# Should the synodic period or the orbital period be used to determine the diameter of the Moon?

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Aristarchus famously used a lunar eclipse to determine the diameter and the distance of the Moon, given that a lunar eclipse can last up to 3 hours and the angular diameter of the Moon is 0.5°.

Using this method, the diameter of the Moon is

$$frac{( extrm{earth diameter}) imes ( extrm{moon angular diameter})}{frac{ extrm{duration eclipse}}{ extrm{orbital period of the Moon}} imes 360}.$$

Is it correct to use the orbital period here, rather than the synodic period? Should the orbital period or the synodic period of the Moon or some other period be used in this expression, and why?

What you need, as Aristarchus, is a frame of reference in which the Earth's shadow is stationary. Otherwise you are trying to deal with a moving thing (the Moon) relative to another moving thing (the shadow cast by the light of the Sun as it orbits the Earth).

In a stationary-shadow frame of reference, the period you need is the time it takes the Moon to orbit from the middle of the Earth's shadow to the middle of the Earth's shadow. That is to say, from new moon to new moon: which is to say, the synodic month.

## Measuring Titan's Orbital Period

Just got a new telescope and one of my first projects was to measure Titan’s orbital period around Saturn. Titan is Saturn’s brightest moon and should be visible in most small telescopes. All of my measurements were done by sight and estimation of Titan’s distance compared to the horizontal diameter of Saturn. Obviously accuracy could be improved with some actual measurement tools or the use of a camera. Unfortunately it was difficult to get a long string of clear nights, so there is one noticeable 4 day gap in data. I also did not wait for the moon to make it all the way back around to the starting position, I figured I had taken enough data to get a decent fit.

Each night I would observe at roughly the same time, and make a drawing in my notebook. After observing every clear night for 11 days I was able to create the above simulated image (drawn in Photoshop) showing Titan’s change of position. I then measured the coordinates for each night’s measurement and plotted the results in a graph. To make the math a bit easier I assumed that Titan’s orbit was circular, thus the position of the moon along the X axis (the plane of orbit) would vary sinusoidally with time (see Wikipedia’s article on Simple Harmonic Motion). Using scipy and matplotlib I fitted the data to the equation x(t) = A cos(ωt + φ), producing the plot below. The result for the fit indicated a period of 380.4 HRS, off slightly from the known value of 382.68 HRS. Still, not bad for just rough measurements!

## Moon Phases Simplified

It's probably easiest to understand the moon cycle in this order: new moon and full moon, first quarter and third quarter, and the phases in between.

As shown in the above diagram, the new moon occurs when the moon is positioned between the earth and sun. The three objects are in approximate alignment (why "approximate" is explained below). The entire illuminated portion of the moon is on the back side of the moon, the half that we cannot see.

At a full moon, the earth, moon, and sun are in approximate alignment, just as the new moon, but the moon is on the opposite side of the earth, so the entire sunlit part of the moon is facing us. The shadowed portion is entirely hidden from view.

The first quarter and third quarter moons (both often called a "half moon"), happen when the moon is at a 90 degree angle with respect to the earth and sun. So we are seeing exactly half of the moon illuminated and half in shadow.

Once you understand those four key moon phases, the phases between should be fairly easy to visualize, as the illuminated portion gradually transitions between them.

An easy way to remember and understand those "between" lunar phase names is by breaking out and defining 4 words: crescent, gibbous, waxing, and waning. The word crescent refers to the phases where the moon is less than half illuminated. The word gibbous refers to phases where the moon is more than half illuminated. Waxing essentially means "growing" or expanding in illumination, and waning means "shrinking" or decreasing in illumination.

Thus you can simply combine the two words to create the phase name, as follows:

After the new moon, the sunlit portion is increasing, but less than half, so it is waxing crescent. After the first quarter, the sunlit portion is still increasing, but now it is more than half, so it is waxing gibbous. After the full moon (maximum illumination), the light continually decreases. So the waning gibbous phase occurs next. Following the third quarter is the waning crescent, which wanes until the light is completely gone -- a new moon.

## About Lunar and Solar Eclipses — the Saros and Exeligmos cycles

“The saros is a period of approximately 223 synodic months (approximately 6585.3211 days, or 18 years, 11 days, 8 hours), that can be used to predict eclipses of the Sun and Moon.”

