# How do people measure the distance between the Earth and The Moon?

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How do people measure the distance between the Earth and The Moon? What methods were used? Where can I find the official data?

There are many ways and I'm not entirely sure who you mean with "how do people measure the distance" (does this exclude space observatories like e.g. Clementine, probes currently in lunar orbit like e.g. Lunar Atmosphere and Dust Environment Explorer a.k.a. LADEE, or any other currently available technology?), but one interesting and extremely precise way is by laser ranging, pointing them towards one or more of the retroreflectors that were left on the surface of the Moon by the Apollo landers, in what is known as the Lunar Laser Ranging Experiment. Since they are laid on the Moon's near side that is always pointed towards the Earth, these retroreflectors are available for measurements to any properly equipped researcher, no matter which country they're from.

By measuring the time it takes for the light to reflect back, we can infer distance by simply multiplying the c (speed of light in vacuum) with the time taken between the light signal being transmitted and received and then dividing all of it by two to get a single leg of the distance. I'll let you read the linked to Wikipedia page on LLRE for more information, but if you find that too boring, here's a YouTube video on Mythbusters: Moon Hoax Retroreflectors that explains it from the practical perspective.

Some of the findings of this long-term experiment are:

• The Moon is spiraling away from Earth at a rate of 3.8 cm (about 1.5 inches) per year. This rate has been described as anomalously high.
• The Moon probably has a liquid core of about 20% of the Moon's radius.
• The universal force of gravity is very stable. The experiments have put an upper limit on the change in Newton's gravitational constant G of less than 1 part in $10^{11}$ since 1969. The likelihood of any "Nordtvedt effect" (a composition-dependent differential acceleration of the Moon and Earth towards the Sun) has been ruled out to high precision, strongly supporting the validity of the Strong Equivalence Principle.
• Einstein's theory of gravity (the general theory of relativity) predicts the Moon's orbit to within the accuracy of the laser ranging measurements.

So these distance measurements are so precise they're a lot more interesting and reliable than merely telling us how far away the Moon is. For one, it also blows any Moon landing hoax theories right out of the water. Links to many more official results can be found in references and external links sections of Wiki on LLRE.

There is also a plethora of other ways to measure the distance to the Moon, with different precision and equipment requirements, but I will let others add some other methods in their answers. If however you had some specific "people measuring the distance" equipped with specific measuring devices in mind, don't forget to mention that in your question. ;)

By measuring the length of the shadow in Alexandria at noon on the Summer Solstice when there was no shadow in Syene, he could measure the circumference of the Earth!

High Noon on the Summer Solstice

[Click on the image to view full size (34k)] (Graphic by R. Pogge)

At Syene: The Sun is directly overhead, no shadows are cast at that moment.

At Alexandria: The Sun is 7 12 /60 degrees south of overhead, casting shadows.

Since a full circle is 360 degrees, the arc from Alexandria to Syene is thus approximately 1/50th of a full circle (the sun angle above divided by 360).

Therefore, the circumference of the Earth is 50 times the distance from Alexandria to Syene. Question 1: How far is Alexandria from Syene? 5000 stadia

Question 2: How big is 1 stadion? 600 Greek Feet (length of a foot race in a Greek "stadium")

185m/stadion that I quote in these notes.]

Putting Eratosthenes result into modern units, his estimate of the circumference of the Earth is as follows:

The modern measurement is 40,070 kilometers.

Eratosthenes' estimate is only about 15% too large!

## Earth-Moon System to Scale

The illustration below shows the Earth-Moon system to scale, with the same scale used for size and distance. I’ve seen a physical model of this once, with the Earth and the Moon in opposite corners of a room, and found it striking how far away (relative to its size) the Moon is from the Earth.

Earth-Moon system to scale

The diameter of the Earth is 12742 km, while the diameter of the Moon is 3475 km. So, to draw the Earth-Moon system to scale on a screen, there have to be 3.67 “Earth”-pixels for each “Moon”-pixel. This ratio of 3.67 is actually quite small compared to the ratio of the sizes of other planets and their moons in our solar system. In other words, the Earth has a large moon. The (average) distance between both is 384400 km. This distance translates into 111 “Moon”-pixels between (the center of) both objects.

Showing the Sun-Earth system to scale in the same way is not reasonably possible on this page. The diameter of the Sun is 1391000 km. Hence, there have to be 109 “Sun”-pixels for each “Earth”-pixel, which is manageable. However, since the distance to the Sun is 149597900 km, the distance between the Earth and the Sun to the same scale would be 11741 “Earth”-pixels.

The enormous size of the Sun becomes apparent in the following illustration, which shows the Sun with the Earth-Moon system to the same scale (all sizes and the distance between the Earth and the Moon). Drawn with the Earth in its center, the Sun is so large that it extends beyond the orbit of the Moon.

