Help with Eratosthenes' Astronomical Measurement!

[monkeymocha]

Astronomy Help! :(

I'm sure that compared to all the other problems posted here, this Astronomy problem is incredibly easy, but I'm TERRIBLE at sciences and mathematics and really need some help.

The problem is: "Although the distance to the Sun is obviously much larger than the distance between Alexandria and Syene, it is not infinite. This results in a small error in Eratosthenes' measurement, which is given by the difference in the angle of the Sun between Alexandria and Syene. What is the difference in the angle in degrees? (assume a distance between Alexandria and Syene of 860 km. It may help to draw a diagram.)"

The answer choices are:
a) 3.29 x 10^-4 degrees
b) 2.07 x 10^-3 degrees
c) 5.75 x 10^-6 degrees
d) 0.329 degrees
e) 3.29 x 10-2 degrees


This problem is referring to how Eratosthenes measured the circumference of the Earth by using the distance from Alexandria and Syene and the angles of the sun. See http://www.eg.bucknell.edu/physics/astronomy/astr101/specials/eratosthenes.html for details.

I honestly have no idea where to even begin. I'm not even sure what the problem is even asking. If anyone could just point me on the right track, I would really be happy! Thank you!

 

[Borek]

Draw the diagram (Sun and both cities). There is a triangle that contains angle in question.

--
methods

 

[Blueshift5]

Hello,

Don't worry, astronomy and mathematics can be challenging for many people. Let's break down this problem and see if we can make it easier to understand.

Eratosthenes' measurement was based on the fact that the Sun's rays are essentially parallel when they reach Earth. However, because the Earth is not a perfect sphere, the distance between Alexandria and Syene is not on the same parallel as the distance between the Sun and Earth. This means that the angle of the Sun's rays at Alexandria and Syene will be slightly different.

To solve this problem, we need to use some basic geometry. Draw a diagram with the Earth, the Sun, and the two cities. The angle we are looking for is the difference between the angle of the Sun's rays at Alexandria and Syene.

Next, we need to use the distance between Alexandria and Syene, which is given as 860 km. We also need to know the distance between the Sun and Earth, which is approximately 149.6 million km.

Using the basic formula for calculating angles in a triangle, we can set up the following equation:

tanθ = opposite/adjacent

In this case, the opposite side is the distance between Alexandria and Syene (860 km) and the adjacent side is the distance between the Sun and Earth (149.6 million km). We can rearrange this equation to solve for θ, the angle we are looking for.

θ = arctan(opposite/adjacent)

Plugging in the numbers, we get:

θ = arctan(860 km/149.6 million km)

Using a calculator, we get θ = 0.329 degrees. This is the angle at which the Sun's rays hit the Earth at Alexandria.

Now, since we are looking for the difference in angles between Alexandria and Syene, we need to subtract the angle at Syene from the angle at Alexandria. This will give us the error in Eratosthenes' measurement.

θ difference = 0.329 degrees - 0 degrees (since the angle at Syene is 0)

Therefore, the difference in the angle of the Sun between Alexandria and Syene is 0.329 degrees.

Looking at the answer choices, we can see that the correct answer is d) 0.329 degrees.

I hope this helps and good luck with your studies!