Note that the Saros cycle of 6585.3211 days is nearly equal to 16 Full Moon cycles of 411.78433 days:

6585.3211 / 16 ≈ 411.5825 days

Now, the 18-year Saros cycle is just part of a longer more complete Triple Saros cycle of 19,756 days. This is known as the “Exeligmos” (Wikipedia link).

“The Mesopotamians, and in particular the Babylonians, were one of the first civilisations to keep records of their astronomical observations. Because of this, they were also the first to notice a remarkable pattern: that eclipses of a particular type are repeated every 18 years, and more closely repeated every 54 years. The 18 year period became known as the Saros, and the 54 year one as the Triple Saros or Exeligmos.”

On the Saros by Kevin Clarke (1999), InconstantMoon.com

As a 54-year Exeligmos is completed, our Moon returns to its start position, which means that a lunar or solar eclipse will recur over almost the same geographic region as it did 54 years earlier. It is highly important to note that at the completion of one Exeligmos the eclipse will return to a place positioned 90 minutes earlier in our celestial sphere.

1440 minutes / 90 minutes = 16

That is, the Moon will gain exactly 1/16th of a full revolution of right ascension (RA).

When we consider Earth’s 1 mph motion, it gets really interesting. The distance covered by Earth and the Moon, in unison, in the course of a 54-year Exeligmos cycle, turns out to be very close to the orbital diameter of our Moon (ca. 763,000 km).

19756 days X 38.43 km = 759,223.08 km

This is just about 3800 km shorter than the Moon’s orbital diameter. However, one may reasonably consider that this discrepancy may be accounted for by the diameter of the Moon itself (3476 km).

It would seem intuitively logical that an Exeligmos cycle will be completed when Earth and the Moon have together covered a distance almost equal to the Moon’s orbital diameter.

The Triple Saros cycle or “Exeligmos” comprises ca. 19,756 days, into which one can find nearly 48 Full Moon cycle lengths.

19,756 / 411.78433 days (a Full Moon cycle) ≈ 48

So 16 shows up again (3 X 16 = 48) telling us just how much the unsung “16 factor” pervades the arithmetic regulating our system’s celestial bodies. Add to this our realization that orbits share resonance at various multiples of the Moon’s TMSP period of 29.22 days. The Moon appears to be in every aspect a body of central importance to our binary solar system.

Let us briefly recap the ubiquitous appearances of the “16 factor”.

• As Mars completes one of its orbits, it processes by 1/16th of a solar year (22.828 days).

• Every 32 years (2 X 16), Mars very nearly conjuncts with all of our system’s celestial bodies.

• Mercury regularly retrogrades for an average period of 1/16th of a solar year (22.8 days).

• The Sun’s orbital speed (107.226 kmh) is extremely close to 16 X its rotational speed (6670 kmh).

• Our Moon’s Saros and Exegilmos cycles appear to be multiples of 16 Full Moon cycles.

• As it completes one Exegilmos cycle our Moon gains 1/16th of a full 1440 minutes of RA.

## I need help finding the orbital period of the moon.

The angular size of the Moon, as seen from Earth, is roughly half a degree.

This can be measured with instruments like a sextant (since roughly the 16th century) and it was calculated well before that by covering the Moon with standard-sized objects (e.g., coins) then measuring the distance form the coin to the eye.

The average size could be calculated using the diameter of the Moon and the average distance:

3475 km / 384,400 km = 0.00904 radians = 0.518 deg. = 31'

By observation, you can tell that the Moon moves (relative to the fixed stars) by its own apparent diameter in one hour. To find out how many hours it takes for an orbit, you have to divide a full orbit (360 degrees) by the rate per hour (0.518 deg. per hour) to get:

360 / 0.518 = 695.0378 hours = 695 h 02 m 16 s = 28 days 23 h 02 m 16 s

Of course, this is just an approximation (the real average time it takes for the Moon to move by its own diameter changes with where the Moon is in its orbit: when it is closer to Earth, it goes faster AND it has a larger apparent diameter.

The real average sidereal month is 27 d 07 h 43 m 12 s
You could get this accuracy by measuring the average time for "one diameter" over very long periods. You would discover that the average time it takes for the Moon to move by its average apparent diameter - relative to fixed stars - is actually 56 m 28 s, not one hour.

This is how ancient astronomers did it. No instrument needed, but you would have needed lots of observations over very long periods (years) to get this kind of accuracy.