Sun, Earth, and Moon, all to scale [Sun photo: Thomas Bresson]

I’ve written a practical follow-up article on the Earth-Moon system, which uses a basketball and a tennis ball to represent the Earth and the Moon. You can find more numbers on the Solar System on NASA’s Solar System Exploration site.

vivek monteiro (not verified)

Tom
Do take a look at our daytime astronomy website www.daytimeastronomy.com and also at www.navnirmitilearning.org. There is an interesting discussion on the 'nano solar system' in the section on resources. Basically, what happens when we scale down by 10^9. It's a very popular exhibit and activity with kids here in India.
Vivek

In reply to tom by vivek monteiro (not verified)

Hello Vivek,
I normally don't allow promotional comments, but I'll make an exception since you seem to have a very nice thing going! I am a member of the Urania (in Dutch) public observatory, where we try to do similar things, including with kids and in schools. Consider something like I show in the article Earth-Moon System Scale Model for one of your events. And then send me a photo! :-)
Tom

In the comment below, first sentence of the second paragraph, Earth and Moon should be reversed. As written it states that the moon is 3.67 times the diameter of earth.
"each “Earth”-pixel has to correspond to 3.67 “Moon”-pixels"
should be written as
"each “Moon”-pixel has to correspond to 3.67 “Earth”-pixels"

In reply to In the comment below, first by DCR (not verified)

Thanks for pointing this out, it was indeed a mistake to put it like that. I've corrected it by rephrasing the sentence.

Tomas Telensky (not verified)

Hi Tom!
Excellent, thank you for this! Just a note: you wrote "to draw the Earth-Moon system to scale on a screen, there have to be 3.67 Earth-pixels for each Moon-pixel". This is very misleading, since this is length ratio, whereas "number of pixels on screen" is an area unit. So, the ratio of number of pixels that both objects take would actually be a square of the length ratio: 3.67 * 3.67 = 13.47. So, to avoid confusion it's better to speak not of the pixels, but for example "size ratio".

Anyway, thanks for this excellent work! Before I googled your site I was already very surprised by the ratio, how big the distance is compared to the size of Earth. and this web confirms by beautiful picture. Thanks!!

## Method

Eratosthenes placed a vertical pole in Alexandria and another in Syene during the summer solstice, and noticed that the pole in Alexandria cast a shadow at noon, meaning that the Sun was not directly above the pole, but slightly to the south. After determining the distance between the two cities and accounting for the Earth’s curvature, Eratosthenes determined the angle of the shadow from the vertical pole to be 7.12°, which represented about one-fiftieth of the circumference of a circle. The distance between the two cities was about 5,000 stadia (500 mi), and Eratosthenes concluded that if one-fiftieth of the circumference was 5,000 stadia, then the full circle was 250,000 stadia (25,000 mi) or 40,000 km. Roughly 2,000 years later, modern equipment calculated the Earth's circumference to be 24,901.461 mi at the equator and 24,859.734 mi at the poles.

## Lab 1

Sky objects have properties, locations, and predictable patterns of movements that can be observed and described.

Students will conduct a series of inquiries about the position and motion of Jupiter’s moons using prescribed Internet simulations.

#### Computer Setup and/or Materials Needed:

Access NASA Jet Propulsion Laboratory - California Institute of Technology: Solar System Simulator and
a) Select THE MOON in the “Show me _______ “ drop down menu
b) Select THE SUN in the “as seen from _______ “ drop down menu
c) Select the radio button “I want a field of view of ____ degrees” and set the drop down menu to 0.5
d) UNCHECK all the "Options" check boxes except for EXTRA BRIGHTNESS (**this makes it easier to see!!**)
e) Click “Run Simulator”

### Phase I: Exploration

After completing the above steps, answer the following questions. If you are taking the course for credit, complete the open-ended responses within the 'Lab 1' Module link in Canvas.

1. The resulting image shows what one would see looking through a special telescope. In this picture, where is the observer with the special telescope located?

2. How does the image change if you INCREASE the field of view?

3. What is the exact date of the image?

4. Astronomers typically mark images based on the time it currently is in Greenwich, England, called UTC. What is the precise time listed on the image?

5. Using a ruler to measure the distance on the screen between the middle of Earth and the middle of the Moon, what is the measured distance? You do NOT need to know the exact number of kilometers, but simply a ruler-measurement you can compare with other measurements you make later. Alternately, you can use the edge of a piece of lined paper held in the landscape orientation and count the lines, or mark the locations of Earth and Moon along the edge of a piece of blank paper and hold the paper up next to the arbitrary "Squigit" ruler (Links to an external site.) (Note: ruler pops up in a different window. WARNING: This window is resizable, so be sure not to resize the squigit ruler window AFTER you have begun using it!) to get a measurement. You will be making many measurements in this lab, so pick a method that is efficient for you and allows reasonable precision and accuracy.

