# The Last Total Solar Eclipse…Ever-Updated

One year ago, I posted a fun problem of predicting when we will have the very last total solar eclipse viewable from Earth. It was a fun calculation to do, and the answer seemed to be 700 million years from now, but I have decided to revisit it with an important new feature added: The slow but steady evolution of the sun’s diameter. For educators, you can visit the Desmos module that Luke Henke and I put together for his students.

The apparent lunar diameter during a total solar eclipse depends on whether the moon is at perigee or apogee, or at some intermediate distance from Earth. This is represented by the two red curved lines and the red area in between them. The upper red line is the angular diameter viewed from Earth when the moon is at perigee (closest to Earth) and will have the largest possible diameter. The lower red curve is the moon’s angular diameter at apogee (farthest from Earth) when its apparent diameter will be the smallest possible. As I mentioned in the previous posting, these two curves will slowly drift to smaller values because the Moon is moving away from Earth at about 3cm per year. Using the best current models for lunar orbit evolution, these curves will have the shapes shown in the above graph and can be approxmately modeled by the quadratic equations:

Perigee: Diameter = T2 – 27T +2010 arcseconds

Apogee: Diameter = T2 -23T +1765 arcseconds.

where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5.

The blue region in the graph shows the change in the diameter of the Sun and is bounded above by its apparent diameter at perihelion (Earth closest to Sun) and below by its farthest distance called aphelion. This is a rather narrow band of possible angular sizes, and the one of interest will depend on where Earth is in its orbit around the Sun AND the fact that the elliptical orbit of Earth is slowly rotating within the plane of its orbit so that at the equinoxes when eclipses can occur, the Sun will vary in distance between its perihelion and aphelion distances over the course of 100,000 years or so. We can’t really predict exactly where the Earth will be between these limits so our prediction will be uncertain by at least 100,000 years. With any luck, however, we can estimate the ‘date’ to within a few million years.

Now in previous calculations it was assumed that the physical diameter of the Sun remained constant and only the Earth-Sun distance affected the angular diameter of the Sun. In fact, our Sun is an evolving star whose physical diameter is slowly increasing due to its evolution ‘off the Main Sequence’. Stellar evolution models can determine how the Sun’s radius changes. The figure below comes from the Yonsei-Yale theoretical models by Kim et al. 2002; (Astrophysical Journal Supplement, v.143, p.499) and Yi et al. 2003 (Astrophysical Journal Supplement, v.144, p.259).

The blue line shows that between 1 billion years ago and today, the solar radius has increased by about 5%. We can approximate this angular diameter change using the two linear equations:

Perihelion: Diameter = 18T + 1973 arcseconds.

Aphelion: Diameter = 17T + 1908 arrcseconds.

where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5. When we plot these four equations we get

There are four intersection points of interest. They can be found by setting the lunar and solar equations equal to each other and using the Quadratic Formula to solve for T in each of the four possible cases.:

Case A : T= 456 million years ago. The angular diameter of the Sun and Moon are 1890 arcseconds. At apogee, this is the smallest angular diameter the Moon can have at the time when the Sun has its largest diameter at perihelion. Before this time, you could have total solar eclipses when the Moon is at apogee. After this time the Moon’s diameter is too small for it to block out the large perihelion Sun disk and from this time forward you only have annular eclipses at apogee.

Case B : T = 330 million years ago and the angular diameters are 1852 arcseconds. At this time, the apogee disk of the Moon when the Sun disk is smallest at aphelion just covers the solar disk. Before this time, you could have total solar eclipses even when the Moon was at apogee and the Sun was between its aphelion and perihelion distance. After this time, the lunar disk at apogee was too small to cover even the small aphelion solar disk and you only get annular eclipses from this time forward.

Case C : T = 86 million years from now and the angular diameters are both 1988 arcseconds. At this time the large disk of the perigee Moon covers the large disk of the perihelion Sun and we get a total solar eclipse. However before this time, the perigee lunar disk is much larger than the Sun and although this allows a total solar eclipese to occur, more and more of the corona is covered by the lunar disk until the brightest portions can no longer be seen. After this time, the lunar disk at perigee is smaller than the solar disk between perihelion and aphelion and we get a mixture of total solar eclipses and annular eclipses.

Case D : T = 246 million years from now and the angular diameters are 1950 arcseconds. The largest lunar disk size at perigee is now as big as the solar disk at aphelion, but after this time, the maximum perigee lunar disk becomes smaller than the solar disk and we only get annular eclipses. This is approximately the last epoc when we can get total solar eclipses regardless of whether the Sun is at aphelion or perihelion, or the Moon is at apogee or perigee. The sun has evolved so that its disk is always too large for the moon to ever cover it again even when the Sun is at its farthest distance from Earth.

