Astronomy 10: Lecture 3
Reading Assignment: Arny Essay 1 pg. 57, Essay 2 pg. 177
When the Moon is eclipsed it is visible from everywhere on Earth. It lasts for many hours as the Moon first moves into the Penumbra of Earth's shadow and then finally into the Umbra. It often turns a deep red color because some light is still reaching it. This light is light being lensed by Earth's atmosphere and is red for the same reason that sunsets are red.
Distance and Size of the Moon
Some 75 years before Eratosthenes made his famous measurement of the size of Earth, another great Greek thinker had figured out a way to measure the relative sizes and distances of the Moon and Sun. His name was Aristarchus of Samos.
These calculations again involve some straightforward logic and the application of simple geometry.
The first thing that Aristarchus noticed was that when the Moon was eclipsed by the Earth you could see the Earth's shadow creep across the face of the Moon. Earth's shadow is circular, and if you assume that the Sun's rays are roughly parallel to one another that implies that the size of Earth's shadow at the distance of the Moon is equal to the size of Earth. It's not exactly true, but it's very close. From the arc of Earth's shadow you can estimate the size of it's full circle, then you measure the size of the Moon's circle. Now you know the relative size of the Earth to the Moon. What you are measuring is the angular size of the Earth's shadow and the Moon. In the limit of small angles, physical size and angular size are directly proportional.
Aristarchus measured a ratio of Earth's size to the Moon's size to be:
D/DM = 2.86
(Where the symbol refers to Earth)
The real answer is 3.67. So the Moon is about one-third the size of Earth. Using the angular size of the Moon we can estimate its distance as well using the relationship between angular size (), physical size (D), and distance (d):
Where in this equation is in radians, the natural, unit-less measure of angles. The conversion from radians to degrees is simply:
2 radians = 360°
-> 1 radian = 57.3°
The angular size of the Moon is about 0.5° ( = 0.5°). Thus we have a distance to the Moon of about 30 Earth Diameters (dM = 30D)
Distance and Size of the Sun
Next Aristarchus reasoned that when the Moon was seen to be exactly in either 1st or 3rd Quarter Phase (exactly half lit up as seen from Earth), we can draw a right triangle between Earth, Moon, and Sun. The legs of the right triangle are the distance from the Earth to the Moon (dM) and the distance between the Moon and the Sun (dM). The hypotenuse is then the distance between the Earth and Sun (d). (The symbol refers to the Sun). We can measure the angle between dM and d (), it's the angular separation between the Sun and Moon in the sky. Once again a little simple trigonometry then gives us the distance to the Sun.
Aristarchus measured = 87°. This corresponds to a distance to the Sun of d = 573D. Now, the angular size of the Sun is also about 0.5° so by the same method as for the Moon Aristarchus estimated that the Sun was 5.33 times bigger than the Earth.
This measurement is a difficult one to make, and the real answer is = 89.9°. This corresponds to a distance to the Sun of d = 11,700D and a size of the Sun, D = 109D.
Either way, Aristarchus was able to establish that the Sun was bigger than Earth. Since this was so, he reasoned that it would make more sense for Earth to orbit the Sun rather than the other way around. But his contemporaries pointed to some arguments of Aristotle's that contradicted such an idea. One critical one being that the stars should exhibit parallax if Earth goes about the Sun. They do indeed exhibit this phenomenon but they are so far away that it cannot be measured without a telescope. It would be some 1900 years before Aristarchus would finally be proved correct by the work of Kepler and Galileo.
Navigation by the Stars
As we briefly discussed before, one can use the Sun and stars to determine your location on Earth.
More about Latitude
- Magnetic Compass: As I mentioned before the Earth has a magnetic field produced by the motion of its liquid iron core. The poles of this magnetic field are currently roughly aligned with the geographical poles of the planet. So you can use a compass to get a rough idea of where North is.
- Polaris (the North Star): Better still would be to locate the North Star (Polaris). As we discussed it lies very near the North Celestial Pole and therefore remains motionless in the sky. Recall that if you are viewing Polaris from the North Pole it is at the Zenith, and if you are viewing it from the Equator it appears due North on the horizon. Everywhere between these locations it is at some angle (altitude) above due North on the horizon (azimuth = 0°). The angle above the horizon of the North Star is equal to your Latitude on Earth. From Berkeley, the North Star is an angle of 37.9° above the horizon, and Berkeley is at a Latitude of 37.9° North.
In the Southern Hemisphere there is no bright star that is near the South Celestial Equator. So it's a bit trickier, but still all you need to do it locate the South Celestial Equator (using star trails is one method) and measure its angle above due South on the horizon and that is equal to your Latitude South of the Equator. You could also determine your Latitude by measuring the angular height of a star or the Sun at transit. If you have a star chart that gives the Celestial coordinates of that Star or the Sun at that time of the year you can just do some simple arithmetic to determine your Latitude.
Determining Longitude is much more difficult. You need a precise and accurate watch or clock in order to do this correctly. There are two best ways to do this.
If you wanted to use stars for Longitude determination you would go about it in the same as with the Sun. But now you need to use a Sidereal Clock. Recall that the Sidereal Day is 23 hours 56 minutes.
- Compare the Local Solar Time to the time on a watch that reads the Mean Solar Time of a known longitude. For example: Say your watch is calibrated so that when it reads 12:00 noon the Sun is transiting as seen from a longitude of 0° (the prime meridian). If your watch is reading 24 hour as opposed to 12 hour time (i.e., 1:00 PM = 13:00) then your watch is reading what is called Universal Time. Now, your watch reads 12:00, you know that at 0° Longitude the Sun is at the meridian. But in your sky the Sun is still 2 hours away from transit (your local solar time is 10:00). The Sun is East of the meridian, so you are West of 0° Longitude. Earth rotates 360° in 24 hours and likewise the Sun completes a full circuit in 24 hours (definition). So the Sun moves at a rate of 15°/hour. If the Sun is 2 hours from transit then it is 30° from the meridian (in the plane of its path through the sky that day). You are at a Longitude of 30° West of Greenwich.
- Similarly you could wait for the Sun to transit in your sky (local noon) and read off the time on your watch set to Universal Time. If you read a time of 06:00 then that means that the Sun is 6 hours before noon in Greenwich which places you East of 0°. 6 hours of time corresponds to 90° of angle. So you are at 90° East Longitude.
Before the invention of the clock it was very difficult to navigate on long sea voyages. Nowadays we have network of satellites in orbit of Earth they are all in contact with one another via radio waves. All you need is a cheap, hand-held receiver to tap into their signal. They use triangulation to locate you to an accuracy to within a few meters!
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