Parallax for the rest of us.

"Mathematics are for the mathematicians."

Copernicus de Revolutionibus pref. (as quoted by Kuhn, p. 143)

 

Perhaps the greatest barrier to acceptance of a sun-centered cosmos was the problem of parallax--or rather the lack of any observable parallax in the so-called 'fixed stars'.

If one is to suppose that the earth revolves around the sun, it must therefore travel a great distance; and yet, from any point of its orbit to the opposite (180° and a little over 180 days later), despite the fact that the observer's position has changed by a great distance, no change could be detected in the position of any star against the celestial background. Of course, the inevitable implication is that the stars are at so great a distance that we simply cannot see by the naked eye the parallax=the angle formed at the distant point when the observer changes position.

Ferris has helfpul cartoons using observers in the heavens, pp. 126-7. But to bring this down to earth, try to visualize the following:

(1) Imagine you are standing at the base of a tall building--say, Hammons Tower--and you wonder how high it is. You can determine this roughly by walking out to a point where you look back to the top of the building at a 45° angle (suppose you have an instrument that can measure this, such as surveyors transit).

At that point, where the angle formed by the path you have just walked and your line of vision to the top of Hammons Tower is 45°, the distance from the tower is equal the height of the tower. Those two measures--the path you have walked and the height of the building-- form the equal sides of a right triangle (the angle formed by those two measures is 90°--assuming the building is actually plumb). If the angles are equal, the sides must be equal.

Now common sense would tell you, as you go farther out, your angle of vision to the top of the building will become smaller and smaller (angle a in the following diagram).

For those who have no trigonometry (or have forgotten it), the angles may be thought of as very roughly proportionate to the opposite sides. This is a way of estimating; it will not give accurate computations; it is just a way to get into the idea.

So, if you walk out to a point where you are looking back at the top of the building at a 15° angle, you can know that the other angle b (at the top of the building) is 75° (the 3 angles of a triangle must total 180°); and therefore your distance from the building is something approaching 5X the height (as 75 = 5X 15). So, say you've measured with your pedometer that you're 1000 feet away; you would guess that the height of the building is at least 200 ft. The actual height works out about 268 ft. (tan 15°= .268 X 1000)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2) Now the same principle can be applied to roughly estimating the distance of heavenly bodies--with the observer's position moved from angle a to angle b. More precisely in astronomical applications, where the parallax angle, measured in radians (2*pi radians = 360 degrees), is small compared to one, the tangent of the angle may be very well approximated by the angle itself (provided the angle is measured in radians), and the distance of an object is very accurately inversely proportional to its parallax angle.

Basically, the star you want to measure is where you used to be, at angle a. And the side A (that used to be the building height) is now roughly the distance between two points of observation. As you travel onboard spaceship earth from the point where the star is directly overhead, to the point 90 days later (roughly) where the radius linking earth-to-sun forms a right angle with that first line of observation, you are now in position to use the same basic reckoning to figure the distance of the star...if only you could detect some angle--if only you could see that the star has somehow changed its position from directly overhead amid its fixed background of stars!

If you could detect a change of even 1°, that would allow you to figure that the angle b that you're looking from is 89 --or round to 90°-- and thus that the distance of the star is roughly 90X the distance of Earth from Sun (called 'the astronomical unit').

 

 

 

 

But, of course, there is not even that 1° of difference. So we keep reaching. Assume we can go from one point of observation with the star directly overhead, 180° or roughly 180 odd days later. So from one point to the other we have spanned the diameter of earth's orbit=2 astronomical units.

And if we could see even 1° difference at this point, it would yield a distance twice what we hoped for last measure: roughly 180X Earth-Sun distance.

But still, no observable parallax. In fact with instruments accurate even to one-tenth of 1° (= 6 minutes of arc), there is no observable parallax. And that meant for astronomers in the time of Tycho, that if Copernicus is right, the stars must be more than 1800X the distance from Earth to Sun.