The math of a solar eclipse

The Math of a Solar Eclipse

A brief introduction

The following pages provide an introduction to the 2-dimensional (2-D) mathematics explaining solar eclipses.

This method serves to explain the geometry of solar eclipses, by considering the Moon's position at the time of an eclipse.

Provided many observations are available, it may become possible to make an extremely accurate determination of both the distance to the Moon as well as its size

The Danish Cartoonist Storm Petersen once said “Life is hard, but Math is even harder…”

You do not have to read all of this mathematical in order understand and to participate in this Astronomy On-Line Collaborative Project which is concerned with the Solar Eclipse on August 11, 1999.

You can participate just by doing the easy observations, as described in the “Join these projects”.

When taken together with the observations made by other groups, you will contribute to these calculations.

On the other hand, the mathematics here are not very complex and should be quite comprehensible to secondary shool students who have studied basic trigonometry.

Seasonal Variations

You have (probably) all noticed that during summer time the shadows are short (left):

The Math of a Solar Eclipse

and during winter time the shadows are long (right).

So, the `height' of the Sun (above the horizon) varies from season to season. This height is referred to as the altitude by astronomers.

Astronomers usually describe this variation by considering the conditions at the Earth's Equator:

The Math of a Solar Eclipse

Note that the solar rays hit our planet at a certain angle with respect to the Equator. Astronomers call this angle the solar declination.

In the situation above, the Sun is positioned so that we experience Summer in the Northern hemisphere. Mathematically speaking, at this time our Sun has a positive declination.

The solar declination reaches a maximum at +23.44 degrees around June 22.

During wintertime in the Northern hemisphere, the Sun reaches its minimum declination, -23.44 degrees, around December 22.

Halfway between these two dates, that is, around March 22 and September 22, the Sun's declination is near 0 deg. These topics were discussed in details during the 1989 EAAE-ESA-ESO Sea and Space Project Navigational Chapters.

Daily values for the Sun's declination may be found in the Sea and Space Almanacs, Books for Scouts, etc.

Lunar and solar declination

The Amazing Mechanics of How and When Solar Eclipses Occur

As we are now less than a week away from “The Great American Eclipse” on Aug. 21, there will be countless articles appearing in newspapers, magazines and online talking about what causes a solar eclipse.

I can remember as a young boy, not quite 7 years of age, having my grandfather explain the mechanics of what causes an eclipse using a salt shaker (for the moon), a pepper shaker (for Earth) and his fist for the sun.

By lining them up and placing the salt shaker between the pepper shaker and his fist, my grandfather explained that when such a lineup occurs in space, the moon will “get in the sun's way” and block out some or all of it, producing an eclipse of the sun.

Of course, there is a lot more to it than this. Just when, for example, do these alignments take place? [Total Solar Eclipse 2017: When, Where and How to See It (Safely)]

Eclipse seasons

If the moon orbited Earth in the same orbital plane in which Earth orbits the sun, an eclipse of the sun and an eclipse of the moon would each happen every month.

But the moon's orbit is tilted slightly relative to that of Earth: by 5.15 degrees.

As a result, at new moon phase, from our earthly vantage point, the moon usually appears to pass either above or below the sun in our sky and the moon's shadow misses Earth.

If the new moon crosses directly in front of the sun, then we have an eclipse (or as my grandfather would have said, “the moon will get in the sun's way”). In fact, when we see its silhouette in front of the sun, it's the only new moon that we can actually “see.”

The accompanying table lists the eclipses for 2015, 2016 and 2017. 

Three Years of Eclipses

The Math of a Solar Eclipse

Notice that the dates are not distributed randomly throughout the year, but are grouped at approximately half-year intervals. We cannot, for example, have an eclipse in September and another in December. For 2017, these “eclipse seasons” are in February and August.

Celestial crossing point

In our sky, there is an imaginary line that we call the ecliptic. That is the line that marks Earth's orbital path around the sun. The sun always remains on that ecliptic line, while the moon wanders above and below the ecliptic as a result of its orbital inclination.

When the moon crosses the ecliptic — that is, when it reaches that point in its orbit where it passes through Earth's orbital plane — we call that point a “node.” An eclipse can only occur when a new or full moon is near a node. On Aug.

21, the moon is new only about 8 hours after arriving at its ascending node, and the alignment of the moon and sun is good enough for a total eclipse. When we speak of an “ascending” node, that is the point where the moon crosses the ecliptic going from south to north.

On Monday, that particular point will be in the constellation Leo, while the descending node is in Capricornus. 

Now, go back and look at the table again. Note that the eclipse seasons occur a few weeks earlier each year. This is because the nodes are not stationary points at all. They move slowly westward, or “regress,” along the ecliptic while the moon's inclination remains almost unchanged.

