February 20^{th}, 2010

The editors of the University of Hawaii's Institute for Astronomy (IFA) asked me to write about this topic for the IFA's bi-monthly newsletter. Astronomers talk about how the Sun is 93 million miles away or Jupiter is 5 times that distance from the Sun or that an asteroid will come within 1 million miles of the Earth, but how do we know? Let's explore the ways that people have calculated distances to the Sun and planets since the time of the Greeks all the way up to modern times.

[Sources for this section: http://en.wikipedia.org/wiki/Eratosthenes, http://www.youth.net/eratosthenes/welcome.html, http://www.britannica.com/EBchecked/topic/191064/Eratosthenes-of-Cyrene, http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html]

If we're going to figure out the distances between the Earth, the Sun, and other planets, it's a reasonable question to ask how people first figured out the size of the Earth itself. How might someone going about determining the size of a ball so huge that we can only see such a small part of it at any given time that it essentially looks flat?

The Greek mathematician Eratosthenes (276 - 192 B.C.) invented geography, amongst many other major accomplishments, and was the first person to estimate the circumference of the Earth. Eratosthenes is a fascinating character worth reading up on, if you like that sort of thing. He invented a way to identify all prime numbers, developed a latitude and longitude system for the Earth, and supposedly died by voluntary starvation when faced with the possibility of becoming almost completely blind.

So how did he go about determining the size of the Earth? Eratosthenes learned of a deep vertical well in the city of Seyne (now Aswan, Egypt) that was fully illuminated at noon on only one day of the year, June 21st, the summer solstice. This meant that the Sun was directly overhead on June 21st at noon. (And although this is not a necessary piece of the experiment, that meant that the Sun was directly overhead in Seyne on the day of its most northward position, implying that Seyne is located very close to the Tropic of Cancer.) Eratosthenes also observed that the Sun was not directly overhead on June 21st in Alexandria, Egypt.

In order to calculate the size of the Earth, Eratosthenes assumed that the Earth is round and that the Sun's rays are parallel to each other. Then, knowing that a vertical object doesn't cast a shadow when the Sun is directly overhead, he knew that the angle of a shadow cast by a vertical object on June 21st in Alexandria was the same as the angle between Alexandria and Seyne with respect to the center of the Earth.

The assumption that the Sun's rays are parallel to each other when they arrive at Earth is a good one because the Sun is so far away. Even though the Sun emits light in all directions, the Earth is so far away from the Sun that is looks like a tiny speck that spans an angle of only about 5 thousandths (.005) of a degree (tan θ = earth's diameter/Sun-Earth distance = 7926 miles/93,000,000 miles). Thus, the light rays arriving at the Earth's North Pole are angled only a few thousandths of a degree from those arriving at the Earth's South Pole. Thus, we can consider the rays of light coming in from the Sun to be parallel to each other.

How did Eratosthenes go about measuring that angle? He set up a verticle post and then at noon on June 21st, measured the angle between the tip of the post and the edge of the shadow.

Because he knew that the Sun was directly overhead in Seyne at the very same time, the angle the shadow cast would be the same as the angle between Seyne and Alexandria. Eratosthenes measured that angle to be about 1/50th of a circle. That meant the distance between Seyne and Alexandria was 1/50th of the way around the circumference of the Earth. He estimated that the distance between the two cities was 5,000 stades and that the Earth's circumference was thus 25,000 stades, which comes out to about 23,300 miles. The actual circumference of the Earth is about 25,000 miles, so he was only off by about 6%. The true length of a stade, however, is not exactly known, so he might have been even closer.

