Trace and size of totality path of the 4/8/2024 solar eclipse

On 4/7/2024, we just returned to NJ from a trip to Ha Long Bay of Vietnam. The next day, 4/8, a solar eclipse occurred in the US. As this was over the US mainland, there were numerous American watching the totality of the solar eclipse; they reserved the hotels near the totality path. If hotels were fully booked, they traveled many hours from home to the closest towns where the totality path passed, just wanted to experience how the Sun disappeared during the broad daylight. In the Morristown Astronomical Society for which I am a member, there are people who prepared food and drink and drove 7-8 hours going north to Watertown, NY to witness this great event in the afternoon in some crowded spot, and then drove 7-8 hours to come home. Quite crazy.

The last time a total solar eclipse occurred was 7 years ago on 8/21/2017. The moon¡¦s shadow swept from Oregon in the west coast, taking the direction of south-east toward Casper, Kansas City, Nashville, and finally entered Atlantic Ocean at Charleston, South Carolina. The width of the shadow on the Earth surface was about 66 miles, and the totality path in the US covered about 12 million residents. In comparison, the 4/8/2024 totality path came from Mexico, entered the US in Texas, taking the north-east direction, passing Dallas, Indianapolis, Cleveland, Buffalo, and then into Montreal of Canada, exiting to Atlantic Ocean at New Brunswick of Canada. The width of the Moon¡¦s shadow was about 115 miles and the path in the US covered about 32 million residents. In the 21^st Century, there are 8 total solar eclipses in North America. That means on average, there is one total solar eclipse every 12 years. The next one is going to happen on 8/2/32044.

Because of the interest recently in how astronomy determines the Lunisolar calendar, the author has dug into the ephemeris of the Sun, Moon, and Earth, and has learned the details on how the Lunisolar calendar was constructed. It is natural to get interested in the much-talk-about total solar eclipse of 4/8/2024. Specifically, why the totality path this time is moving from south-west of the US to the north-east; and why the width of the path is what the newspaper said¡C The author has spent about 2-3 weeks focusing on the calculation of moon¡¦s shadow, with the goal of quantifying the trace of the moon¡¦s shadow and its size. Undoubtedly, these calculations can be found somewhere on the internet (e.g., Hong Kong¡¦s HKWW.org), but as the saying goes, ¡§The summit is what drives us, but the climb itself is what matters.¡¨ Other people have done it but doing it yourself step-by-step makes all the difference.

For those interested in Astronomy, a good reference book is ¡§Astronomical Algorithms¡¨ by Jean Meeus and a good website is NASA JPL Horizons. Horizons System (nasa.gov)¡D Meeus has put together a lot of formulas for calculation of the orbit of about any celestial body in the solar system. Any serious reader of the book can write codes in a laptop computer and perform the calculation. We are not going to discuss all the details of these calculations but focus on the simple principles of the orbits of the celestial bodies and specifically on the total solar eclipse.

There are four types of solar eclipses: Total, Annular, Partial, and Hybrid. There are at least two, but not over five solar eclipses a year. On average, there are 2.38 solar eclipses a year. Most of the time, there are only two. It is quite rare to have five. A total solar eclipse is uncommon; it occurs on average every 18 months. To occur in North America is even more uncommon, about every 12 years on average.