If you want to measure the synodic month (cycle of Moon phases) you have to do it relative to the Sun. One way is to measure the average time between transits of the Moon (transit = Moon passing over your meridian).

## Mapping the Orbit and Phases of the Moon

Summary: Track the phases of the Moon, observe it's orbit around the Earth, measure its orbital period, and measure the angle between the plane of the moon's orbit and the ecliptic plane.

Needed Supplies: Observing log, pencil, star map SC001, crossbow, phase of the Moon template.

Start Date: You will have the greatest success with this lab if you follow the moon through 5-6 weeks. We strongly recommend that you start no later than January 20.

### General Description:

Look up! Observe the moon. Hints: if the moon is near 1st quarter or full moon phase, you will be able to make your observations in the early or late evening. If the moon is past full moon or near 3rd quarter, you will have to make your observations either very late at night or in the early morning. This lab requires only simple observations that take only about 10-15 minutes each night but reveals a surprising number of effects. To do this lab, you need to become familiar with the constellations and the use of star maps, a skill also developed in the Constellations and Bright Stars lab.

Frequency of observations: The Moon takes about 1 month to orbit the Earth. The goal is to observe the Moon during one full cycle. You may have to extend this by 1-2 weeks if bad weather prevented you from getting good coverage. You will need 8-12 well-spaced observations during the cycle (that's observing every 3-4 days or almost on every clear night), to get good coverage. Also, for the period after the full Moon, the Moon is visible only late at night or early in the morning. A bit of effort will be necessary to observe the later part of the cycle. It really isn't as much work as it may sound. Remember that each observation takes no more than 15 minutes! This is one of the most interesting labs in Introductory Astronomy.

Curious about why the Full Moon appears so large when it is rising? This is the famous Moon illusion, a well understood phenomenon.

### Procedure

Go out and determine the position of the Moon with respect to the stars and constellations you can identify. Do this as accurately as you can. You can do this by careful "eyeballing" (to the nearest 1 o or better) of the position if there are stars visible near the Moon or better still, using the crossbow to measure its angular separation from two stars. You should use the crossbow when observing during scheduled labs and "eyeball" when observing on your own time.

When using the crossbow: You need to choose 2 stars that are good, fixed references for measuring the position of the Moon on that particular night. Measure the separation 1) between the Moon and each of the two stars and 2) between the two stars (or another pair of stars in that same area of the sky). This latter measurement will allow you to figure out the scale of the star chart (i.e. how many mm correspond to 1 o ). You need to know this to plot the position of the Moon from your measurements.

Note: Star charts are like geographical maps. They are obtained by projecting a curved surface (a portion of a sphere) on a flat surface (sheet of paper). This inevitably leads to distortions, where the scale is not uniform across the entire map. This is why on many maps of the Earth (planispheres) Greenland appears much larger than South America, while in fact it is much smaller, as can be verified with a globe. For this reason, you should not assume that the same scale applies everywhere on the SC001 star map and you need to determine the scale near the observed position of the Moon for each crossbow observation. As you will see, this requires little additional work.

When "eyeballing" the position: It is difficult to do this when the Moon is rising or setting (i.e. low in the sky) as fewer stars are visible low in the sky and all the stars are on the same side of the Moon (the others being blocked by the horizon). As much as possible, make the observation when the Moon is well above the horizon.

Plot this position on your star chart (SC001) by sketching a small moon in its approximate phase at the correct location relative to the stars. If you used the crossbow, check your work by comparing your plotted position with what you see in the sky. Label the position with the date. See the binder of examples kept in the storage shed.

Using the templates provided for the phases of the Moon, draw the phase of the Moon precisely as well as the surface features you can see with your naked eye. Indicate the date and time next to your sketch. See the binder of examples in the storage shed.

In your observing log, record the date and time, weather conditions, phase of the moon, and the constellation in which the Moon appears. If you used the crossbow, record the names of your reference stars and all measurements. Add notes (including personal comments) as appropriate.

Outside of lab, use the star chart (SC001) to find the position of the Moon and of the Sun along the ecliptic (both in degrees) on the date of your observation. This is known as the ecliptic longitude. This is done by simply reading off the degree scale along the ecliptic (wavy curve on the map). The position of the Sun in the sky throughout the year is also shown along that curve. Example: Suppose that you observe the Moon very near the star Spica (in Virgo) on November 21. The Moon's ecliptic longitude would be 204 o and the Sun's would be 238 o .

