Astronomy

When was the variation of apparent diameter of the moon first measured?

When was the variation of apparent diameter of the moon first measured?


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The orbit of the Moon around the Earth isn't quite a circle. Since distance between the Earth and the Moon changes over time, the apparent diameter of the Moon also varies very slightly. With a digital camera, you can observe this by counting pixels. However, this variation is very small and not easily noticed.

Who discovered that the angular diameter of the Moon varies over time?


Track the changing size of the Moon

A more significant variation comes from the Moon’s elliptical orbit. Here the difference between the perigee (closest) and apogee (furthest) distance is significant at around 43,000km.

Lunar perigee occurs once every lunar orbit, about once a calendar month, and on the 3 or 4 times a year this coincides with a full or new Moon it’s known as a supermoon.

To show this variation, you’ll need to photograph the Moon with a telescope or telephoto lens that can produce a lunar image which fills at least half the imaging frame of a digital camera.

Then attempt to image the Moon every day over the course of a month (this is the hard part as the weather will interfere). Make sure the setup is the same for all the shots you obtain.

Once you’ve collected as many phase images as possible, open them in a program like GIMP and rotate each disc so the Moon’s ‘centre-line’ is vertical.

This is easy to determine for crescent Moons, as it’s the imaginary line connecting the crescent’s two points. For gibbous phases it can be trickier, but still fairly obvious.

Arrange the images to show the Moon’s apparent diameters as close as possible to side-by-side, like in our image, lining them up in day order.

Amazingly, this simple comparative arrangement shows something quite incredible: the periodic variation in the apparent diameter of the Moon caused by its continually varying distance from Earth.

Pete Lawrence is an experienced astronomer and a co-host ofThe Sky at Night. This article originally appeared in the December 2020 issue ofBBC Sky at Night Magazine.


The Moon is drifting away from the Earth

According to the giant impact hypothesis, a widely accepted theory, the Moon was created as a result of a catastrophic impact between Earth and a Mars-sized planet (called Theia) about 4.5 billion years ago.

The Moon had orbited much more closely in the past and it is drifting away from the Earth since its formation. This drifting was also confirmed by American and Soviet experiments, using laser-ranging targets placed on the Moon.

It is predicted that the lunar distance will continue to increase until (in theory) the Earth and Moon become tidally locked to each other, just like Pluto and Charon (today, only the Moon is tidally locked, that’s why we see only one side of the Moon).

This would occur when the duration of the lunar orbital period equals the rotational period of Earth. The two bodies would then be at equilibrium, and no further rotational energy would be exchanged. When that happened, you’d see Earth in roughly the same spot from the Moon forever. In other words, you’d see the same face of Earth from the Moon.

However, models predict that 50 billion years would be required to achieve this configuration, which is significantly longer than the expected lifetime of the solar system.

Pluto and Charon are tidally locked to each other. Image: Artist’s conception depicting New Horizons spacecraft near Pluto. Pluto’s moon Charon, also known as (134340) Pluto I is in the background. New Horizons is an interplanetary space probe that was launched as a part of NASA’s New Frontiers program. Engineered by the Johns Hopkins University Applied Physics Laboratory (APL) and the Southwest Research Institute (SwRI), with a team led by S. Alan Stern, the spacecraft was launched on January 19, 2006, from Cape Canaveral Air Force Station by an Atlas V rocket directly into an Earth-and-solar escape trajectory with a speed of about 16.26 kilometers per second (58,536 km/h 36,373 mph). Its primary mission was to perform a flyby study of the Pluto system in 2015. It has a secondary mission to fly by and study one or more other Kuiper belt objects (KBOs) in the decade to follow. It is the fifth of five artificial objects to achieve the escape velocity that will allow them to leave the Solar System (others being Pioneer 10 – launched in 1972, Pioneer 11 – launched in 1973, Voyager 2 – launched in August 1977, and Voyager 1 – Launched in September 1977). Image: NASA Goddard Media Studios

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Moon Games
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Explanation: The Moon's measured diameter is around 3,476 kilometers (2,160 miles). But apparent angular size, or the angle covered by an object, can also be important to Moon enthusiasts. Angular size depends on distance, the farther away an object is, the smaller an angle it covers. Since the Moon is 400,000 kilometers away, its angular size is only about 1/2 degree, a span easily covered by the tip of your finger held at arms length, or a measuring tape held in the distance by a friend. Of course the Sun is much larger than the Moon, 400 times larger in fact, but today the New Moon will just cover the Sun. The total solar eclipse can be seen along a track across northern Canada, the Arctic, Siberia, and northern China. (A partial eclipse is visible from a broader region). Solar eclipses illustrate the happy coincidence that while the Sun is 400 times the diameter of the Moon, it is also 400 times farther away giving the Sun and Moon exactly the same angular size.


