Backyard astrophysics:
Kepler's laws
For a number of years the idea of "backyard astrophysics" has been in the back of my mind. What observations can we as amateur astronomers make to demonstrate astronomical knowledge? One kind of observation I make quite frequently is to take an image of the Sun to count sunspots and measure their positions. Previously [1], I had shown how the brightness of the Sun depends on its altitude above the horizon, and what this tells us about the Earth's atmosphere.
If you make fulldisc drawings of the Sun yourself, you know that you have to adjust the projection such that the solar image is not only in the right place, but also of the right size to fit the standard drawing template. Similarly, When I measure sunspot positions in my digital images I have to determine the centre and radius of the solar image. Since my focal length is fixed, I measure the size of the solar image rather than adjust it to fit a prepared template.
When I reduced the images day after day and week after week, I noticed that the measured size of the Sun did not merely vary at random by a pixel or two. There seemed to be a systematic change over the weeks. I had serendipitously rediscovered Kepler's laws of planetary motion, in the first instance that the Earth does not revolve in a perfect circle around the Sun. Scientists call it serendipity when they discover something by accident. The term derives from an old Persian tale of the three Princes of Serendip  a country we today call Sri Lanka; the legendary princes seemed to have a knack for finding out things they had not set out to discover.

Plotting about three years' worth of data one can very clearly see a sine curve (Fig. 1). This is a very small modulation of 8 pixel (average to extreme) on top of a large solar radius of 494 pixels. One of the instances when the radius has the average value can be read to be JD 2453648. From these starting guesses it takes a few attempts at plotting data minus fit to find that the period is 2π times 58 days.
With a spreadsheet it is then possible to calculate the rootmeansquare (rms) of the deviations of data from fit: For each data point the difference between measured and fit value is squared, all those squares are averaged, and the root of the average is taken. This is a measure of how well the curve fits the data. We vary the fit parameters to improve the fit, i.e. to reduce the rms. The best fit for the data here is when the rms is 0.83 pixel, and the fitted curve for apparent radius ρ in image pixels as a function of Julian Date then is:
ρ/pix = 493.55 + 7.9 sin[(JD/d  2453647.0) / 58.3]
Nice curve, but what does it mean? How does this relate to Kepler's laws [2]:
 The planets move in ellipses with the Sun in one of their focal points.
 The radius vector of a planet encompasses equal areas in equal time intervals.
 The squares of the revolution periods of two planets relate to each other like the cubes of their orbits' semi major axes.
Fig. 2: An elliptic orbit around the Sun. The size of the ellipse is measured by the semimajor axis a, its shape is measured by the numeric eccentricity e. The ellipse has two focal points, with the Sun in one of them. The ellipse shown here has an eccentricity of 0.6. 
Fig. 2 shows an extreme ellipse like a comet might use for an orbit. Perihelion  when the planet is closest to the Sun  and aphelion  when it is furthest from the Sun  both occur on the major axis, top right and bottom left in Fig 2. The Sun is not in the centre of the ellipse, but in one of its foci, a distance e (the numeric eccentricity between 0 and 1, 0.6 in Fig. 2) times a (the semi major axis) from the centre. For e = 0 the ellipse becomes a circle, as e approaches 1 it turns into a parabola.
Our primary results was that the apparent size of the Sun is changing between two extremes, perihelion and aphelion. The ratio of these two extremes can be expressed as a ratio of the sine of the Sun's apparent radius. Fortunately, the apparent radius is so small at 0.27° that the sine function can be approximated in a very simple manner, and the numeric eccentricity is simply the ratio of the size modulation divided by the size:
e = 7.9 / 493.55 = 0.016
This is a tiny eccentricity, but it translates to a difference of 6.5 % in the brightness of the Sun between perihelion and aphelion. Looking at the real orbit of the Earth in Fig. 3 you cannot tell the difference from a circle. But the small separation between the centre and the focus, the small offset of the Sun from the centre can just be seen.