6. In the measurement you just took, which side of the Earth was the Moon on: Enter either "L", "R" or "N/A" (if the Moon was behind the Earth)?

7. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by 1 hour and determine the new distance between the Earth and Moon.

8. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

9. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by one day from when you started and determine the new distance between the Earth and Moon.

10. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

11. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by three days from when you started and determine the new distance between the Earth and Moon.

12. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

13. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by five days from when you started and determine the new distance between the Earth and Moon.

14. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

15. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by 10 days from when you started and determine the new distance between the Earth and Moon.

16. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

17. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by two weeks from when you started and determine the new distance between the Earth and Moon. Be sure to include left or right.

18. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

19. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by one month from when you started and determine the new distance between the Earth and Moon.

20. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

21. Use the browser’s BACK button to return to the Solar System Simulator homepage. Now, advance the time by three months from when you started and determine the new distance between the Earth and Moon.

22. In the measurement you just took, which side of the Earth was the Moon on: "L", "R" or "N/A" (if the Moon was behind the Earth)?

23. Consider the research question of, “how long does it take the Moon to orbit Earth?” It has been said that it takes about one “moon-th” for the Moon to go around Earth. Which of your observations confirms or contradicts this statement? Explain.

### Phase II: Does the Evidence Match a Given Conclusion?

24. Consider the research question, “How long does it take one of Jupiter’s moons to orbit Jupiter?” Set the Solar System Simulator to observe Jupiter from the Sun, where Jupiter takes up 10% of the image, and measure the distance between Jupiter and Io shown on the image

25. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

26. Advance the “time” by one day, and record the distance between Jupiter and Io.

27. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

28. Advance the “time” by two days from when you started, and record the distance between Jupiter and Io.

29. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

30. Advance the “time” by three days from when you started, and record the distance between Jupiter and Io.

31. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

32. Advance the “time” by four days from when you started, and record the distance between Jupiter and Io.

33. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

34. Advance the “time” by five days from when you started, and record the distance between Jupiter and Io.

35. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

36. Advance the “time” by six days from when you started, and record the distance between Jupiter and Io.

37. In the measurement you just took, which side of Jupiter was Io on: "L", "R" or "N/A" (if the Io was behind Jupiter)?

38. If a fellow student proposed a generalization that "Io orbits the Jupiter about every 48 hours," would you agree or disagree with the generalization based on the evidence you collected by noting patterns in the time it takes for Io to return to its original position from where it started? Explain your reasoning and provide specific evidence either from the above tasks or from new evidence you yourself generate using the Solar System Simulator. It is not enough to vaguely reference "the answers to the questions above" - you need to cite specific numeric evidence and explain how it supports your answer.

### Phase III: What Conclusions can you draw from this Evidence?

39. Europa is one of the four largest moons orbiting Jupiter. The others are Io, Callisto, and Ganymede. What conclusions and generalizations can you make from the following data collected by a student in terms of HOW LONG DOES IT TAKE EUROPA TO ORBIT JUPITER? Explain your reasoning and provide specific evidence, with sketches if necessary, to support your reasoning.

Time Measured Distance from Jupiter Appearance Notes
11pm Monday 0 squidgets Not visible, likely behind Jupiter
11pm Tuesday 5.0 squidgets On Jupiter's right side
11pm Wednesday 1.5 squidgets On Jupiter's right side
11pm Thursday 5.0 squidgets On Jupiter's left side
11pm Friday No observations cloudy

Remember, a picture is worth 10 3 words! Optional: Feel free to create and label sketches or graphs to illustrate your response. Please upload any sketches/graphs. (Please do not email your file to the instructor.)

### Phase IV: What Evidence do you need?

40. Imagine your team has been assigned the task of writing a news brief for your favorite news blog about the length of time it takes Ganymede, the largest moon in the entire solar system, to orbit Jupiter once. Describe precisely what evidence you would need to collect, and how you would do it, in order to answer the research question of, "Over what precise period of time does it take Ganymede to orbit Jupiter?" You do not need to actually complete the steps in the procedure you are writing.

Write a Procedure: Create a detailed, step-by-step description of evidence that needs to be collected and a complete explanation of how this could be done - not just "look and see when the Ganymede is first on one side and then on the other", but exactly what would someone need to do, step-by-step, to accomplish this. You might include a table and sketches - the goal is to be precise and detailed enough that someone else could follow your procedure. Do NOT include generic nonspecific steps such as "analyze data" or "present conclusions" -- these are meaningless filler. Be specific! Remember, a picture is worth 10 3 words! Optional: Feel free to create and label sketches or graphs to illustrate your response.

