# 1.19 Testing a Hypothesis

Very few scientists create new theories or discover laws of nature. The day to day business of science consists of gathering data and consolidating existing knowledge. A hypothesis is a proposed explanation for a set of measurements. The mathematical formulation of the hypothesis is called a model. Scientists test hypotheses by acquiring data of more and more scope and accuracy.

Image of heliocentric model from Nicolaus Copernicus' "De revolutionibus orbium coelestium". Click here for original source URL.

Here is an important example of hypothesis-testing from the history of astronomy. The Copernican hypothesis was that the planets travel around the Sun on circular orbits. The planet velocity is the same at every point in a circular orbit. Therefore the velocity of Mars, for example, as seen from the Sun should not change. (The actual measurement is an angular velocity on the sky, which can be converted into a space velocity knowing the distance to Mars. An additional complication comes from the fact that the measurement is made not from the Sun but from the moving Earth.)

Kepler knew that the angular motion of Mars was not constant and he hypothesized that the orbit was elliptical. We can use Kepler’s laws to calculate what this implies about the velocity of Mars in its orbit. Mars has an orbital eccentricity of 0.093. So if r is the mean distance from the Sun, the orbit varies from 1.093r to 0.907r. Elliptical orbits vary in speed with distance from the Sun according to ν ∝ √ r, so the velocity varies smoothly from √ 1.093 ν = 1.045 ν to √ 0.907 ν = 0.952 ν throughout the orbit. In other words, the orbit velocity varies from 4.5% above the mean to 4.5% below the mean; a total variation of 9%. So to detect the elliptical motion — and rule out the hypothesis of a circular orbit — requires measurement with at least this accuracy. Scientists can improve on the testing of a hypothesis either with more measurements, or more accurate measurements, or both.

Now consider an example of great current interest — the detection of extrasolar planets. The wobble motion of a star provides an opportunity to detect an unseen planet by its gravitational influence. Suppose we accurately observe the velocity of a star just like the Sun and look for the undulation in velocity caused by an orbiting planet. If the data show a clear variation the we can calculate the mass of the planet required to cause the variation. In this case, astronomers can confirm the hypothesis that a planet orbits the star and measure that it is a planet like Jupiter.

The radial velocity method to detect exoplanet is based on the detection of variations in the velocity of the central star, due to the changing direction of the gravitational pull from an (unseen) exoplanet as it orbits the star. When the star moves towards us, its spectrum is blueshifted, while it is redshifted when it moves away from us. By regularly looking at the spectrum of a star - and so, measure its velocity - one can see if it moves periodically due to the influence of a companion. Click here for original source URL.

The example of detecting extrasolar planets can be used to look at hypothesis-testing in more detail. The data might not be accurate enough to test the hypothesis of a Jupiter-sized planet. It also might not be possible to test a hypothesis when the effect being looked for is very small. The reflex motion of an Earth-sized planet is only 0.09 m/s. This Doppler motion would be a curve with 100 times less amplitude than the curve for Jupiter, completing a cycle each year. As you can imagine, a large amount of extremely accurate data would be required to test the hypothesis of an Earth-sized planet.

This is an image of the planet?Uranus?taken by the spacecraft?Voyager 2?in 1986. This is how Uranus would look to humans in visible light. But since it orbits the sun every 84 earth-years, we need nearly a century of observation to gain a substantial understanding of this planet. . Click here for original source URL.

Sometimes astronomers suffer from insufficient data. Observations spanning a 5-year interval are insufficient to see an entire cycle of reflex motion. Once again, the hypothesis of a Jupiter-sized planet cannot be tested. The situation would be even worse if we were looking for a planet even further from its star. Uranus has an orbital period of 84 years, so nearly a century’s worth of data would be needed to look for such a planet. A final possibility is that accurate data shows not significant variation from a constant velocity, at a level smaller than the prediction for a Jupiter-sized planet. This data allows us to reject the hypothesis of a planet with that mass.

The general description of hypothesis-testing has three features: the number of observations, the error in each observation, and the amount by which each observation deviates from the prediction of a hypothesis or model. Mathematically, we can say:

c^{2} = Σ [(x_{data} - x_{model})^{2} / σ (x)^{2}]

In this equation, take the square of the difference between the data and the model at every point, divide it by the square of the error in the observation, and sum this quantity over all data points. (A sum over quantities is represented in mathematics by the Greek letter capital sigma, Σ.) Essentially, this is a calculation of the amount by which the data deviates from the model. The quantity x could be any measurable quantity: light intensity, velocity, magnetic field strength, and so on. A good model closely represents the data and has a low value of c^{2}. A bad model poorly represents the data and has a high value of c^{2}. This method of testing a model or a hypothesis is at the heart of what scientists do.