Telescopes

What can you see right now with both your eyes? It’s not so much what you see as how much you can see. If you have healthy eyesight you can probably see about 120° in front and slightly to the sides, so basically a third of what is around you. This is your field of view.

It is what is in front of you that you see best. We look through a telescope and see a small are of sky and this is the field of view we refer to in astronomy. It’s usually abbreviated to FOV.

An eyepiece will let us see more of less of the sky. A wide FOV lets us see more. This is useful if we want to look at a particular constellation. A smaller FOV covers a smaller area so we might want to look at some of the Moon rather than all of it. The eyepiece will also let us magnify the image. We will come to that later.

There are two types of FOV apparent and real.

My 26mm eyepiece has an apparent FOV of 52°. This is the apparent FOV as it’s not connected to telescope yet.

True FOV is when it is attached to a telescope. How do we figure out what this is then? With sums…

True FOV = Apparent FOV / Magnification
So we take 52 and divide it by the magnification. For which we need another formula (sorry about this).

Magnification = Telescope Focal Length / Eyepiece Focal Length
I know my telescope’s focal length is 1470mm (as it says next to the lens) and divide that by 26 = 56 magnification.
True FOV = Apparent FOV / Magnification so 52/56 = 0.9°.

A good reference when measuring the sky is the Moon. A full moon's diameter is half a degree or thirty arc minutes. With my 26m eyepiece I could see almost two moon diameters width of sky.


When was the variation of apparent diameter of the moon first measured? - Astronomy

AST 301
Homework #1
Due Friday Jan. 24

1. This weekend the Moon should be near full and visible in the evening (rising later each night). I want you to measure the angular diameter of the Moon using the ruler I xeroxed on this paper. Hold the paper or another ruler at arm's length and measure the apparent width of the Moon. Also measure the distance of the ruler from your eye.

Use these numbers to calculate the angular diameter of the Moon. The angle is approximately:

where w is the measured width of the Moon on your ruler, and d is the distance from your eye to the ruler. (w and d must be measured in the same units.)

The book says the angular width of the Moon is 0.5 . Your answer is probably different from this number. (Most people get a bigger number.) What errors might have occured in your measurement?

2. From observations of the shadow of the Earth on the Moon during a lunar eclipse we know that the diameter of the Moon is about 3.5 times smaller than the diameter of the Earth. Find the diameter of the Earth in your book and calculate the diameter of the Moon.

We can now calculate the distance to the Moon. We could solve the formula given above for the distance, taking w to be the diameter of the Moon and d to be its distance, but it's a little easier to note from the diagram below that the ratio of the diameter of the Moon to its distance equals the ratio of the measured diameter on your ruler to the distance of the ruler from your eye. Use either method, and calculate the distance to the Moon using your measurements and the diameter of the Moon you just calculated.


Diameter of the Moon

The moon is Earth's nearest neighbor in space. Since ancient times, people have measured time by the phases of the moon, from new moon to new moon again (about 29½ days). However, people also thought the moon was a goddess or god (Luna and Diana were the "moon goddesses"), powerful enough to control humans and their actions. To this day, many countries use the moon to determine their months of the year through use of a lunar calendar.

The moon orbits the earth, and completes one rotation after 27 and a third days. The distance between the Earth and the Moon is 384,385 km. In comparison to the Earth however, the moon's weight and diameter are much smaller. The weight of the moon is 1/81 that of the Earth, and its diameter is ¼ of the Earth's. According to the data, the diameter of the moon is relatively centered, at about 3,479 km. This can be determined by a very simple proportion setting circumference equal to 360°, and the diameter of the moon equal to 0.5° (as this is how much of an angle the moon occupies):

r = 3.844 × 10 8 m
d = diameter of the moon

Solving for d gives 3.479 × 10 6 m

On the moon itself, no life exists, and there is no wind, air or water. While during the day, temperatures rise to over 173 °C and at night, temperatures drop way below freezing.

People have been fascinated by the moon ever since 1609, when Galileo formed a crude telescope to examine it. While as time progressed, drawings of the lunar surface improved, it was never fully explored. When the NASA space program began in 1958, space exploration became a reality. On July 20, 1969, Neil Armstrong became the first human being to land on the moon with Apollo 11 and explore it. Since then more information has been gathered on the moon (i.e., its density, formation, and effect on earth), and experiments on it are still being conducted to this day.

As the only natural satellite of the Earth, the Moon has always captured the attention of scientists across the world. Aristarchus, (circa 240 BC) often referred to as the "Copernicus of antiquity", was the first to estimate the size of the Earth, the size and distance to our Moon, and the size and distance of our Sun. His ideas were truly radical for his time period, but were eventually accepted by future scientists.

Aristarchus applied modern geometric methods in order to measure the size of celestial bodies such as the moon.

Where S is the object's diameter, D is the object's distance, and θ (theta) is the angular size of the object in the sky.