### Phase V: Formulate a Question, Pursue Evidence, and Justify Your Conclusion

Your task is to design an answerable research question, propose a plan to pursue evidence, collect data using using Solar System Simulator (or another suitable source pre-approved by your instructor), and create an evidence-based conclusion about some motion or changing position of a moon or planet of the solar system, that you have not completed before. Remember, a picture is worth 10 3 words! Optional: Feel free to create and label sketches or graphs to illustrate your response. Please upload all sketches/graphs here. (Please do not email your file to the instructor.)

### Research Report:

41. Write your specific research question.

42. Write your step-by-step procedure, with sketches if needed, to collect evidence. (Do NOT include generic nonspecific steps such as "analyze data" or "present conclusions" -- these are meaningless filler. Be specific!)

43. Provide your data table and/or results.

44. Provide your evidence-based conclusion statement.

### Phase VI: Summary

45. Create a PITHY 50-word summary, in your own words, that describes the motions, orbits, or rotations of Jupiter’s moons (or other moons or planets in our solar system that you might have studied). You should cite what you learned from doing each of the phases of this lab, not describe what you have learned in class or elsewhere. Include a word count at the end of your answer. (Remember, 50 words is not much! This is intended to keep you mindful of making your answers BRIEF and PITHY.)

### Submit your work in Lesson 3

This lab assignment is not due in Canvas until the due date indicated on our course calendar during Lesson 3.

## How Much Trash Is on the Moon?

Moon-based detritus includes leftover urine-collection kits, an olive branch and tons of robotic equipment from lunar probes.

There are dozens more pieces of lunar debris. But how much garbage, exactly, have humans left or sent to the moon?

It's challenging to say, but the trash on the moon likely weighs upward of 400,000 lbs. (181,000 kilograms) on Earth. This weight is taken from Wikipedia, but it sounds about right considering that quite a few heavy artifacts, such as five moon rangers, are still there, said William Barry, NASA chief historian.

Much of this moon litter was left by NASA astronauts who landed on the lunar surface between 1969 and 1972 during the Apollo program. The other rubbish comes from crewless missions from space-exploring agencies, including those from the United States, Russia, Japan, India and Europe, Barry said.

Many of the older pieces are lunar probes that were sent to the moon to learn about it, such as whether spaceships could land on its surface. In the 1960s, some scientists thought that the moon might have a quicksand-like exterior because so many space rocks had pummeled and pulverized it over the years. These robotic probes, which stayed on the moon after their missions ended, showed that this idea was wrong, and that human-made gear could land on the moon's surface, Barry said.

The moon is also home to lunar orbiters that mapped its terrain before they crashed into its surface, adding to the garbage heap.

Other gear in the growing landfill has helped scientists learn about the moon. For instance, the Lunar Crater Observation and Sensing Satellite (LCROSS) was sent to the moon to study the hydrogen there and to confirm the existence of water. Its mission was successful, and LCROSS is still hanging out on the moon's surface, Barry said.

As for the objects left by the Apollo astronauts, there wasn't a lot of thought put into bringing back unneeded equipment, Barry said. Moreover, doing so would have used up precious resources, such as fuel, he added.

"On any engineering project, like landing on the moon, you design the mission to do what you need it to do and not a whole lot more," Barry told Live Science. "The real concern was: Can we get the crew safely to the moon, can they get the samples they need and can we get them back in one piece?"

But, as the saying goes &mdash one person's trash is another's treasure. Although many people might call the odds and ends humans have left on the moon "garbage" (what else would you call a used urine-collection assembly?), NASA takes a kinder view.

Researchers can study these objects to see how their materials weathered the radiation and vacuum of space over time, Barry said. Moreover, some of the objects on the moon are still being used, including a laser-range reflector left by the Apollo 11 crew. [What Does the Top of the Moon Look Like?]

Researchers on Earth can ping this reflector with lasers, which allows them to measure the distance between Earth and the moon, according to NASA. These experiments helped scientists realize that the moon is moving away from the Earth at a rate of 1.5 inches (3.8 centimeters) a year, NASA reported.

The so-called trash left on the moon also has archaeological merit, Barry said. Future lunar visitors may want to view the old Apollo sites and see gear from NASA, the European Space Agency, the Russian space agency Roscosmos and other countries, Barry said.

You can find a full list of the abandoned objects on the moon here. However, the list hasn't been updated since 2012, Barry noted, and is missing more recent objects, such as Ebb and Flow, two NASA lunar probes that helped researchers analyze the moon's gravitational field.

## $M = 6.0 imes 10^ mathrm$

Note: I updated this answer to include a description of the historical techniques.