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This regression of the moon's nodes takes 18.6 years for one complete circuit; a complete nodal cycle. If, on the other hand, the orientation of the moon's orbit were fixed, then the eclipse seasons would come at the same time every year.

[Science-Savvy Grandparents Explain Solar Eclipse in New Children's Book]

Earth versus sun: Moon caught in the middle

We can thank the gravitational tug-of-war between Earth and the sun for the changes wrought on the moon's orbital motion. Earth, of course, has greater influence on the moon because it is much closer to it than the sun. However, the sun still manages to perturb the lunar orbit in many other ways.

Along with the continual turning of the moon's orbital plane, even when the moon passes squarely across the face of the sun, the eclipse in question might not necessarily be total but annular; because the moon travels around the Earth in an elliptical orbit, it sometimes can be too far away from Earth for its dark umbral shadow to touch down on Earth's surface. The result? An annulus, or ring of sunlight that remains in view around the moon's silhouette. The major axis of the ellipse (the technical term is the line of the apsides) is also perturbed by the sun, and rotates eastward in the orbit plane in 8.85 years. Each month the dates of apogee (the moon's farthest point from Earth) and perigee (its closest point) occur a few days earlier relative to the dates of the moon's phases.

And just as the moon orbits Earth in an elliptical orbit, so does Earth similarly orbit the sun. We get as close as 91.4 million miles in early January and pull as far away as 94.5 million miles in early July. The difference amounts to less than 3.

5 percent, but the sun's gravitational attraction varies inversely as the square of the distance, in this case about 7 percent.

This annual oscillation in the sun's gravitation pull causes a perturbation in the moon's orbit known as the annual equation, and causes the moon to be slightly ahead of schedule from July to January and slightly behind at other times. 

The sun not only alters the orientation of the moon's orbit, however, but also its shape; a perturbation called the evection.

Also, because the moon is about a half million miles closer to the sun at new moon compared to full moon, the moon runs slightly ahead between new and first-quarter phases, and again between full and last quarter, an irregularity known as the variation. [Astronauts on the Space Station Will See the Solar Eclipse 3 Times]

Calculation complexities

You might think that these ever-so-slight differences in the motion of the moon have only been recently discovered, but, believe it or not, both the annual equation and the variation were known to the great Danish astronomer Tycho Brahe around the year 1600.

The invention of the telescope was still a decade or so into the future, so Tycho used naked-eye instruments. Tycho was an assiduous observer of the sky, and was able to detect minute variations in the movements of the moon and the planets.

And knowledge of the evection was known some 2,000 years ago to ancient Greek astronomers such as Ptolemy and Hipparchus.

That’s Maths: The Great American Solar Eclipse

Next Monday, August 21st, the shadow of the Moon will bring a two-minute spell of darkness as it sweeps across the United States along a path from Oregon to South Carolina. The eclipse is one of a series known as Saros 145.

If the Moon moved in the plane of the Earth’s orbit about the Sun, we would have a solar and a lunar eclipse each month. But the Moon’s orbital plane is tilted by about five degrees.

For a solar eclipse, the Moon must pass between Earth and Sun, with the three bodies in a line. This can happen only at points called nodes, where the orbital planes intersect.

The Moon passes through two nodes every month, but normally these are not located directly between Earth and Sun, so no eclipse occurs.

For more than 2,500 years, astronomers have observed a regularity in the pattern of eclipses.

From time to time the geometry of the Earth-Moon-Sun system is repeated to the extent that solar and lunar eclipses can be predicted: once the date and time of a particular eclipse is known, it is possible to predict similar events by extrapolating forward in time. The dominant period is Saros, recurring every 223 lunar months or about 18 years (more precisely, 18 years 11 days eight hours).

The eclipse near Paris in August 1999 was the 21st event in the Saros series 145, which started in AD 1639. The eclipse next Monday is Number 22 in this series. The last event, number 77, will be in 3009. Thus, the total duration of Saros series 145 is about 1,370 years.

Suppose we start the clock at a solar eclipse: the Moon is new and at a node. In a period called a draconic or nodical month, averaging 27.21 days, the Moon orbits the Earth, returning to its original position relative to the stars.

But the Sun has moved on, so we have not reached the next New Moon. Realignment with the Sun occurs a few days later, but now the Moon is no longer at a node, so there is no eclipse.

The time for the Moon to return to the same position relative to the Sun is called a synodic month, lasting 29.53 days.

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Eventually, a time is reached which is both a whole number of nodical months and a whole number of synodic months. Then the Moon is back between Earth and Sun and at a node, so another solar eclipse occurs. It turns out that 242 nodical months is close to 223 synodic months. Both are of duration 6,585 days or about 18 years. This is the Saros period.

To complicate matters, the Moon must be close to Earth (near perigee) to fully block the Sun’s light. Perigees recur every anomalistic month or 27.55 days, and it turns out that 239 such months fit the pattern of the Saros beautifully.