[Sources used for this section: http://history.nasa.gov/SP-4211/ch11-4.htm, http://en.wikipedia.org/wiki/Jean_Picard, http://en.wikipedia.org/wiki/Triangulation]

Tycho Brahe developed the method of triangulation to accurately calculate long distances. In triangulation, a series of triangles are used to measure distance. The first step is to measure one side of the first triangle. It's a small distance, so it's easy enough to do with high accuracy. Now create a triangle. By measuring the angles of that triangle, the other sides of the triangle can be found using the law of sines. Now make a new triangle using one side of the first triangle. Since you have already calculated the length of one side of your new triangle, you simply need to measure an angle again to find the lengths of the other sides. Continue with this method until you've gone the distance that you want to know. Since people can measure angles with far more accuracy than measuring long distances, this method of piggybacking triangles results in more accurate distances than used previously. An example of triangulation to calculate long distances is shown in the diagram below.

The French mathematician, Jean Picard (1620 - 1682), calculated the radius and circumference of the Earth using triangulation. His calculations resulted in a radius of 6328.9 km. The modern value of the Earth's radius at the poles is 6357 km, only a .44% difference!

[Sources for this section: http://history.nasa.gov/SP-4211/ch11-4.htm, http://en.wikipedia.org/wiki/Global_Positioning_System]

In the modern times, we can measure the size of the Earth with extreme accuracy using satellites. Initially, satellites aided in distance calculation through triangulation. Multiple stations would have to look at the same satellite at the same time in order to calculate an accurate distance between them.

Eventally, when satellite orbits were calculated with high precision, it became possible to find the distance between two points on the Earth's surface by knowing the time at which each location observed the satellite. It was determined that distances - even intercontinental distances - could be calculated to accuracy better than 10 km.

Nowadays, we have a network of GPS satellites in orbit around the Earth. A person on Earth with a GPS receiver can get an accurate position by connecting to four different satellites. Because the satellites know where they are in their orbits to high precision and the GPS unit can determine how long it took the signal from each satellite to reach it, it is possible to calculate an extremely accurate position on the Earth's surface. This, of course, lends itself to measuring distances across the Earth's surface in high accuracy and measuring the size of the Earth to that same high accuracy.

[Sources for this section: http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html, http://www.astronomy.ie/eclipse08.html, http://www.universetoday.com/guide-to-space/the-moon/distance-to-the-moon/]

The Greeks could not use triangulation to measure the distance to the Moon because the angles measured from two cities only a few hundred miles apart were not good enough to give a result. Aristarchus of Samos (310 - 230 BC) developed a measure of the distance to the Moon by using geometry and observations of eclipses.

During a lunar eclipse, the Moon moves into the shadow of the Earth and is no longer illuminated by the Sun. The shadow cast by the Earth into space can be thought of as a cone that extends behind the Earth. The shape of the cone depends on the sizes of the Sun and the Earth and is shown in the figure below as the umbra.

Image Source: http://www.astronomy.ie/eclipse08.html

It is possible to find the distance at which the shadow of the Earth comes to a point. This distance depends on the angular size of the Sun in the sky. Because the shadow's geometry depends only on the angular size of the Sun, we can actually determine the shape of the cone by using any disk or ball-like object here on the surface of the Earth. If we hold up a quarter and block out the Sun's light, we are creating a shadow that looks similar to that of the Earth's shadow in space. The quarter's shadow is much smaller, but it has the same geometry. (Of course, NEVER look directly at the Sun. You can go blind. The first website listed above suggests you use the Moon instead since it has the same angular size of the Sun.) By moving the quarter forwards and backwards until it just blocks out the Sun's light, you will find that the pointy end of the shadow (i.e. the location of your eye) is 108 quarter diameters from the quarter. Same for a dime or any other ball or disk size. This means that the Earth's shadow is a cone that extends 108 Earth diameters behind the Earth.

By carefully watching the Moon pass through the Earth's shadow during a total lunar eclipse, the Greeks determined that the width of the cone was about 2.5 times bigger than the angular diameter of the Moon (which means the diagram above is not drawn correctly to scale). This still wasn't quite enough information because the Moon could be very large but close to the Earth or very small and far from the Earth and give the same observations.