Historical Techniques

Newton developed his theory of gravitation primarily to explain the motions of the bodies that form the solar system. He also realized that while gravity makes the Earth orbit the Sun and the Moon orbit the Earth, it is also responsible for apples falling from trees. Everything attracts everything else, gravitationally. That suggested that one could in theory measure the gravitational attraction between a pair of small spheres. Newton himself realized this, but he didn't think it was very practical. Certainly not two small spheres (Newton 1846):

Whence a sphere of one foot in diameter, and of a like nature to the earth, would attract a small body placed near its surface with a force 20000000 times less than the earth would do if placed near its surface but so small a force could produce no sensible effect. If two such spheres were distant but by 1 of an inch, they would not, even in spaces void of resistance, come together by the force of their mutual attraction in less than a month's time and less spheres will come together at a rate yet slower, namely in the proportion of their diameters.

Nay, whole mountains will not be sufficient to produce any sensible effect. A mountain of an hemispherical figure, three miles high, and six broad, will not, by its attraction, draw the pendulum two minutes out of the true perpendicular : and it is only in the great bodies of the planets that these forces are to be perceived, .

Newton's idea on the impracticality of such tiny measurements would turn out to be incorrect. Little did Newton know that the scientific revolution that he himself helped propel would quickly make such tiny measurements possible.

Weighing the Earth using mountains

The first attempt to "weigh the Earth" was made during the French geodesic mission to Peru by Pierre Bouguer, Charles Marie de La Condamine, and Louis Godin. Their primary mission was to determine the shape of the Earth. Did the Earth have an equatorial bulge, as predicted by Newton? (The French had sent a different team to Lapland to accomplish the same end.) Bouguer used the trip as an opportunity to test Newton's suggestion that a mountain would deflect a plumb bob from surveyed normal. He chose Chimborazo as the subject mountain. Unfortunately, the measurements came up completely wrong. The plumb bob was deflected, but in the wrong direction. Bouguer measured a slight deflection away from the mountain (Beeson, webpage).

The next attempt was the Schiehallion experiment. While surveying the Mason-Dixon line, Charles Mason and Jeremiah Dixon found that occasionally their calibrations just couldn't be made to agree with one another. The cause was that their plumb bobs occasionally deviated from surveyed normal. This discovery led to the Schiehallion experiment conducted by Nevil Maskelyne. Unlike Bouguer, Maskelyne did get a positive result, a deflection of 11.6 arc seconds, and in the right direction. The observed deflections led Maskelyne to conclude that the mean density of the Earth is 4.713 times that of water (von Zittel 1914).

It turns out that Newton's idea of using a mountain is fundamentally flawed. Others tried to repeat these experiments using other mountains. Many measured a negative deflection, as did Bouguer. There's a good reason for this. For the same reason that we only see a small part of an iceberg (the bulk is underwater), we only see a small part of a mountain. The bulk of the mountain is inside the Earth. A huge isolated mountain should make a plumb bob deviate away from the mountain.

Weighing the Earth using small masses

So if using mountains is dubious, what does that say about the dubiousness of using small masses that would take months to approach one another even if separated by mere inches?

This turned out to be a very good idea. Those small masses are controllable and their masses can be measured to a high degree of accuracy. There's no need to wait until they collide. Simply measure the force they exert upon one another.

This idea was the basis for the Cavendish experiment (Cavendish 1798). Cavendish used two small and two large lead spheres. The two small spheres were hung from opposite ends of a horizontal wooden arm. The wooden arm in turn was suspended by a wire. The two large spheres were mounted on a separate device that he could turn to bring a large sphere very close to a small sphere. This close separation resulted in a gravitational force between the small and large spheres, which in turn caused the wire holding the wooden arm to twist. The torsion in the wire acted to counterbalance this gravitational force. Eventually the system settled to an equilibrium state. He measured the torsion by observing the angular deviation of the arm from its untwisted state. He calibrated this torsion by a different set of measurements. Finally, by weighing those lead spheres Cavendish was able to calculate the mean density of the Earth.

Note that Cavendish did not measure the universal gravitational constant G. There is no mention of a gravitational constant in Cavendish's paper. The notion that Cavendish measured G is a bit of historical revisionism. The modern notation of Newton's law of universal gravitation, $F=frac$, simply did not exist in Cavendish's time. It wasn't until 75 years after Cavendish's experiments that Newton's law of universal gravitation was reformulated in terms of the gravitational constant G. Scientists of Newton's and Cavendish's times wrote in terms of proportionalities rather than using a constant of proportionality.

The very intent of Cavendish's experiment was to "weigh" the Earth, and that is exactly what he did.

Modern Techniques

If the Earth was spherical, if there were no other perturbing effects such as gravitational acceleration toward the Moon and Sun, and if Newton's theory of gravitation was correct, the period of a small satellite orbiting the Earth is given by Kepler's third law: $left( frac T <2pi> ight)^2 = frac$ . Here $T$ is the satellite's period, $a$ is the satellite's semi-major axis (orbital radius), $G$ is the universal gravitational constant, and $M_E$ is the mass of the Earth.

