Astronomical Unit: Prequel – three remarkable measurements

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I must apologize for the long silence since I last posted. You may remember that I had promised to answer how in ancient days scholars got to know about the astronomical unit or rather the distance between our earth and the sun. Before I answer that, it is important to know how each measurement was made based on other earlier measurements. Therefore before I talk about measuring the sun-to-earth distance, i.e. the Astronomical Unit, I’ll briefly discuss about other two great measurements. Therefore three historically great measurements– related to the earth, the moon, and the sun:

The diameter of the earth:

A Greek librarian Eratosthenes (276 – 195 BC)
living in Alexandria, Egypt realized that the sunrays do not fall in the same way in every place on the
earth. This is an indirect proof of the fact that our earth’s surface is rounded rather than being a flat one since in the latter case we should always expect an equal size of an object in different places.
But that is not true and Eratosthenes realized the fact that during the summer solstice (June 20/21), the sun casts a shadow on a pillar in Alexandria whereas there is no shadow in a town named Syene (presently known as Aswan, which is situated at the tropic of cancer) where the sun appears exactly at the zenith at the middle of the day, i.e. it makes a 90 degree angle of incidence. Myth says that there was a well in Syene which used to be perfectly lit by the sun in the summer solstice noon.

Now Eratosthenes measured the diameter of the earth by measuring two things: (1) The angle \theta made to the centre of the earth by Alexandria and Syene (of course he was capable to imagine that our could be a sphere in shape.). (2) The arc made by Alexandria and Syene s, i.e. the distance between the two places.

Eratosthenes first measured the angle of incidence in Alexandria by measuring the height and shadow of any vertical object on the ground:
angle of incidence=arctan(shadow length/height).
Then he figured out that this angle is the same angle \theta by using the fact that any intercept between two parallel lines always makes the same angle with each of them (equality of alternating internal angles) considering that the sunrays are effectively parallel as they arrive from a very far distance.

Eratosthenes found that the angle is about 7^\circ.

Now the knowledge of the arc s he gathered from the caravans who used to travel from Syene to Alexandria was 5000 stadia, which is about 800 km. Now the whole circumference of the earth makes 360^\circ angle.
Therefore the circumference S he estimated by taking the ratio of these two angles: \frac{S}{s}=\frac{2\pi}{\theta}

Thus S=\frac{2\pi}{\theta}s=\frac{360}\theta(360/7)\times 800\, \text{km}\simeq 40,000\, \text{km}. And we also get the radius to be 40,000/2\pi\simeq 6366 km.

These are remarkably quite close to the modern accepted values
(equatorial circumference=40,075.017 km and equatorial radius=6,378.1 km).

Below you can see a beautiful clip from one of the famous Carl Sagan’s Cosmos episodes.

[By the way, Erastothenes’ contribution is not only the estimation of the radius of the earth, mathematicians will never forget the Sieve of Eratosthenes , the first algorithm to generate prime numbers up to any limit. Interested readers can also look for the book The Librarian Who Measured the Earth by Kathryn Lasky.]

The distance to the moon and its size:

Here the credit goes to Aristarchus (310 – 230 BC), another Greek genius. He just waited for a lunar eclipse to happen. What he noticed was that how much time (t_1) the moon takes to go into the full eclipsed phase (when it is completely dark) starting from the commencement of the eclipse
and how long (t_2) it remains in the total eclipse phase. In the total eclipse phase, the moon spends time inside the main shadow part (umbra) of the earth. Thus the ratio t_2/t_1 gives the ratio of the width of the umbra to the diameter of the moon. Aristarchus found that t_2/t_1\simeq 2.5. Thus the umbra region in the moon’s path during the lunar eclipse
2.5 times bigger than moon’s diameter.

Now there is one important fact that was not at all overlooked by the Greek scholars: the apparent sizes of the moon and the sun are quite close to each other. This means that if we send the moon
to the distance of the sun keeping the same solid angle created by the moon, the moon will become as big as the sun, i.e. the angles subtended by the moon and the sun on a point on the earth are the same. For this reason, during the total solar eclipse the moon can perfectly fit to block almost all the rays coming from the sun and the umbra tapers to the earth’s surface. The coincidence of same apparent sizes has been beautifully mentioned, in a way of story-telling to the kids, in the film Agantuk by the famous Indian filmmaker Satyajit Ray. The story-teller (acted by Utpal Dutta) says that it is an unresolved magic
of the universe (see the youtube link below or http://mynethome.net/2009/01/sun-moon-and-machu-picchu/).

