine the latitude () and longitude () of the center of the moons shadow on the earths surface. Once these coordinates are calculated for various time values, you can plot the path of the eclipse.

NOTE: In these equations, longitude is measured positively to the west from the prime meridian. Latitude is measured positively to the north from the equator.

Plotting the Path

Once you obtain the coordinates of the eclipse path for various times, you can plot it on a world map by modifying the script below.

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Write a MATLAB program to determine the path of the eclipse. Write your program in such a way that when executed, ONLY the final plot is displayed.

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Polynomial Besselian Elements for: 2023 Apr 20 4.000 TOT (te) 12 0.0268500 -0.4273660 11.4117899 0.5468840 0.0096630 240.242935 1 0.4950182 0.2441992 0.0137410 e0001216 e.0001210 15.003420 2 8.0803135 -8.8000494 -8.8838030 -0.8800116-0.0800115 8.008000 30.0000071 -0.000037 e tan f1 0.0046550 tan f2 0.0046318 At time t1 (decimal hours), each Besselian element is evaluated by: where: a . x, y, d, 11, 12, or ; t . t1 . te (decimal hours), and te . 4.000 TOT