The answer to our initial question is that the last total solar eclipse is likely to occur about 246 million years from now when we include the slow increase in the solar diameter due to its evolution as a star.

Once again, if you want to use the Desmos interactive math module to exolore this problem, just visit the Solar Eclipses – The Last Total Eclipse? The graphical answers in Desmos will differ from the four above cases due to rounding errors in the Desmos lab, but the results are in close accord with the above analysis solved using quadratic roots.

# Landscape Dimming During a Total Solar Eclipse

During a solar eclipse, the lansdcape will slowly dim until it is nearly complete darkness along the path of totality. other observers wil see te landscape dim a bit but then brighten to normal intensity. If you didn;t know that an eclipse was going on you might not even notice the dimming, mistaking it for a cloud passing across the sun. The geometric condition for this dimming have to do with the area of exposed solar surface and how this changes as the disk of the moon passes across it. Below is a simple mathematical model for ambient light dimming that you can put to the test the next time a solar eclipse passes over your geographic location.

I have reanalyzed the geometry and defined it in terms of the center-to-center distance, L, between the sun and moon, and their respective radii Rs and Rm as the figure of the upper half-plane of the intersection shows, with the yellow area on the left representing the disk of the sun and the white area on the right the disk of the moon. This problem was previously considered in 2000 by British astronomer David Hughes who used the distance defined by the segment FE, which he called alpha, but L = 1+M-a. The figure shows the moon overlapping the disk of the sun in a lens-shaped zone whose upper half is represented by the area AFDE.

The basic idea is that we want to compute the area of the lunar arc cap AFD by computing the area of the sector BAF and subtracting the triangle BAE from the sector area. That leaves the area of the cap as the left-over area. We perform the same calculation for the solar sector CAE and subtract the triangle CAD from this.  The resulting area of the full lens-shaped overlap region is then

Occulting Area = 2x(AreaAFD + AreaAED).

Because of the geometry, the resulting area should only depend on the center-to-center separation and the radii of the sun and moon. You should not have to specify any angles as part of the final calculation. In the following we will use degree measure for all angles.

The area of the sector of a circle is just A = (Theta/360)piR2 so that gives us the first two relationships:

To simplify the problem, we are only interested in the fraction of the full sun disk that is illuminated. The full sun has an area of pi Rs2, so we divide Am and As by pi Rs2 , and if we define Rs=1.0 and M = Rm/Rs we get:

Although M is fixed by the solar-lunar ratio, we seem to have two angular variables alpha and theta that we also have to specify. We can reduce the number of variables because the geometry gives a relationship between these two angles because they share a common segment length given by h.

so that the EQ-1 for A can be written entirely in terms of the center-to-center distance, L,  and moon-to-sun disk ratio M = Rm/Rs. This is different than the equation used by Hughes, which uses the width of the lens (the distance between the lunar and solar limbs) segment FDE=a as the parameter, which is defined as L = 1+M-a.

During a typical total solar eclipse lasting 4 minutes, we can define L as

L = 1900 – 900*(T/240) arcseconds where T is the elapsed time from First Contact in seconds. Since L is in units of the current solar diameter (1900 asec) we have

EQ 3)          L = 1 – T/480.

If we program EQ 1, 2, and 3 into an excel spreadsheet we get the following plot for the April 8, 2024 eclipse.

First Contact occurs at 16:40 UT and Fourth Contact occurs at 19:57 UT so the full duration is 197 minutes. During this time L varies from  -(1+M) to +(1+M). For the April 8, 2024 eclipse we have the magnitude M = 1.0566,  so  L varies from -2.0566 (t=0) to +2.0566 (t=197m). As the moon approaches the full 4-minute overlap of the solar disk between L=-0.05 and L=+0.05 (t =97m to t=102m), we reach full eclipse.

We can re-express this in terms of the landscape lighting. The human eye is sensitive to a logarithmic variation in brightness, which astronomers have developed into a ‘scale of magnitudes’. Each magnitude represents the minimum change in brightness that the human eye can discern and is equivalent to a factor change by 2.51-times. The full-disk solar brightness is equal to -26.5m, full moon illumination is  -18.0m on this scale. The disk brightness, S, is proportional to the exposed solar disk area, where E is the solar surface emission in watts/m2 due to the Planck distribution for the solar temperature of T=5770 k.  This results in the formula:

m = -26.5 – 2.5log10(F)

where F is the fraction of the full disk exposed and is equal to Equation 1.  For a sun disk where 90% has been eclipsed, f=0.10 and the dimming is only 2.5log(1/10) = 2.5m. How this translates into how humans perceive ambient lighting is complicated.

