Why Did Aristarchus Measure the Sun's Distance During a Half Moon?

[pbody]

1. Why did Aristarchus make his measurements of the Sun's distance at the time of the half moon?

Aristarchus watched for the phase of the Moon when it was exactly half full, with the Sun still visible in the sky. Then the sunlight must be falling on the Moon at right angles to his line of sight. This meant that the lines between Earth and the Moon, between Earth and the Sun, and between the Moon and the Sun form a right triangle.

A rule of trigonometry states that, if you know all the angles in a right triangle plus the length of anyone of its sides, you can calculate the length of any other side. Aristarchus knew the distance from the Earth to the Moon. At this time of the half Moon he also knew one of the angles, 90°. All he had to do was measure the second angle between the line of sight to the Moon and the line of sight to the Sun. Then the third angle, a very small one, is 180° minus the sum of the first two angles ( the sum of the angles in any triangle = 180°)

Measuring the angle between the lines of sight to the Moon and Sun is difficult to do without a modern transit. For one thing, both the Sun and Moon are not points, but are relatively big. Aristarchus had to sight on their centers (or either edge) and measure the angle between - quite large, almost a right angle itself. By modern day standards his measurement was very crude. He measured 87° while the true value was 89.8°. He figured the Sun to be about 20 times the Moon's distance, when in fact it is about 400 times as distant. So although his method wand ingenious, his measurements were crude. Perhaps Aritarchus found it difficult to believe the Sun was so far away, and her erred on the nearer side. We don't know.

Today we know the Sun to be an average of 150,000,00 kilometers away. It is somewhat closer to the Earth in December (147,000,000 km), and somewhat farther in june (152,000,000 km).

2. Well these are relevant to me but perhaps not to get the answer, I was wondering how he knew the distance from the moon was it the moon trick where it was 1/110 which is 1 diameter of the moon divided by 110 moons in itself?

As well I was wondering about the half moon when the sun is in the sky? What is meant by this?



3. To get a right angle

 

[Vorde]

Your post is a bit sloppy, but I'll answer the question as best as I can.

Basically if we see a half moon, it means a straight line from us to the moon is perpendicular to a straight line from the moon to the sun (if you think about it you'll see why). This means that a triangle between us the moon and the sun has a 90 degree angle at the moon.

You seem to understand the rest of the process, though I'd note that the distance to the moon was roughly calculating using trig where two points of the triangle were at different points on the Earth and the third was at the moon.

 

[pbody]

Your post is a bit sloppy, but I'll answer the question as best as I can.

Basically if we see a half moon, it means a straight line from us to the moon is perpendicular to a straight line from the moon to the sun (if you think about it you'll see why). This means that a triangle between us the moon and the sun has a 90 degree angle at the moon.

You seem to understand the rest of the process, though I'd note that the distance to the moon was roughly calculating using trig where two points of the triangle were at different points on the Earth and the third was at the moon.

I don't know how to organize to well and that is why I am asking here so I don't have to see anybody eye to eye and be embarrassed just get to ask away so my apologize but I feel like all of my thoughts are kind of that way.

Well why couldn't we formulate a straight line from a full moon?

and I understand that he got his calculations threw trig but how would he know the distance I thought you would have to know more than just the angle I thought you needed to know the length of at least one distance at least that is what I got out of the exert!

 

[Vorde]

We could get a straight line to a full moon, or any other moon, but the angle between the earth-moon line and the moon-sun line won't be 90 degrees. It's really important that the triangle is a right triangle because it means all you need is one angle and one side.

If you sketch a top-down view of the moon orbiting the Earth you'll see why a right angle only happens during a half moon.

As for measuring the distance to the moon, they measured the distance between the two points on the surface of the earth, and then they measured the angle from each point and did a little bit of geometry/trig to figure out the distance. They didn't account for a bunch of things so they got the value wrong, but the method was theoretically fine.

 

[pbody]

I see what your saying if I sketch something but every time I attempt to sketch something I fail and I will spend hours trying to figure it out, do you have a web address or something where I can get a better idea (full moon vs. half moon) angular differentials?

So you can measure distance by measuring angles?

 

[Vorde]

Yes, look at the trigonometry wikipedia page to learn about the wonders of trig. As I sas having a right triangle is so useful because you only need one side and one angle, if it's not a right triangle you need more information than that.

 

[pbody]

Yes, look at the trigonometry wikipedia page to learn about the wonders of trig. As I sas having a right triangle is so useful because you only need one side and one angle, if it's not a right triangle you need more information than that.

Vorde I am sorry I wasn't clear in one of my questions. How was the measurement determined? Or where is the best place to gain that information.I don't mean the angle but the distance?

 

[Vorde]

Distance to what? The distance to the moon was measured by devising a triangle with two points on Earth and one on the moon. By measuring (using any classical way of measuring) the distance between the two points on Earth, and by measuring the angle from the horizon to the moon from each of the two points, you can use trig to calculate the distance to the moon.

The distance to the sun was calculated the same away, except the three points are now moon-sun-earth instead of moon-earth-earth, this works because you know the distance to the moon from the previous calculation.

Remember, they did all of this incorrectly because practically it's more complicated than that.