The time of day for each eclipse is eight hours later than the one before, so it is displaced about one third of the way around the Earth. This explains why an eclipse over the United States in 2017 follows one over Europe in 1999. A range of maps of the eclipse is available on the NASA site – eclipse2017.nasa.gov

For those looking to add to the mathematical side of life, booking is open for an evening course, Sum-enchanted Evenings, online at ucd.ie/all/study – or by phone (01- 7167123).

Peter Lynch is emeritus professor at the school of mathematics & statistics, University College Dublin. He blogs at thatsmaths.com.

Activities

One of the reasons that ancient peoples could not predict total solar eclipses was because they did not appreciate the mathematics involved in forecasting.

Also, many of the parameters needed to accurately predict eclipses had not been astronomically measured until the first century CE.

If you are taking a trip to visit Grandma in another town and want to predict at what time you will arrive, it really helps to know how, many road miles you will be traveling and how fast you will go!

Here are a selection of math challenges that will take you through some of the basic mathematics related to the August 21, 2017 eclipse. The mathematics level span all grades and abilities from elementary proportions and algebra all the way up to trigonometry and, yes, the calculus!

Challenge 1 – Working with Geographic Coordinates

How to use longitude and latitude measure.

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Challenge 2 – X Marks the Spot

Find where two eclipse tracks cross by graphing them

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Challenge 3 – X Marks the Spot

Find where two eclipse tracks cross by solving two linear equations for their intersection point.

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Challenge 4 – X- Marks the Spot with Quadratic Equations

Use the Quadratic Formula to find the intersections points for two eclipse tracks.

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Challenge 5 – Estimating the Speed of the Lunar Shadow

From the distances between observers along the path of the 2017 eclipse, and the eclipse times, estimate the speed of the moon’s shadow.

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Challenge 6 – As the Crow Flies on a Spherical Planet

Use a trigonometric formula involving sines and cosines to calculate the distance in kilometers between any two points on Earth from their longitudes and latitudes.

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Challenge 7 – Exploring the Lunar Shadow Cone

From basic angular measure and proportions, estimate the length of the lunar shadow cone during the August 21, 2017 eclipse.

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Challenge 8 – Exploring Angular Diameter

Use angular measure and proportions to determine the apparent size of the sun and moon.

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Challenge 9 – Lunar Shadow Size on Earth’s Surface

Estimate the diameter of the lunar shadow on Earth’s surface using geometry and proportions.

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Challenge 10 – Shadow Speed and Earth’s Rotation

Use the cosine law for Earth’s rotation speed to estimate the ground speed of the lunar shadow at different latitudes.

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Challenge 11 – Modeling Shadow Speed, Diameter and Duration along the Path of Totality

Use an algebraic model of the moon’s shadow speed and diameter to estimate how long totality will last.

Read More

Challenge 12 – Shadow Ground Speed

Use a physics-based model of the earth-moon system to determine the ground speed of the lunar shadow, and the distance to the moon from Earth.

Read More

Challenge 13 – The Last Total Solar Eclipse on Earth

Use linear equations to predict how long we will continue to have total solar eclipses as the sun grows larger and the moons gets farther from Earth.

Read More

Challenge 14 – Time on Mars

Illustrative Mathematics

  • For a total eclipse to occur, the sun must be completely within the two tangent lines
    from the point on the earth to the moon as shown in the picture. Suppose we
    call $P$ the point on the earth and we label the center of the moon $Q$ and
    the center of the sun $R$.

    These three points will be collinear when the moon is optimally placed to produce an eclipse at point $P$. Suppose $S$ is the point where the upper tangent line from $P$ to the moon meets the moon.

    This is all
    pictured below, with the moon removed from the picture in order to show triangle
    $PQS$ more clearly:

    Also pictured is
    ray $RB$, with $B$ on line $overleftrightarrow{PS}$ chosen so that angles $PQS$ and $PRB$ are congruent. Angle $PSQ$ is a right angle because $overleftrightarrow{PS}$ meets the circle
    representing the moon tangentially at $S$.

    Triangles $PSQ$ and $PBR$ share corresponding congruent angles $PQS$ and
    $PRB$ by hypothesis as well as angle $P$ which belongs to both triangles.
    Therefore angles $PSQ$ and $PBR$ are also congruent and angle $PBR$ is a
    right angle. By the AAA criterion for similarity, triangles $PSQ$ and $PBR$
    are similar.

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    Since corresponding sides of similar triangles have proportional
    side lengths this means that
    $$
    frac{|RB|}{|PR|} = frac{|QS|}{|PQ|}.
    $$

    If $B$ is not on the sun or is
    on the boundary of the sun, then the moon will block all of the sun's rays.
    We know that $|PQ|$, the distance from the earth to the center of the moon
    is the distance of the earth to the moon plus the radius of the moon. At its
    smallest, this is about $226,680$ miles and at its largest about $253,180$
    miles.