From this, it's easy solve for the product $G M_E$ if the period $T$ and the orbital radius $a$ are known: $G M_E = left( frac <2pi>T ight)^2 a^3$. To calculate the mass of the Earth, all one needs to do is divide by $G$. There's a catch, though. If the product is $G M_E$ is known to a high degree of accuracy (and it is), dividing by $G$ will lose a lot of accuracy because the gravitational constant $G$ is only known to four decimal places of accuracy. This lack of knowledge of $G$ inherently plagues any precise measurement of the mass of the Earth.

I put a lot of caveats on this calculation:

• The Earth isn't spherical. The Earth is better modeled as an oblate spheroid. That equatorial bulge perturbs the orbits of satellites (as do deviations from the oblate spheroid model).
• The Earth isn't alone in the universe. Gravitation from the Moon and Sun (and the other planets) perturb the orbits of satellites. So does radiation from the Sun and from the Earth.
• Newton's theory of gravitation is only approximately correct. Einstein's theory of general relativity provides a better model. Deviations between Newton's and Einstein's theories become observable given precise measurements over a long period of time.

These perturbations need to be taken into account, but the basic idea still stands: One can "weigh the Earth" by precisely observing a satellite for a long period of time. What's needed is a satellite specially suited to that purpose. Here it is:

This is LAGEOS-1, launched in 1976. An identical twin, LAGEOS-2, was deployed in 1992. These are extremely simple satellites. They have no sensors, no effectors, no communications equipment, no electronics. They are completely passive satellites. They are just solid brass balls 60 cm in diameter, covered with retroreflectors.

Instead, of having the satellite make measurements, people on the ground aim lasers at the satellites. That the satellites are covered with retroreflectors means some of the laser light that hits a satellite will be reflected back to the source. Precisely timing the delay between the emission and the reception of the reflected light gives a precise measure of the distance to the satellite. Precisely measuring the frequency change between the transmitted signal and the return signal gives a precise measure of the rate at which the distance is changing.

By accumulating these measurements over time, scientists can very precisely determine these satellites orbits, and from that they can "weigh the Earth". The current estimate of the product $G M_E$ is $G M_E=398600.4418 pm 0.0009 ext^3/ ext^2$. (NIMA 2000). That tiny error means this is accurate to 8.6 decimal places. Almost all of the error in the mass of the Earth is going to come from the uncertainty in $G$.

## Shape of the Moon's Orbit

Kepler's first law implies that the Moon's orbit is an ellipse with the Earth at one focus. The distance from from the Earth to the Moon varies by about 13% as the Moon travels in its orbit around us. This variation can be measured with a telescope we will make a series of measurements and combine them to study the Moon's orbit.

The most useful laws of nature can be applied in many different situations. Kepler's three laws, invented to describe the orbital motion of planets about the Sun, are very useful: with minor modifications, they also describe the Moon's motion about the Earth, the orbits of Jupiter's satellites, and even the orbital motions of binary stars. The Moon provides a natural laboratory for orbital motion we can use it to make a simple test of Kepler's first law.

Kepler's three laws of planetary motion are:

A planet travels around the Sun in an elliptical orbit with the Sun at one focus.

In effect, the first law describes the shape of a planet's orbit, the second says how a planet's speed varies at each point on its orbit, and the third law provides a way to compare different orbits.

These same three laws can also describe the Moon's orbital motion around the Earth: just substitute Earth for Sun and Moon for planet. (Of course, the Earth has only one Moon, but we could use the third law to compare the Moon's orbit with the orbit of the Space Station or other artificial satellite.)

### THE MOON'S ORBIT

Kepler's first law says that planets have elliptical orbits. As a result, the distance between a planet and the Sun changes rhythmically as the planet moves in its orbit. In many cases, this rhythmic change is rather subtle for example, the Earth's distance from the Sun varies between 98.3% and 101.7% of its average value. (By the way, the Sun is closest in January, and furthest in July, so this change doesn't explain the seasons!) In contrast, the ellipticity of the Moon's orbit is fairly dramatic the Moon's distance from the Earth varies between 92.7% and 105.8% of its average value of 384,400 km.

This variation in distance produces several effects which we can observe here on Earth. For example, when the Moon is closest to the Earth (perigee), it moves faster, while when it is furthest from the Earth (apogee), it moves slower. The Moon also appears to nod back and forth a bit as it orbits the Earth. But the most dramatic effect is the change in the Moon's apparent diameter: when the Moon is close, it looks larger, and when the Moon is far, it looks smaller. We will use this effect to study the change in the Moon's distance.

### OBSERVATIONS

To measure the Moon's apparent diameter, we use a 25 mm eyepiece equipped with a measuring scale. Looking through this eyepiece, you can see the scale, which is something like a ruler, superimposed on the Moon's image. The basic idea is to point the telescope at the Moon, align it so the scale goes right across the Moon at its widest point, and measure the Moon's diameter in the units on the scale.