The final piece of information that they used was that the Moon was the correct size and distance to cause a solar eclipse. In other words, the Moon just so happens to be the angular size of the Sun. Using the geometry in the diagram below, it's possible to find a relationship between the angular size of the Sun and Moon, the distance of the Moon, and the distance to the tip of the Earth's shadow.

We have two triangles above. One due to the Earth's shadow starting at the orbit of the Moon and extending to the tip of the shadow (orange) and one created by the Moon as viewed from the Earth (green). Both triangles are similar isosceles triangles with an angle of θ at their vertex. Again, this is because, viewed from Earth, both the Sun and Moon have the same angular size. We see that the bigger triangle is 2.5 times larger than the smaller triangle. This means that the distance from the Moon to the tip of the shadow is 2.5 times longer than the distance from the Earth to the Moon. OR, and here's where we get the answer, the distance from the Earth to the tip of its shadow is 3.5 times the distance to the Moon. We already know the distance to the tip of the shadow! It's 108 Earth diameters. Thus, 108 Earth diameters / 3.5 = 30.86 Earth diameters. At the time that they did this calculation, the Greeks had only a rough estimate of the size of the Earth. If we were to employ this answer using modern numbers, we would find: Earth's diameter = 7900 miles. The distance to the Moon is then 30.86 x 7900 miles = 243,770 miles. The actual orbit of the Moon is not circular, so its closest approach is 225,622 miles and farthest approach is 252,088 miles. That's not too shabby!

[Sources for this section: http://legacy.signonsandiego.com/news/science/20060713-9999-lz1c13laser.html]

Image source: http://spacespin.org/article.php/90962-nasa-goddard-shoots-moon

In modern times, we are able to bounce a radio or laser signal off of the surface of the Moon to find the distance. Since we know the speed at which light travels, it is easy to measure the time it takes for the light beam or radio wave to travel from the Earth to the Moon and back again. Simply the time for the round trip by the speed of light gives us the distance to the Moon. Things like the effects of relativity must be taken into account for very high accuracy, but this simple method gives you the general idea of how it's done. This method can give the distance to the Moon with an accuracy of millimeters!

[Sources for this section: http://www.astronomynotes.com/history/s7.htm, http://en.wikipedia.org/wiki/Nicolaus_Copernicus, http://en.wikipedia.org/wiki/Tycho_Brahe, http://en.wikipedia.org/wiki/Johannes_Kepler, http://en.wikipedia.org/wiki/Isaac_Newton, http://en.wikipedia.org/wiki/Giovanni_Domenico_Cassini, http://www.astro.princeton.edu/~clark/ParBkgd.html]

Measurements of the very large distances to the Sun and other planets were very difficult to make through geometric observations. Additionally, the Greeks believed that the Earth was the center of the Universe, so it was even more difficult for them to come up with a model that accurately described their observations of the heavens. It wasn't until Copernicus, Tycho Brahe, Kepler, and Newton that great strides were made in discovering the correct model of the solar system and the distances to the heavenly objects in it.

Copernicus (1473 - 1543) was the first to suggest that the Earth was not the center of the solar system, but that it was, in fact, the Sun around which the planets revolved. Born shortly after Copernicus' death, Tycho Brahe (1546 - 1601) compiled the most extensive and accurate measurements of the positions of the planets and stars of his time. This extremely accurate data attracted Kepler (1571 - 1630), who began to work as an assistant astronomer for Tycho. Kepler's detailed analysis of Tycho's planetary data lead to the discovery of Kepler's three laws. 1) Planets orbit the Sun in ellipses. 2) A line connecting a planet to the Sun sweeps out equal areas in equal times, a rule we now know describes the conservation of angular momentum. 3) A planet's orbital period is related to its distance from the Sun - the square of the period is equal to the cube of the distance (P^{2} = a^{3}).