    We also know that $|PR|$ is the distance of the earth to the sun plus
    the radius of the sun. From the given information this is about $93,000,000 + r$
    if $r$ denotes the radius of the sun. Since total eclipses do occur this tells
    us that $|RB| geq r$ and the largest possible value of $|RB|$ would be
    $|RB| = r$.

    Plugging in what we know to the proportion at the
    end of the first paragraph gives
    $$
    frac{r}{93,000,000 + r} = frac{1080}{|PQ|}
    $$
    and we know that $226,680 leq |PQ| leq 253,180$. Cross-multiplying and
    rearranging gives
    $$
    (|PQ| – 1080)r = 1080 imes 93,000,000
    $$
    or
    $$
    r = frac{1080 imes 93,000,000}{|PQ|-1080}.

    $$
    The largest possible value of $r$ will come when $|PQ| – 1080$ is as small as possible, that is when the moon is as close to the earth as possible: this can be seen both from the structure of the expression for $r$ above or using geometric intuition as the closer the moon is to the earth the more of space it will obstruct.

    This corresponds to a value of $225,600$ miles for $|PQ| – 1080$. Plugging
    this into the above expression, the largest the radius of the sun could be
    while still allowing for a total eclipse is about $445,000$ miles.

    Because
    the moon and sun have been modeled by circles, reflecting this picture about the
    line $overleftrightarrow{PR}$ the same reasoning applies to rays emanating from the lower half
    of the sun being blocked (or not) by the lower half of the moon.

    Note that the expression
    $$
    frac{1080 imes 93,000,000}{225,600} ,,mbox{miles}.
    $$
    for the maximum size of the sun, allowing for a total eclipse, makes intuitive sense:
    if the moon were twice as far away and twice as large it would ''appear'' identical.

    The fraction $frac{93,000,000}{225,600}$ represents how many times as
    far away the sun is than the moon. So if it were this many times as large as
    the moon it would ''appear'' the same and therefore be perfectly blocked out
    by the moon (if correctly positioned).

    This plausible argument is made rigorous
    with the language of similar triangles used above.

  • We start with the equation from part (a)
    $$
    frac{r}{93,000,000 + r} = frac{1080}{|PQ|}.
    $$
    We are given that the radius of the sun is about $432,000$ miles so this means
    that $r = 432,000$.

    Recall that this equation holds when the moon blocks out the
    sun exactly with no room to spare, that is when the point $B$ of the picture lies
    on the boundary of the sun. Solving for $|PQ|$ gives
    $$
    |PQ| = frac{1080 imes (93,000,000 + 432,000)}{432,000} = 233,580.

    $$
    This is much closer to 225,600 than to 252,100 and so it will only be during a very small period of the moon's orbit around the earth that it will be close enough to the earth to block out the entire sun.

    In addition, the relative position of earth, moon, and sun need to be in just the right order for a total eclipse to occur, so it is not suprising that these are so rare.

  • How to Predict an Eclipse Without a Computer

    A composite image shows a lunar eclipse sequence from December 2010 in Jacksonville, Florida. Tuanna2010/CC BY 3.0

    Human beings have been predicting solar and lunar eclipses for almost 2,000 years, long before they knew what exactly was happening or why it is meaningful.

    These days, we have a pretty good understanding of when they’ll take place: NASA has plotted out every single one for the next 1,000 years or so (and the previous 4,000).

    But if you don’t want to rely on the experts, what are your options for predicting a solar eclipse yourself?

    Counting

    Perhaps the easiest way to predict a solar eclipse doesn’t require any sophisticated knowledge of the universe whatsoever‚ just the ability to watch, and to count, for a long time. In fact, people have been watching and counting—making calendars, essentially—for almost as long as they’ve farmed the land.

    “Every organized, agriculturally based civilization develops a calendar, because that’s what you need to determine planting times,” says Ramon Lopez, a space physicist at the University of Texas at Arlington.

    “You don’t need a heliocentric model, you just need to know rising and setting times of various models in the sky.”

    Ancient people may not have known that the Earth is round, or spinning, or orbiting the Sun, but once they started recording when things happened, they began to notice patterns.

    One of the clearest patterns for predicting solar eclipses is the Saros cycle, first observed by the ancient Mesopotamians. Within a Saros series, solar eclipses occur at intervals of 223 lunar months.

    These cycles each last around 1,000 years—eclipse after eclipse after eclipse.

    Unfortunately, the subsequent Saros eclipses don’t happen in the same place, but one-third of the way around the Earth. It takes three Saros cycles for an eclipse to recur in a similar place as the first one—approximately 54 years later.

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