Our sine curve fit to the apparent radius of the Sun reaches a maximum a quarter period after the fitted reference date. That is to say, the Earth went through perihelion at
T = 2453739 d = 20060103
Fig. 3: The Earth's orbit around the Sun according to my measurements. Note the extra cross at the centre of the orbit just below the Sun. At the time of vernal equinox the Earth is on the left. The Earth moves counterclockwise once a year. At summer solstice it is at the bottom, at autumn equinox on the right and at winter solstice at the top. Shortly after that it passes through perihelion (the point closest to the Sun), indicated by the line from the Sun pointing just left of upwards. This happens in early January. 
Our data do not say very much about Kepler's second law. It is a statement about how the planet moves faster when it is closer to the Sun. We would have to measure the position of the Sun over the year to confirm this.
But we can talk a little bit about the third law, because we have measured the revolution period of the Earth, what we call a year. We find that the length of the year and the mean daily motion of the Earth around its orbit are:
P = 2π 58.3 d = 366.3 d
n = 360° / P = 0.983°/d
= 1.987 10^{7} rad/s
Kepler wrote down his laws before Newton had told us all about force and counterforce, and how the gravitational force diminishes with the square of the distance. Kepler's second law is a consequence of the balance between the attractive gravitational force and the centrifugal force. Knowing the law of gravity we can write Kepler's second law in terms of the gravitational constant G and the combined mass of the Sun (M) and the planet (m):
n^{2} a^{3} = G (M + m)
Wow! If we plug in the astronomical unit and the universal constant G, we have in effect weighed the Sun! It is not a weight, but a mass; and it is not the mass of the Sun but the combined mass of Sun plus Earth (and Moon). But it is still amazing that we can do this just by measuring how long a year is. With [3]
a = 149.59787 10^{9} m
G = 6.672 10^{11} m^{3} / (kg s^{2})
we find
(M + m) = 1.98 10^{30} kg
How well have we done compared to current best knowledge [3]? Our eccentricity of 0.016 is 4 % smaller than the real value of 0.0167 (for 2000). The 2006 perihelion was actually on 20060102, one day earlier than we figured. The year we have measured is the anomalistic year (perihelion to perihelion). Its true length is 365.259635 d, our value is 1 d (0.3 %) too long. The mass of the Sun is 1.9891 10^{30} kg, the mass of the Earth plus Moon is about 330,000 times less and hence insignificant. Our value for the solar mass is then 0.5 % low.
The accuracy of our values is remarkable. Historically, much depends on the accurate measurement of the astronomical unit and of the gravitational constant. Our length of the year and time of perihelion are not terribly accurate, but it is clear that perihelion is in the middle of northern winter. Hence, summer and winter must be determined mostly by factors other than proximity to the Sun.
The Sun radiates away energy at a rate of 3.846 10^{26} W [2]. If it made heat like I do at home, it should be a mix of 20 % methane and 80 % molecular oxygen (by mass). This could burn to carbon dioxide and water, releasing 6.12 10^{30} kWh. That seems a lot to the domestic consumer, especially when converted to £ or € at current prices. Indeed it would last the highdemand consumer Mr. Sun for 1800 years. But our recorded history is longer than that and life on Earth has existed for over 3500 million years. So the Sun must be doing something much, much more efficient to make heat and light. The processes known to atomic physics (aka chemistry) are not sufficient, it took insight into nuclear physics to understand how the Sun manages to shine for as long as it has. The Sun does not have enough uranium to use nuclear fission and plutonium does not occur naturally anyway. But hydrogen  which makes up most of the universe  happens to be a good fuel for nuclear fusion. The result is a substance we call helium, because it was first detected in the spectrum of Helios, the Sun.
References
 Horst Meyerdierks, 2008, Atmospheric extinction, ASE Journal, 56, 9
 Albrecht Unsöld, Bodo Baschek, 1999, Der neue Kosmos, 6. Auflage, Springer
 1998, Astronomical Almanac for the year 2000, United States Naval Observatory and Rutherford Appleton Laboratory