Fig. 1. Measurement of Moon's apparent diameter on 02/20/03 at 06:55 HT (16:55 UT). At this time, the image of the Moon's disk was 5.8 mm + 5.7 mm = 11.5 mm in diameter.

Fig. 1 shows how the measurement is made. Notice that this scale, unlike a ruler, has its zero point in the middle. So to determine the diameter of the Moon's image, you measure from the midpoint to each side of the Moon's disk, and add these two values to get the total. The scale is calibrated in millimeters, so your result should be expressed in millimeters. Also, notice that the eyepiece has been rotated so the scale crosses the disk of the Moon at widest point. If the scale had been rotated any other way, the measured diameter would have been less than the true value. It's always possible to turn the scale to span the Moon's true diameter, no matter what the Moon's phase for example, the diameter of a crescent Moon is measured from horn'' to horn''.

The most efficient procedure is to use the Earth's rotation to slowly move the scale across the face of the Moon. First, rotate the eyepiece in the holder until the scale is parallel with the widest part of the image (if the eyepiece doesn't rotate easily, loosen the screw holding it in place). Second, point the telescope a little to the west of the Moon - you can easily tell which is west since that's the direction the Moon appears to move as a result of the Earth's rotation. Try to place the dividing line somewhere in the middle of the Moon's disk, but don't worry about centering it exactly. Third, wait while the Moon's image drifts past the scale, and make a measurement when the widest part of the image falls on top of the scale. Record the distances from the dividing line to the two sides of the Moon's disk separately then add them and record the total.

Repeat these steps at least three times, making three sets of measurements! This includes the initial step of rotating the eyepiece in the holder. Repeated measurements yield better accuracy they also give you a fighting chance of spotting any errors you may have made.

Weather permitting, we will make measurements each time the Moon is visible this semester.

The three measurements you've made each night give you three independent (and probably different) values for the total diameter of the Moon's image. Don't worry if these values differ by 0.1 or 0.2 mm or so that's normal measurement uncertainty. But if one value is very different from the other two, you probably made some kind of mistake while taking that measurement. You should drop any obviously incorrect measurements before going on to analyze your observations.

For example, suppose you made three measurements, and found total diameters of 11.0 mm, 11.1 mm, and 11.2 mm. These values are all pretty close to one another, and you can average them to get 11.1 mm. On the other hand, suppose you found diameters of 10.1 mm, 11.0 mm, and 11.2 mm while two of these values are reasonably close together, the other is very different. In this case, it's likely that the 10.1 mm value is incorrect, while the others are reliable and can be averaged to get 11.1 mm.

For each night, average all the values you think are reliable the result is your best measurement of the diameter of the Moon's image that night. Call that average value d. Now to calculate the Moon's distance, use this equation:

Here F is the focal length of the telescope's main mirror, which is F = 1200 mm. Because d and F both have units of millimeters, D is a pure number -- the units of d and F cancel out. In fact, D is the Moon's distance in units of the Moon's actual diameter.

An example may help make this clear. In Fig. 1, the Moon's image is d = 11.5 mm across. Using this value in the equation, we get D = 104.3 for the Moon's distance, in units of the Moon's diameter. To express the Moon's distance in units of, say, kilometers, you can multiply D by the Moon's actual diameter in kilometers (3,476 km) the result is about 363,000 km, which is a reasonable distance for the Moon when it's near perigee. But for this assignment, the Moon's diameter provides a perfectly good yardstick, so there's no need to go through the final step of expressing the distance in kilometers.

Once you've calculated D for each night, you should make a plot showing how the Moon's distance varies with time, using the blank graph we'll hand out in class. Unfortunately, the data points you'll have won't look like a smooth curve there's too much time between measurements, and your graph won't include the half of each month when the Moon rises late at night. So we will take photographs of the Moon at other times which you can measure in class. With these additional measurements, your graph should show a smooth variation in the Moon's distance with time.

To actually plot the Moon's orbit as an ellipse we would need more information. It's not enough to know how far away the Moon is we also need to know the direction from the Earth to the Moon.

### WEB RESOURCES

Use these worksheets to record and organize your data.

Use this chart to make a graph of D over time.

Web page describing the variation in the Moon's apparent size as a result of its elliptical orbit. Created by John Walker.

JavaScript program to calculate dates of lunar perigee and apogee. Created by John Walker.

Animation showing the Moon as seen from the Earth from 07/31/05 at 14:00 HT to 12/31/05 at 08:00 HT (08/01/05 at 00:00 UT to 12/31/03 at 18:00 UT). Note the rhythmic variation in the Moon's apparent diameter and the wobbling'' motion known as libration. Generated using Solar System Simulator (Courtesy NASA/JPL-Caltech).