By observing a lunar eclipse, Aristarchus found that the Moon moved across the sky an amount equal to 2.5 times the Moon's angular diameter. Consequently, the Earth's shadow on the Moon loses a distance equal to the Moon's diameter. He then concluded:

2½ Moon Diameter = Earth's Diameter − 1 Moon Diameter
Earth's Diameter = 7/2 Moon Diameter
Moon's Diameter = 2/7 Earth's Diameter

Substituting the Earth's diameter into the equation reveals an answer of 3.64 × 10 6 M. This answer does not match the figure that my sources use, but is very close. The slight deviation is most likely the result of inaccurate tools and measurements.

Bibliographic Entry Result
(w/surrounding text)
Standardized
Result
Earth Science. Glencoe McGraw Hill, 2002: 753. "Radius (km): 1737.4" 3474.8 km
World Atlas. USA: Rand McNally & Co., 1968: XXIV. "In comparison to our planet, the moon is considerably smaller its diameter is only 2,160 miles." 3476.2 km
Moore, Patrick, & Wil Tiron. Guide To Stars And Planets. Cambridge, UK: Cambridge University Press, 1997: 24. "The moon is officially ranked as Earth's satellite, but since it is relatively large and massive, with a diameter of 3,476 km, and a mass of 0.012 that of Earth, it may better be regarded as a companion planet." 3476 km
"Moon." The Columbia Viking Desk Encyclopedia. New York: Viking Press, 1953: 838. "Moon, Earth's only satellite (diameter c.2,160 mi)" 3476.2 km
Moon: Physical Characteristics. The Columbia Electronic Encyclopedia, 2000. "The moon's diameter is about 2,160 miles, somewhat more than ¼ the Earth's diameter." 3476.2 km

The moon has been a mystery to man for centuries. Thankfully, in more recent years, we have had the ability to explore this vast world of huge mountain peaks, enormous valleys, and gigantic craters that satellites our planet. The moon which measures 3,476 km, is about ¼ the size of Earth. In comparison to the largest moon in our solar system, Ganymede, which orbits Jupiter and has a diameter of 5,262 km, and the smallest major moon in our solar system, Miranda, which orbits Uranus and has a diameter of 472 km, our moon is considered to be rather large. It takes 27 1/3 days for the moon to fully orbit the Earth's barycenter (center of gravity), and contrary to popular belief, it rotates in an elliptical orbit, not circular.

Most of the characteristics we know about the moon are things that Galileo determined. In 1610, he invented the telescope and studied the moon's various visible characteristics. Since then, science and technology have allowed us to actually send man into space to walk on the moon. Perhaps one of the more interesting things I found out is that even the older books, published before 1969, when the first man landed on the moon, had the correct measurement of the moon's diameter. This shows that most of the observations made about the moon were made through telescopic use, so that most of what we know about the moon can indeed be attributed to Galileo.


How to Use the Planet Chart

  • Using the four buttons at the top, select either Distance from the Sun, Distance from the Earth, Size in the Sky, or Brightness to control how the planets are displayed.
  • Press the Play button at the bottom of the chart to make time move in fast forward mode. You can also move backward and forward in time by sliding the hand cursor along the red timeline.
  • If the red dot next to NOW is flashing, the chart is showing the current distances and the UTC time. You can always return to this view by clicking NOW.
  • Press the calendar icon to choose a different date. Enter the time in the fields to the right of the clock icon, if you want to be very specific. The time is indicated in UTC. Find the time difference between UTC and other locations

Distance from the Sun

  • The distance of each planet from the Sun varies because all the planets orbit the Sun on different elliptical paths.
  • The distances displayed below the planets are in kilometers or miles, depending on your settings.
  • The bar below the planets illustrates their relative distance from the Sun and each other.

Distance from the Earth

  • The distance of each planet from the Earth varies because all the planets orbit the Sun on different elliptical paths.
  • Keeping in mind that you are "seeing" the planets from Earth in this chart, you will notice that the Sun, Mercury, Venus, and Mars swap order as time passes. The distance between Earth and Jupiter, Saturn, Uranus, and Neptune also varies, but they always remain in the same order as they are all so far away from each other and from our planet.

Size in the Sky

See how large the planets appear in the sky. For local times and where to look etc., try the night sky in your location. The apparent sizes of the planets are measured in arcseconds ("). For comparison, the Sun and the Moon measure about 1800 arcseconds.

Brightness

We measure the apparent brightness of celestial bodies in magnitude. The brighter a planet shines, the lower the magnitude value. Negative numbers indicate that the planet is very easy to spot in the night sky, even with ambient light. The planets also have phases, like the Moon, but these are not indicated in this chart. The two planets which are closer to the Sun than Earth&mdashMercury and Venus&mdashhave the most easily visible planetary phases, but you need a telescope to see them.


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