Earth and Moon, same angular size, at one point. I think I made a mistake...

[Erik te Groen]

Hello everyone,

I am new here. I just registered.
This is my first post.
I actually have only one question.

Recently I was watching a documentary about the Apollo Lunar program. The astronauts were filming the Earth and the Moon, while on route.
Then a question suddenly popped up... At which point, between the Earth and the Moon, will both bodies have the same angular size ?
I started making some calculations and came up with an answer (see my drawing).
I also made this drawing to have a better visualization. To my surprise, I ended up with two quite different answers. There is quite a large
discrepancy between the two answers.
The drawing - to scale - showed me the exact angle and the point where both bodies have the same angular size.

Therefor I think I made a mistake in my calculations.
But...where, how, what and why? I can't find the mistake.

So... Please help.

Cheers,
Erik.

 

[Dr.Peterson]

We can't find your error without seeing your work. Please show your calculations.

 

[Erik te Groen]

Look at the drawing.
Text in black, left bottom part.

 

[Dr.Peterson]

Look at the drawing.
Text in black, left bottom part.

Okay, now I've enlarged it enough to read. But you didn't really explain your thinking. And I can't tell what the yellow zone represents.

What you need is something like this (which can be explained in terms of similar triangles):

[MATH]\frac{diam_{earth}}{diam_{moon}} = \frac{dist_{earth}}{dist_{moon}}[/MATH]​

What you did amounts to something like

[MATH]\frac{diam_{earth}}{diam_{moon}} = \frac{dist_{earth}+dist_{moon}}{dist_{moon}}[/MATH]​

That is, your distance from each sphere has to be proportional to its size; you used the total distance between earth and moon. (I think -- you didn't say exactly what each number represents.)

 

[Erik te Groen]

I divided the Earth's diameter by the Moon's diameter and came up with a factor of 3.66 .
I then divided the distance between Earth and Moon by 3.66. So, 384400 km / 3.66 = 105027 km.
I then subtracted that number from the distance (384400 - 105027) and came up with 279373 km.
At that distance from Earth both planets should have the same angular size. But my drawing shows a very different number.
It is significantly closer to the Moon than what the calculation shows.
I tried changing numbers and different ways of calculation, and came up with different distances (yellow area), but never
anything close to point zero, where the drawing says it should be.
I trust the drawing, because it gives a clear visual representation of the situation.

The difference is more than 16,000 km. More than the Earth's and Moon's diameter added up....!

So, I'm confused. Where did I make the mistake?

_______
Erik

 

[Dr.Peterson]

I already told you: You are assuming the wrong proportionality. You haven't explained why you think you need to do what you did, so I can't say beyond this where you went wrong in your thinking. Did you try using similar triangles as I suggested?

Here's a different way to say the same thing: The distance from the earth to your point, as a fraction of the distance from earth to moon, is equal to the diameter of the earth as a fraction of the SUM of the diameters.

The latter fraction, using your numbers, is 12754/(12754+3475) = 0.785877; that times 384400 is 302,091. Is that close enough to your 296,000? I'm not sure how you found the latter, and how accurate you think it is.

 

[Erik te Groen]

302000 is close enough.
Actually, using real life numbers, 302000km is the number that popped up, often, in my drawings.
Measured from the Earth's surface, I get ~296000 km.
I now know that my drawing is correct and that my calculations were wrong. I think I now see
where I went wrong.
Thanks for the help.

_______
Erik

 

[Cubist]

I divided the Earth's diameter by the Moon's diameter and came up with a factor of 3.66 .
I then divided the distance between Earth and Moon by 3.66. So, 384400 km / 3.66 = 105027 km.
I then subtracted that number from the distance (384400 - 105027) and came up with 279373 km.

It might help to consider an alternative scenario - what if the two bodies have equal diameter. Let's call them Earth2 and Moon2. Would you...

Divide the Earth2's diameter by the Moon2's diameter and came up with a factor of 1.00.
Then divide the distance between Earth2 and Moon2 by 1.00. So, 384400 km / 1.00 = 384400km.
Then subtract that number from the distance (384400 - 384400km) and came up with 0 km. This is obviously incorrect. This should hopefully help you to see that your method isn't correct.

Let's use one of @Dr.Peterson 's suggestions to get a better answer. This is adapted from post#4...

[MATH]\frac{diam_{Earth2}}{diam_{Moon2}} = \frac{dist_{Earth2}}{dist_{Moon2}}[/MATH]
[MATH]\frac{3,500}{3,500} = 1 = \frac{dist_{Earth2}}{dist_{Moon2}}[/MATH]
But the total distance between Earth2 and Moon2 is 384,400km therefore

[MATH]dist_{Earth2}+dist_{Moon2}=384,400km[/MATH]
[MATH]dist_{Earth2}=384,400km-dist_{Moon2}[/MATH]
Combine the above with the following

[MATH]1 = \frac{dist_{Earth2}}{dist_{Moon2}}[/MATH]
To obtain

[MATH]1 = \frac{384,400km-dist_{Moon2}}{dist_{Moon2}}[/MATH]
[MATH]dist_{Moon2} = 384,400km-dist_{Moon2}[/MATH]
[MATH]2 \times dist_{Moon2} = 384,400km[/MATH]
[MATH]dist_{Moon2} = \frac{384,400km}{2}[/MATH]
...this is what you'd expect, the spacecraft has to be midway between Earth2 and Moon2 for equal angular size.

 

[Erik te Groen]

Thank you, too.

Funny thing is, that I got quite close using the wrong method. Close, but never close enough. That's
why I could not figure it out. I could not figure out where I went wrong.
I should have been able to solve this problem, but I couldn't. After all, it's not rocket science.....

Oh wait...........

;-)
_______
Erik