## How can the weight of Earth be determined?

There is an easy answer to this question, and it involves some good news and some bad news. Imagine this scenario: step on the scale in your bathroom and weigh yourself. (Hopefully this isn't bad news.) Now suppose you take the scale, travel to the moon, and stand on it again. What will you weigh there? The new number will be about 1/6 of what you weighed on Earth. Finally, imagine traveling out into deep space and weighing yourself once more. You will weigh nothing. (This is the good news.) Your weight is variable because weight is a force that depends on something pulling on you. Specifically, it is the force of gravity, which depends on the mass of the object that is attracting you. If you pushed Earth out into deep space, it, too, would weigh nothing. (So the bad news is that you would weigh as much as Earth.)

That said, I think that this reader was really asking how the mass of Earth can be determined. This is a bit more complicated, though only slightly so. What is more, unlike the previous question, it does have a straightforward answer, and you can still get the answer using your bathroom scale. As is often the case in physics, fairly complicated things can be described very well with a simple equation. In the case of gravitational attraction, the equation is as follows:

The value G is called the gravitational constant, and it has been ascertained through many decades of careful experiments. We know how far a person standing on the surface is from the planets center (about 6,371 kilometers), so all we need to know is his mass, and then we can calculate Earth's mass. Finding a persons mass will only involve counting all of the atoms in his body. (But dont forget to take into account that burrito he ate last night, too.) This is quickly turning out to be a complicated problem. As an alternative, perhaps we could use a block of some material for which it is easier to make an estimate of the number of atoms it contained. Maybe, but this is not the sort of experiment you can carry out in your bathroom.

Fortunately, a bathroom scale can still aid in solving this problem (albeit in an unexpected way) by using a simpler equation. It turns out that the rate at which an object accelerates due to the force of gravity, called "g," depends of the mass of the object doing the pulling. In the case of Earth, we have:

So, you can open your bathroom window, hurl your scale out the window, and count how many seconds it takes to hit the sidewalk. Then measure the distance from your window to the ground, and you can compute the acceleration of the scale. The answer you will get is 9.8 m s -2 . Knowing this value of g for Earth's surface, along with the constant G and the 6,731-kilometer distance to Earth's center, you can then calculate Earth's mass to be 6 x 10 24 kilograms. (You also won't be bothered by bad news from your scale anymore.)

## Scale Model of theEarth, Moon, Mars, Phobos, Deimos

Humans have known about the existence of our Moon since the time that we first looked towards the heavens. It was unique among the objects in our night sky until the arrival of the telescope in the early 1600's. It was with the telescope that Galileo first discovered moons orbiting the planet Jupiter. The four moons he discovered in 1610 were Io, Europa, Ganymede and Callisto. The moons of Mars survived undetected for almost 300 years following the invention of the telescope.

It wasn't until 1877 that we discovered that Mars had two moons of its own. The discovery of these moons was made by American astronomer Asaph Hall at the US Naval Observatory. He had at his disposal one of the best telescopes of the day and was observing Mars during a favorable opposition, a time when Mars is much closer to the Earth than is usually the case. The two moons were named for two sons of Ares, the God of War: Phobos (Greek for fear) and Deimos (Greek for terror).

There are two reasons why it took so long for us to discover the moons that orbit Mars. The first reason is that these moon are very small. Our Moon has a diameter of 3,474 kilometers. Phobos is so small that gravity isn't large enough to mold the moon into a sphere. Rather, it is shaped more like a potato about 26 kilometers long by about 22 kilometers across. Deimos, the smaller of the two, is only about 15 kilometers long by 12 kilometers across.

The second reason they were so hard to discover is because they orbit so close to Mars. While our Moon orbits at a healthy distance of 384,000 kilometers from the Earth, Phobos is only 9,378 kilometers distant from Mars and Deimos is a little farther out at 23,459 kilometers. This proximity means that when an astronomer looks through a telescope at Mars, the light being reflected by Mars overpowers the light being reflected by Phobos and Deimos.

To give you an idea of the relative scale of the Earth-Moon and Mars-Phobos-Deimos systems, take a look at the graphic to the left of this text. At the top you can see the planets Mars (left) and Earth (right) drawn to the same scale. Immediately below Mars is a dot labeled Phobos. While the pixel you see makes Phobos larger than it really is, it is drawn at the correct distance from Mars. A little further down, you can see a pixel identified as Deimos. Again, the one pixel size makes Deimos appear larger than it really is but accurately shows you the distance of Deimos from Mars.

Depending on the font size you are using and the width of your browser, you may or may not have already passed the Moon. The size and distance of the Moon as drawn are both to the correct scale. As you can see, someone observing the Earth from Mars would have little difficulty in spotting our Moon, both because of its large size and because of its greater distance from the Earth.

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