[N.B. The story-teller also says that the size of the shadow of the earth on the moon is also the same during the lunar eclipse and this is not true. Since the moon is quite small compared to the earth, it just requires to be inside the umbra of the earth, no need to maintain the same solid angles! He tells that the moon is 500,000 miles away from earth while in reality, the distance between the earth and the moon is 384,400 km or 238,855 miles (almost half of the distance he mentions).]

Anyway, back to Aristarchus again.

If the sun and the moon makes same angle, then if we draw a parallel line to one of the edges of the conical umbra, the adjacent cone to the other edge will perfectly fit another moon (see the figure below).

Thus Aristarchus finally concluded that the earth is 2.5+1=3.5 times bigger than the moon.

Now like the moon fits to block the sun, we can use a coin, tape it to a stick and move it away so that it blocks the moon in our vision. If we do so, we shall find that the distance will be roughly about 110 coin diameters away. This again means that if we extend the coin to the distance of the moon keeping its
boundary inside the same solid angle, the coin will reach to the size of the moon. Since the
angle remains the same, we can safely write
l_{\text{moon}}/l_{\text{coin}}=D_{\text{moon}}/D_{\text{coin}}
where l‘s and D‘s denote distances and diameters respectively.

Thus the moon is 110 moon-diameter away from us (since l_{\text{coin}}=110\, D_{\text{coin}}). Now Aristarchus already estimated the moon diameter to (1/3.5)th of earth
diameter. Thus the distance to the moon he found:
l_{\text{moon}}=110/3.5\, D_{\text{earth}}=\frac{110\times 2\times 6366}{3.5}\, \text{km} = 400,148.57\, \text{km}. Wikipedia says that the value in the apogee (farthest) position 406,700 km, which seems not very bad compared to Aristarchus’ estimation.

Distance to the sun and its size:

Now we cannot do the same coin experiment since it could be dangerous for our eyes. However, we can exploit the knowledge that the sun and the moon make same angle to the earth, i.e. like the moon the sun is 110 sun-diameter away from us. However, the problem is that we do not know the diameter of the sun. Facing the same problem Aristarchus adopted an alternative method to find the distance to the sun. He waited for a situation where the moon appeared exactly as half-full in the day time. The half-moon appears when the sunrays to the moon become perpendicular to the line joining the earth and the moon (see figure below). Thus the earth, moon, and the sun form a right-angled triangle and once the angle subtended at the earth by the moon and the sun ($latex\alpha$ in the figure below) can be measured, the distance to the sun (hypotenuse of the triangle) can be found:
l_{\text{moon}}/l_{\text{sun}}=\cos\alpha.

Similar to the method used by Eristothesen, Aristarchus found
the incident angle to be 3^\circ and hence
\alpha=(90-3)^\circ=87^\circ.
Therefore
l_{\text{sun}}=l_{\text{moon}}/\cos\alpha=19\, l_{\text{moon}}

Certainly this is a quite inaccurate value estimated by Aristarchus since it was very difficult to correctly measure \alpha for him (Instead of calculating from the angle of incidence, he
needed to consider the angle made with line passing to the earth’s center). More accurate measurement done in later time showed \alpha\simeq 89^\circ, which gives l_{\text{sun}} \simeq 400\, l_{\text{moon}}=400\times 400,148.57 \mbox{km}=1,60,059,428 \mbox{km}.

Now once we know the distance, we can find the diameter of the sun by using the magic ratio 110, i.e. diameter of the sun D_{\text{sun}}=l_{\text{sun}}/110\, \mbox{km}=1.45 \times 10^6\, \mbox{km} (modern accepted value 1.392\times10^6 km).

The first scientifically accurate measurement of the earth-to-sun distance was made by Giovanni Cassini in 1672 (two years before Ole Rømer estimated the speed of light) by parallax measurements. I shall discuss this in my next post.

Image courtesy:

http://www.juliantrubin.com/bigten/eratosthenes.html
http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html

Reference: Conceptual Physics (10 th ed.) by Paul G. Hewitt

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