The Moon blocks some light rays from the Sun all the time (except when the Earth is in between the Sun and the Moon); this creates the Moon¡¦s shadow. Most of the time, this shadow sweeps through empty space and no one pays attention. But when the shadow sweeps on the Earth¡¦s surface, we have a solar eclipse. As shown in Figure 1, there are two types of moon¡¦s shadow formed by the Sun: one is a black converging cone (called ¡§umbra¡¨) and the other a gray diverging cone (called ¡§penumbra¡¨). If the Moon¡¦s umbra reaches the Earth and you are within the umbra area on the Earth¡¦s surface, you cannot see the Sun and this is the total solar eclipse, but if you are in the penumbra area, you see partial sun. Sometimes the distance between the Moon and the Earth is too large, and the Moon¡¦s umbra does not reach the Earth¡¦s surface, but its extended diverging cone (called ¡§antumbra¡¨) reaches the Earth. In such case, if you are within antumbra, you see an annular solar eclipse. The hybrid solar eclipse is rare. It is a phenomenon where an eclipse changes from annular to total or vice versa. This happens when the tip of umbra is kissing the Earth¡¦s surface. Because the Moon is moving and the Earth is rotating with uneven surface, the tip of the Moon¡¦s umbra sometimes touches the earth and sometimes misses it. It is noted that for a solar eclipse to occur, the Sun, Moon, Earth should be lined up and this is at the time we have a New Moon, which defines the first day of any month in the Lunisolar Calendar. Consequently, when a solar eclipse occurs, it should happen during a New Moon, but not every New Moon results in a solar eclipse. The reason is that the plane of the Earth¡¦s orbit and the plane of the Moon¡¦s orbit are different planes with a 5^o tilt. This tilt sometimes can make the Moon¡¦s shadow miss the Earth, and there will be no solar eclipses. Trace of Totality of Solar Eclipse on Earth Surface
It is recognized that there are about 93 million miles between the Sun and the Earth and 0.24 million miles between the Moon and the Earth. The most important concept to get the trace of the Moon¡¦s umbra on the Earth¡¦s surface is the 8.3 minutes for the light to travel from the Sun to the Earth/Moon. Additionally, we need to know the positions of the Sun and the Moon (the ephemeris) relative to the Earth at any instant in time. As an observer, we are on the Earth and consider the Earth to be stationary, whereas the Sun and the Moon are moving along their respective orbits. The details on how they move about their orbits can be found in several chapters in the book by J Meeus, ¡§Astronomical Algorithms¡¨. Alternatively, one can also get the ephemeris of the Sun and the Moon from NASA JPL Horizons website (Horizons System (nasa.gov). If you use the Horizons website, the Earth is the observer, the Sun or the Moon is the target, and the apparent RA (Right Ascension) and DEC (Declination) are the parameters. Once you enter the date and the range of time (68 minutes in the US for the 4/8/2024 total eclipse) when the total solar eclipse occurs, say over the US, the website outputs the ephemeris of Sun or Moon almost immediately. The next step is to take the position of the Moon at one instance and match the Sun¡¦s position at the instance 8.3 minutes ago and draw a straight line. If the line hits somewhere on the Earth¡¦s surface, then it is the area that an observer will see the total solar eclipse. Do the same for all the instances that the total eclipse is known to be on the US territory, you get the trace of the totality (but remember to add the effect of the Earth¡¦s rotation of about 2 degrees in the 8.3 minutes duration). This author has done these steps and found the trace of the totality is as shown in Figure 2 above: the totality path entered Eagle Pass in Texas from Mexico, moving in the north-east direction and exited the US territory in Houlton, Maine. This is different from the last total solar eclipse on the US territory which occurred on 8/21/2017 where the totality path entered the US territory in Oregon and moved south-east to exit in South Carolina in 90 minutes. The agreement of Figure 2 with the actual totality trace of 4/8/2024 published on newspapers is quite good.

The Width of the Totality Path:

The calculation of the width of the totality path can be found on the internet, such as https://www.mathscinotes.com/2017/08/calculate-eclipse-path-width/. On the left of Figure 3, the Moon¡¦s umbra is shown and the right shows the umbra touching the Earth surface. Using the radii of the Sun (435,000 miles), Earth (3,986 miles) and Moon (1,086 miles), and the distances of the Earth to the Sun (93,639,659 miles) and Moon (224,880 miles) at the time the 4/8/2024 solar eclipse occurred, you can compute the diameter of the umbra on the Earth¡¦s surface to be about 120 miles. This width is consistent with the 108 ¡V 122 miles published by NASA on its website.

Conclusion: The mysteries of the universe are everywhere. The two celestial bodies closest to us and closely related to our lives, the sun, and the moon, can have different fat and thin matches, but when we look at the sky, the sun and moon are miraculously almost the same. This resulted in formidable solar eclipses. Solar eclipses are wonderful celestial phenomena that humans have often experienced since ancient times. Accurate predictions of the time and location of solar eclipses have only occurred in the past few hundred years. Since the invention of computers, predictions of solar eclipses have become increasingly accurate. Of course, these accurate predictions are backed by profound astrophysical knowledge. NASA collects this knowledge extensively and then uses its huge computing power to accurately predict all solar eclipses in the next few thousand years. This article is just a simple summary, with analysis of the astronomical spectacle of the total solar eclipse we have just experienced in April this year (2024) and share it with friends.

Epilogue: As I finished this short article on April 23, I was shocked to hear two more earthquakes of magnitude 6 or above on the same day in Hualien, Taiwan. Of course, no one could predict that Hualien would experience two consecutive earthquakes in only 20 days from the magnitude 7 earthquake on April 3. The difference between a solar eclipse and an earthquake lies in the accuracy of our predicting capability for future events. While we can accurately predict the precise location and the precise time (date, hour, minute) a solar eclipse will occur in the next thousands of years, for the earthquake, we can only shrug our shoulders at our ability to predict where and when an earthquake is going to happen with any accuracy. It is so different!

(2024-04 in New Jersey¡^