Goals: The student will|
- To use "pre-trigonometry" to calculate the distance to a far-away point.
- The "thumb method" of estimating distances in the field, and the reason it works.
- How astronomers used the diameter of the Earth's orbit around the Sun as a baseline for estimating the distance of some stars, and the meaning of "Parsec" and "light year."
Terms: Parallax, baseline, light year, parsec; degrees, minutes and seconds of arc.
Stories and extras: The 1838 measurements of the distance to the nearest stars. Optional discussion of methods by which pictures and displays can present a 3-dimensional view.
Start of the lesson:
When you observe a distant object from two separate locations, you get slightly different views, because you view it from two different angles.
The information you get from such a pair of observations should be sufficient to calculate how far the object is, without having to go there. Say the two points are named A and B, and the location of the distant object is C (draw on the board to illustrate). All you need is to know the distance AB--that is your "baseline"--and the angles between AB and the lines to the object C from both its ends.
Once you have these angles, you can draw on paper a triangle with the same angles as the triangle ABC--a triangle which has the same proportions. Comparing in that triangle the line representing AB to the actual baseline AB you have measured, you get the scale of the drawing, the ratio between distances in the "real world" and distances on your drawing. You can then measure on the drawing (say) the distance AC to the distant object, and by multiplying this by the scale factor, find what the actual distance is.
But why bother with scale drawings on paper (which anyway have limited inaccuracy) if we can just as easily (or even more so) calculate distances such as AC?
Such calculations belong to the field of trigonometry--literally, "the measurement of triangles." [The teacher may expand this part to cover "Trigonometry--What is it good for?," Section M-7 of "Stargazers".]
Today we will not need "real" trigonometry, in which ABC can be a triangle of any shape, but will confine our attention to triangles which are long and skinny. You get such triangles in astronomy, where the distances can be, well, astronomical, while the baseline may be a distance on Earth, which is much shorter.
In such triangles, you do not make a great error if you regard them as pie-shaped slivers from a big circle. The ratio between the baseline and the circumference of the circle [use the board to illustrate] is then the ratio between your parallax--the angle between your two lines of view, from the two ends of the baseline b--and 360°, the angle covering the entire circle.
However, the length of that circle is 2πR, where R is the distance AC. So what you wind up with is the ratio between b and R, which is what we wanted.
We will go over that again, with various applications.
Guiding questions and additional tidbits
(With suggested answers and extensions).
-- What is the "baseline" in a parallax measurement?
To measure the distance to a far-away, inaccessible object, it has to be viewed from at least two points. The baseline is the line connecting those points.
-- What does the word "parallax" mean?
The parallax of a distant object, viewed from a given baseline, is the angle between its directions of view from the two ends of the baseline. The term is usually applied only when those two points are much closer to each other than to the distant object.
The parallax depends on the size of the baseline: the bigger the baseline, the bigger the parallax
-- When we use a yardstick to measure the distance between two points on a wall, we sometimes close one eye and put the other right above the mark with which we want to line up some division on the yardstick. Why?
We do so to avoid an error due to parallax. We are trying to line up markings on the yardstick with some mark on the wall, but because of the thickness of the stick, the marks on the stick are a bit closer to our eyes than the wall. It could be that each eye lines up the mark slightly differently!
To make sure this does not happen, we close one eye and place the other so that our line of sight is perpendicular to the wall.
-- Describe how you use your outstretched thumb to measure distance outdoors.
(The answer is found in text.)
-- Why does this method work?
[Use a drawing on the board.]
The thumb forms the tip of two triangles, of identical proportions, "similar" triangles.
The triangle formed by the thumb and the eyes has a base (the distance between the eyes) about 1/10 of the distance to the thumb. Therefore the same ratio also holds for the other triangle, between the thumb and two distant points it covers, when viewed from either eye.
Mention that one reason why humans and animals have two eyes is that by viewing objects from two slightly separated positions, we get a feeling of 3-dimensional depth.
Has anyone ever walked towards an eye level clothesline? It can be very disconcerting because you get no feeling for its distance, since both eyes see the clothesline the same way.
Does anyone own a stereo viewer? Bring one to class, if possible! These are little toy viewers, with a window for each eye and with a place into which one can slide a special frame with transparencies--usually a wheel with about 6 pairs of transparencies, which can be changed by pressing a lever.
. Each eye sees a slightly different picture, giving one a 3-D impression. Around 1900, a cruder version of the viewer, the "stereoscope" or "stereo-opticon," was very popular.
Simpler 3-D viewers use "glasses" in which you look at the picture through two pieces of colored cellophane--a red one for one eye, a green one for the other. Without the glasses you see two overlapping pictures, one red, one green, but the eye looking through the green foil cannot see the red picture, and the other one does not see the green one. Since the pictures are slightly different, you get a 3-D effect, though in unusual color.
Nowadays computers exist connected to special glasses which give a 3-D view, combining polaroid material with "liquid crystals" similar to the ones used in electronic watches. Polaroid material only transmits light that vibrates in one direction (it still looks normal to the eye), and by "turning on" the liquid crystal of either eye with a pulse of electricity, it can be made to block that direction, with the result that all the light coming to that eye is blocked.
Liquid crystals can be switched on and off very rapidly. The pulse of electricity is linked to a computer or TV, which shows a rapid alternation of two sets of images--one for the left eye, one for the right one. The pulses are sent out in such a way, that each eye is blocked from seeing the picture not meant for it. A person not equipped with the glasses will see both sets of pictures with both eyes and will received a blurred view.
-- What baseline is used to measure distance to far-away stars?
The baseline is the diameter of the Earth's orbit around the Sun, some 300 million kilometers. We look at the star at two times 1/2 a year apart and check whether its position relative to other stars--assumed to be much more distant--has moved.
The observation must be made over several Earth orbits because stars also have their own "proper motion" relative to Earth, which must be separated from the periodic back-and-forth motion due to the Earth going around in its orbit.
-- What is a parsec?
A parsec is the distance of a star whose apparent position moves by one second of arc when viewed from Earth on two occasions half a year apart.
-- The space shuttle orbits at an altitude of about 200 miles or 320 kilometers. How big would an object at that distance have to be, to cover 1 second of arc?
A circle of 320 km radius has length 640 π> = 2010.6 km.
One second of arc is 1/(360 x 60 x 60) = 1/1,296,000 of that circle or 1.55 meters, the height of a short person
-- How many stars are known to be within 1 parsec of Earth?
There are no stars are as close as one parsec: they are all more distant.
-- What is a light year?
The distance which light, moving 300,000 km/sec (186,000 miles/sec), covers in one year.
For comparison: the Sun is 8 light-minutes away, Pluto about 5 light-hours.
-- What is bigger- a light year or a parsec? And by how much?BR>
A parsec is about 3.26 light-years.
-- How far is Alpha Centauri, the nearest Sun-like star?
Alpha Centauri is about 4.3 light years. A small star associated with Alpha Centauri is slightly closer.