Thanks to Kepler, it was suddenly possible to get the distance to a planet from the Sun just by measuring its orbital period. The only problem was that these distances were expressed in terms of the radius of the Earth's orbit (called an Astronomical Unit or AU), which was unknown. Kepler basically found the ratios of the planets' distances, but still needed the distance to another planet or the distance to the Sun to set his solar system to a scale. When Newton (1643 - 1727) developed his universal law of gravitation, he was able to explain why Kepler's Laws were correct. While Kepler had found his three laws by analyzing data, he did not have an explanation for the physics behind them. When Newton realized that mass exerts a force of gravtiy that influences how other objects move, he was able to explain the movements on the planets around the Sun as well as the elliptical orbits that they took. Now there was a firm mathematical and physical understanding of the motions of the planets. This information alone was not enough to find distances.

In 1673, Cassini (1625 - 1712) was the first to get a distance to another planet. He sent his assistant astronomer to French Guiana while he remained in Paris. Each one measured the observed position of Mars from their two locations on opposite sides of the world then compared them. The difference in Mars' position as observed from either side of the Earth Cassini to calculate the distance to Mars using the method of parallax. Cassini's distance was off by about 7%, but this was a first step in getting the true size of the solar system.

Image source: http://www.astro.princeton.edu/~clark/ParBkgd.html

[Sources for this section: http://en.wikipedia.org/wiki/Astronomical_unit, http://en.wikipedia.org/wiki/Aberration_of_light, http://www.jb.man.ac.uk/~slowe/transit2004/science_dist_radar.html]

Over the centuries since the 1700s, a number of methods to calibrate the distance to the planets were attempted. Astronomers repeatedly observed transits of Venus across the face of the Sun simultaneously from different locations across the Earth in attempts to measure the Sun's parallax. While good in theory, the observations proved to be problematic. An astronomer named Simon Newcomb (1835 - 1909) measured the parallax of the Sun by using a method called aberration. Aberration occurs when an observer is moving perpendicular to the light source. (I'm running out of time for this particular write-up, so you can read about it by clicking on the Wikipedia link above.)

Despite the different attempts to measure the distance to the Sun, a good distance wasn't calculated until 1961 during Venus' closest approach. The powerful radio telescope at the Jodrell Bank Observatory sent out pulses of radio waves towards Venus, then turned on its receiving mode and detected the pulses on their return from Venus. Knowing the speed of light, it was possible to determine the distance to Venus with a high degree of accuracy and finally determine the size of the solar system to scale.

Since then, two NASA satellites and two Soviet satellites have visited Venus. The time delay of the transmissions sent by the satellites back to Earth provides another direct measure of the distance to Venus.

[Sources for this section: http://www.bautforum.com/archive/index.php/t-2717.html, http://www.keplersdiscovery.com/Asteroid.html, http://en.wikipedia.org/wiki/Orbit_determination]

In the present day, as discussed above, it is possible to find distances to nearby objects in the solar system through radar bounce. This applies to nearby comets and asteroids as well. In fact, the technology has gotten so good that astronomers can actually create images of asteroids with radar. The emitted radio waves act like a flashlight on the asteroid, "lighting" it up so that we may see many details. The image below shows radar imaging done by scientists at JPL.

Image source: http://echo.jpl.nasa.gov/

There are a couple of problems with this method, however. First off, it only works with inner solar system objects that are close to the Earth. Radar can't be used with good results for objects in the outer solar system. Secondly, radar imaging is resource intensive and there aren't that many radar systems available to calculate distances to all the objects that we can observe. Even in these modern times, astronomers rely on methods to determine an object's orbit developed by Gauss (1777 - 1855) in 1800. His method requires three Earth-based observations of an asteroid in order to completely identify its orbital path. Even using the smaller telescopes available to astronomers today, it is very easy and convenient for astronomers to take multiple images of a comet or asteroid then apply the mathematics developed by Guass to uniquely determine its orbit.