The concept of a Just Noticeable Difference is an active research area in psychophysics. In assessing heaviness, for example, the difference between two stimuli of 10 and 11 grams could be detected, but we would not be able to detect the difference between 100 and 101 grams. As the magnitude of the stimuli grow, we need a larger actual difference for detection. The percentage of change remains constant in general. To detect the difference in heaviness, one stimulus would have to be approximately 2 percent heavier than the other; otherwise, we will not be able to spot the difference. Psychologists refer to the percentages that describe the JND as Weber fractions, named after Ernst Weber (1795-1878), a German physiologist whose pioneering research on sensation had a great impact on psychological studies. For example, humans require a 4.8% change in loudness to detect a change; a 7.9% change in brightness is necessary. These values will differ from one person to the next, and from one occasion to the next. However, they do represent generally accurate values.

The minimum perceivable light intensity change is sometimes stated to be 1%, corresponding to +5.0m, but for the Weber Fraction a 7.8% change is required in brightness corresponding to only -2.5log(0.078) = +2.7m. This is compounded by whether the observer is told beforehand that a change is about to happen. If they are not informed, this threshold magnitude dimming could be several magnitudes higher and perhaps closer to the +5.0m value.

# The Heliophysics Big Year

Heliophysics is an area of space science, named by NASA, which focuses on the matter and energy of our Sun and its effects on the solar system. It also studies how the Sun varies and how those changes pose a hazard to humans on Earth and in space.

The Heliophysics Big Year is a global celebration of solar science and the Sun’s influence on Earth and the entire solar system.During the Heliophysics Big Year, you will have the opportunity to participate in many solar science events such as watching solar eclipses, experiencing an aurora, participating in citizen science projects, and other fun Sun-related activities. For details, have a look at the NASA video that describes it in more detail [HERE].

This 14-month series for science and math educators focuses on heliophysics topics with related math problems at three levels: elementary, middle, and high school. It is sponsored by NASA’s Heliophysics Education Activation Team.

Each month, I will be hosting a webinar on the theme-of-the-month and also providing some math-oriented activities that go along with he theme. Here is a list of the Webinar viewing dates. When each webinar is completed, you can view a recorded version of it at the link provided. Visit this blog page a day before the scheduled program to register.

HBY Webinar programs:

December 19, 2023 – Citizen Science Projects – Do Auroras Ever Touch Ground?

January 16, 2024 – The Sun Touches Everything – Solar Panel Math and Sunlight Energy.

February 20, 2024 – Fashion – How do color filters work?

March 19, 2024 – Experiencing the Sun – Predicting Solar Storms.

April 16, 2024 – Total Solar Eclipse – The Last Total Solar Eclipse!

May 21, 2024 – Visual Art – Is the Sun Really Yellow?

June 18, 2024 – Performance Art – Do Songs About the Sun Follow the Sunspot Cycle?

July 16, 2024 – Physical and Mental Health – How Old is Sunlight?

August 20, 2024 – Back to School – Can You Accurately Draw the Solar Corona in Under 5 Minutes?

September 17, 2024 – Environment and Sustainability – Interplanetary solar electricity for spacecraft.

October 15, 2024 – Solar Cycle and Solar Max – Predicting the Next Sunspot Cycles and Travel to Mars.

November 19, 2024 – Bonus Science – TBD

December 17, 2024 – Parker’s Perihelion – TBD

To participate in a webinar, held at 7:00pm Eastern Time, you first need to register for it by clicking on the appropriate link below and filling out the registration form. You will then receive via email the link to the live program.

2/20/24 Fashion, Color, and Light https://us02web.zoom.us/webinar/register/WN_4w9xTzxHTO-fcOMJpSjWvA

3/19/24       Experiencing the Sun

https://us02web.zoom.us/j/82743230881

4/16/24       Total Solar Eclipse

https://us02web.zoom.us/webinar/register/WN_MJt-5gG-S-SskTPmYFb25g

5/21/24       Visual Art

https://us02web.zoom.us/webinar/register/WN_XqFErgloQECPkTn_uNEreA

6/18/24       Performance Art

https://us02web.zoom.us/webinar/register/WN_Z4QRFZM2SCSX2QmPC7DMWg

7/16/24           Physical and Mental Health

8/20/24           Back to School

9/17/24           Environment and Sustainability

1015/24          Solar Cycle and Solar Max

11/19/24         Bonus Science

12/17/24         Parker’s Perihelion

View recordings and PowerPoint slide deck of completed webinars at: