Estimating Distance from Earth to Sun Using Trigonometry

[efekwulsemmay]

Homework Statement

The problem from the book states

Suggest a way to measure the distance from the Earth to the Sun.

The problem I am having is actually trying to figure out how someone else's suggested solution. His idea was to take a value R for the mean radius of the sun:

[tex]R=6.96x10^{5} km[/tex]

And use trig to find out a rough distance.

Homework Equations

I drew up a diagram to help figure out how he thought it would work (see attached file Diagram 1).

The Attempt at a Solution

Now my thoughts were that since the adjacent side is equal to R+X and the opposite side of theta is just R, that we set the whole thing equal to tangent theta and solve for X.

[tex]\tan\Theta=\dfrac{R} {R+X}[/tex]

Solve for X:

[tex]X=\dfrac{R} {\tan\Theta} - R[/tex]

This is where I get stuck. I cannot think of a way to find a value for theta without having a value for X, and without a value for theta I cannot seem to find X.

Another thought I had was to somehow try and find the hypotenuse of the triangle but again I cannot figure out a way without either angle theta or a value for X. This is all the information that was assumed and given and this is what I have to work with. I don't understand how exactly he did it. Unless he withheld some other assumption.

 

[diazona]

I can't see your picture, but couldn't you just look at the sun and measure (or estimate) its angular size?

 

[efekwulsemmay]

I can't see your picture, but couldn't you just look at the sun and measure (or estimate) its angular size?

I am not exactly sure what you mean or how to do that. Could you explain?

 

[kuruman]

What are you supposed to know? For example,

You can use Kepler's 3rd Law if you know the appropriate constant and the period of the Earth around the Sun.

 

[efekwulsemmay]

I realize there are many different ways to find the distance. For instance what I did to answer the problem was use the speed of light constant and the fact that it takes roughly 8min for light to reach the Earth from the sun. However, I am trying to understand this specific solution. This is what the guy in class gave and I don't understand what he did in order to solve the problem of finding X. I gave all the information I have on his solution and am asking for help with finding X.

 

[kuruman]

I don't have access to your diagram, yet. What does X represent? It seems to me that R + X is the desired Earth-Sun distance.

 

[ideasrule]

I think you have it backwards. We know light takes 8 min. to reach us because we know the Earth-Sun distance, not the other way round.

 

[efekwulsemmay]

I think you have it backwards. We know light takes 8 min. to reach us because we know the Earth-Sun distance, not the other way round.

I do have it backwards. I used it that way to answer the problem. Literally my quote from the book is exact, thus there were no limitations on what we could use to answer the question. We were just supposed to answer it.

I don't have access to your diagram, yet. What does X represent? It seems to me that R + X is the desired Earth-Sun distance.

X represents the distance from the surface of the sun to the earth. X is the value that needs to be solved. R is the radius of the sun.

 

[kuruman]

Sorry, I got confused. The problem asks "Suggest a way to measure the distance from the Earth to the Sun." Usually that statement is interpreted to mean the center-to-center distance.

** Edit **
So if you know R and you are looking for X and you don't know θ, you have one equation and two unknowns, X and θ. You should either find an additional equation involving θ, or measure θ and plug in.