Make a table of your observations. For each entry, give the date and time, the ecliptic longitude of the Moon and Sun and the angle between the Moon and the Sun in the sky, obtained from the values you got in 5). In the above example, the Sun-Earth-Moon angle would be (204 o -238 o ) + 360 o =326 o . This angle is 0 o at the new Moon (Sun and Moon are lined up in the sky), 180 o at the Full Moon and grows during the cycle to 360 o .

Once you have completed your observations, use your table of Sun-Earth-Moon angles, your plotted positions on the star chart, and your sketches of the phase of the Moon during the cycle to answer the following questions (based on your observations!):

a) Write a brief summary describing the changes you observed in the phase of the moon. Your summary should include the times of the day (or night) when you would now expect to observe a first quarter moon, full moon, last quarter moon, and new moon.

b) Why does the moon go through phases? How does the phase relate to the Sun-Earth-Moon angle?

c) Do your observations indicate that we are indeed always seeing the same face of the Moon at all times? What does this mean?

d) What is the period for the completion of one cycle of lunar phases, as determined from your measurements? There are some subtleties here. To find when one cycle is completed, you need some point of reference (i.e. where is the statring line of the lap?). There are two fairly obvious choices: 1) the Sun and 2) the stars. The synodic period of the Moon (cycle of phases) is measured with respect to the Sun. The synodic cycle is completed when the Sun-Earth-Moon angle has changed by 360 o . The sidereal period is measured with respect to the stars and is completed when the Moon has returned to the same ecliptic longitude. Determine each period based on your tabulated angles. You can do this by making a plot of the angle measured vs. the time in days, drawing a smooth curve through your data and seeing where it crosses the point of completion of the cycle. Make a separate graph for each of the synodic and sidereal periods and determine the period to the nearest 0.1 day. Are the synodic and sidereal periods different? Why or why not? Finally, compare your values with those given in your textbook and comment.

e) Describe the orbital path of the moon through the stars. What constellations does it pass through? How does the path of the Moon in the sky compare with that of the Sun (i.e. the ecliptic)? If you find a deviation between the ecliptic and the path of the Moon, what does it reveal?

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## Lunar cycle and the orbital period

A lunar cycle takes 29.5 days, while a full rotation of the Moon around Earth takes 27.3 days. Common sense would state that these two numbers should be the same. They are not because the Moon has to catch up to its starting position, which has changed. This change was a result of the Earth rotating around the sun.

You may have personally observed that it takes the moon one month to go through a Moon phase cycle. It is not quite a month from new Moon to new Moon, but rather 29.5305882 days. This is called a synodic orbital period, and is the time it takes as we observe from Earth for the Moon to move back to the same position.

If you were to view the same Moon cycle from some point inside our solar system above Earth, then the time it would take for a full cycle would be 27.3217 days, roughly two days less. This is called a sidereal orbital period.

So why are these two different? Well, from Earth, we are viewing the Moon from a moving platform. We are on Earth and Earth is moving. This alters our angle of view with respect to the Moon, and as a result alters the phase. The Earth's orbital direction is such that it lengthens the period for earthbound observers.

Although the synodic and sidereal periods are exact numbers, the Moon phase can't be precisely calculated by simple division of days because the motion of the Moon is affected and perturbed by various forces of different strengths. In this case, the Moon’s motion is referred to as its orbital speed and position. Hence, complex equations are used to determine the exact position and phase of the Moon at any given point in time. As a simple rule when determining rotational speed however, it can be assumed that the Moon rotates at 10 miles per hour, compared to the Earth, which rotates at about 1000 miles per hour.

As a final note, the lunar cycle is incredibly predictable. However, on a scale beyond our lifetime, it is changing very slowly. Every year, the Moon moves an addition 3.8 centimeters further away. This is happening because the Moon is “stealing” some of Earth’s rotational energy, using it to propel itself further and further from Earth’s gravitational well. It is hypothesized that shortly after its formation, the Moon was only 14,000 miles away from Earth. Currently, the Moon is 280,000 miles away. Eventually, the Moon will escape from Earth’s gravitational field entirely.

1. Launch the Stellarium program: Your default location should be Conway, and the default time should be right now. Because the program uses the computer's clock, your time will always be local to Conway&minusdon't be surprised when you change locations and the sun rises and sets at very odd times!
2. Set your location: In the search box of the location window, type in "Pisa," and select Pisa, Italy as your location. However, do not set this as your default location, since you can return to it quickly.

