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NASA
Course: NASA > Unit 2
Lesson 2: Measuring the solar system- A flat earth
- Arc length
- Circumference of Earth
- Occultations
- Occultation vs. transit vs. eclipse
- Size of the moon
- Angular measure 1
- Angular measure 1
- Trigonometric ratios in right triangles
- Angular Measure 2
- Angular Measure 2
- Intro to parallax
- Parallax: distance
- Parallax method
- Solar distance
- Solve similar triangles (advanced)
- Size of the sun
- Scale of solar system
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Parallax: distance
How far away is the Moon?
In the second century BC Hipparchus derived a very good estimate of the distance to the moon using lunar parallax. It is based on how much the moon shifts relative to the background stars when we observe it from different vantage points on earth.
To develop our measurement we first need to setup a triangle. Think of the moon as a point in space with two straight lines connecting it to points on Earth:
In this example the two vantage points are Selsey, UK and Athens, Greece which are separated by 2360 km.
This gives us a triangle. We can simplify things by assuming the moon is exactly between the two points (isosceles triangle).
Now we need to determine the angle p using the parallax effect.
Finding parallax angle
Here are two photos taken at the same time from Athens and Selsey. We can assume the star (Regulus) near the moon is fixed since it’s 78 light years away.
The Moon has appeared to shift position. Our goal is to figure out the angular distance of this shift. To do so we combine these images as a stereo pair. To see this, simply cross your eyes (using above images) until the Moons overlap.
If you do this correctly you will see something like this:
The angular distance between the stars turns out to be approximately 1100 arcseconds, or 0.30 degrees. This looks about right since we know the Moon has an angular diameter of 0.5 degrees. We now have the angle needed. The Moon appears to shift 0.3 degrees when we observe it from two vantage points 2360 km apart.
Finally, we can split our triangle in half to create a right triangle. This allows us to apply our trigonometric functions to find the distance d directly.
tan(angle) = opposite/adjacent
tan(0.15) = 1180/distance
1/381.9 = 1180/distance
distance = 1180*381.9
this gives us our estimated distance to the Moon:
Estimated lunar distance = 450 642 km
This estimate is off by only 17% of the actual distance, which is pretty good for a rough estimate! Compare this to the actual average value: 384,000 km
Challenge question: What were the sources of error in our method above? How could we improve?
Want to join the conversation?
I see that there is error from using the length of the arc (aka surface distance) between the two points on earth rather than the chord length (as the mole digs), but the challenge question cites "sources" of error. What other sources can other students come up with?(5 votes)
Earth is curved. Distance measured over the surface, that is between the cities would be more than the actual distance between the two vantage points. if we take curvature into account and use the actual shorter underground distance the error would be less than 17 percent.
Another point might be that the star in question is taken as a fixed point. Although negligible but it should undergo some parallax as well which might cause a slightly incorrect calculation.
Also the calculations have been rounded off a lot to give a much higher value for the distance to the moon.(8 votes)
I am not getting how to measure the angle.(1 vote)
Another questioner had the same basic question, so I will answer it: First, astronomers can measure it using the equivalent of a theodolite.. You can measure angles with instruments. However, there is a simpler way to think about it. Imagine a line running across the heavens like an equator that goes through the moon (its called the ecliptic, and defines which constellations form the zodiac... all planets and the moon move along this one track in the sky, more or less, because they are all in the same plane). Now, in your head imagine drawing loads of moon-sized circles along that line until you come back to where you started. Turns out you can fit 720 moons along such a line, which means that in each degree of sky, you can fit 2 moons; that is, each moon diameter fills 0.5 degrees in the sky. Now, look at the distance between the star images in the picture. How does that distance compare with the diameter of the moon? About 3/5, right? So the angle between the star images should be 3/5 of 0.5 degrees, or 0.3 degrees. So thats how you can measure it by eye. But really, its measured using instruments, like a sextant or something..., or by running the images through a computer. That is, its technical. The greeks probably measured it by comparing with the moon.(10 votes)
How did we calculate that tan(0.15)= 1/381.9?(4 votes)- tan(0.15) = opposite/adjacent according to trig functions. Hence, you find out the angle by substituiting the values for the base and the height(3 votes)
- is there any other way to find the parallax angle? without using a reference behind the object?(3 votes)
- " We can simplify things by assuming the moon is exactly between the two points (isosceles triangle)." But wont the calculations all go wrong when we assume like that?(2 votes)
No, it is simply defining an isosceles triangle for the measurement.(1 vote)
- can someone please explain me from where the equation base/2/tan(p/2) derived(2 votes)
Isn't the reason the parallax occurs when we look into the distance is because we see from different angels. Your saying it's the same with the sun rays to the earth. That's why the moon shows up red.(2 votes)
THIS IS THE STATEMENT WRITTEN IN ARTICLE
This estimate is off by only 17% of the actual distance, which is pretty good for a rough estimate! Compare this to the actual average value: 384,000 km
my question is how do we calculate the error percentage or what calculation is used so the answer is estimated by only 17% of actual distance ?(1 vote)- 450,642/384,000 = 1.1735...
So rounding to nearest percentage point, 450,642 is about 117% of 384,000, which is a 17% difference.(2 votes)
- When measuring the distance between two vantage points on the Earth shouldn't the bottom line of the triangle be curved instead of a straight line since the Earth is round? Would a curved line change the math needed to correctly answer the question?(1 vote)
If they thought the world was flat, then does that mean that is why their measurement was off?(1 vote)