## Moon Phases Simplified

It's probably easiest to understand the moon cycle in this order: new moon and full moon, first quarter and third quarter, and the phases in between.

As shown in the above diagram, the new moon occurs when the moon is positioned between the earth and sun. The three objects are in approximate alignment (why "approximate" is explained below). The entire illuminated portion of the moon is on the back side of the moon, the half that we cannot see.

At a full moon, the earth, moon, and sun are in approximate alignment, just as the new moon, but the moon is on the opposite side of the earth, so the entire sunlit part of the moon is facing us. The shadowed portion is entirely hidden from view.

The first quarter and third quarter moons (both often called a "half moon"), happen when the moon is at a 90 degree angle with respect to the earth and sun. So we are seeing exactly half of the moon illuminated and half in shadow.

Once you understand those four key moon phases, the phases between should be fairly easy to visualize, as the illuminated portion gradually transitions between them.

An easy way to remember and understand those "between" lunar phase names is by breaking out and defining 4 words: crescent, gibbous, waxing, and waning. The word crescent refers to the phases where the moon is less that half illuminated. The word gibbousrefers to phases where the moon is more than half illuminated. Waxing essentially means "growing" or expanding in illumination, andwaning means "shrinking" or decreasing in illumination.

Thus you can simply combine the two words to create the phase name, as follows:

After the new moon, the sunlit portion is increasing, but less than half, so it is waxing crescent. After the first quarter, the sunlit portion is still increasing, but now it is more than half, so it is waxing gibbous. After the full moon (maximum illumination), the light continually decreases. So the waning gibbous phase occurs next. Following the third quarter is the waning crescent, which wanes until the light is completely gone -- a new moon.

## Calculate the Mass of the Earth

Mass is a measure of how much matter, or material, an object is made of. Weight is a measurement of how the gravity of a body pulls on an object. Your mass is the same everywhere, but your weight would be vastly different on the Earth compared to on Jupiter or the Moon.

G, the gravitational constant (also called the universal gravitation constant), is equal to

Where a Newton, N, is a unit of force and equal to 1 kg*m/s 2 . This is used to calculate the force of gravity between two bodies. It can be used to calculate the mass of either one of the bodies if the forces are known, or can use used to calculate speeds or distances of orbits.

Orbits, like that of the moon, have what is called a calendar period, which is a round number for simplicity. An example of this would be the Earth has an orbital period of 365 days around the sun. The sidereal period is a number used by astronomers to give a more accurate description of time. The sidereal time of one spin of the Earth is 23 hours and 56 minutes, rather than a round 24 hours. The time period of an orbit, which you will use in your calculations in this exercise, will have a great effect on the outcome of your answers.

### Procedure

1. Use a calendar to determine how long it takes for the moon to orbit the Earth. Do some research on the internet to find the sidereal period of the moon.
2. Use the following equation to calculate the average velocity of the Moon

Where v is the average velocity of the moon,

r is the average distance between the moon and the Earth, taken as 3.844 x 10 8 m,

and T is the orbital period, with units of seconds.

1. Calculate the mass of the Earth using both the calendar period of the moon and the sidereal period of the moon. Why are they different? Which is a more accurate calculation and why?

Where Me is the mass of the Earth, in kilograms,

v is the average velocity of the moon,

r is the average distance between the moon and the Earth

and G is the universal gravitation constant.

The sidereal period of the moon, which is 27.3 days, will give you a calculation of Earth's mass that's more accurate than the calendar period of the moon. The mass of the Earth is 5.97 x 10 24 kg.

That is 5,973,600,000,000,000,000,000,000 kg!

Sir Isaac Newton&rsquos Law of Universal Gravitation states that all masses in universe are attracted to each other in a way that is directly proportional to their masses. The universal gravitation constant gives the relation between the two masses and the distance between them. For most things, the masses are so small that the force of attracted is also very small. This is why you don't get pulled by your friends' gravity enough to get stuck to them!

These gravitational forces are extremely useful, as they keep the plants in orbit around the Sun, and the Moon in orbit around the Earth. They also keep the satellites in orbit that bring us information from space and allow us to communicate with people across the world instantaneously.

For further projects, you can use the same ideas to calculate the mass of the Sun, the center of our solar system, using information for any of the planets or other objects that consistently orbit the Sun (such as the planetoid Pluto).

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