Length-of-year lab

The purpose of this lab is to determine the length of a year based on how far a star moves from its apparent position at a constant time (sun-based) every night. The device that I plan to use to measure this change will measure the angle of elevation (above the horizon), and the 'flat' angle (left or right on the horizon), of the star relative to a given point on earth.

My measurement device consists of a camera attached to a tripod; I can center the star in the camera and measure the exact angles (both flat and vertical) of the camera on the tripod with a protractor. I could have used a simple block of wood, but it was easier to look through the camera and center the star in it, so I just used the camera. I can fasten protractors to the front and the side of the tripod, and using protractors attached correctly it is fairly easy to measure the relative angle of the target in the viewframe of the camera.

Data chart:

Date | Time | Angle of elevation | 'Flat' angle |

October 5, 2002 | 9:00 pm | 25^{0} |
0^{0} |

October 19, 2002 | 9:00 pm | 26^{0} |
4^{0} |

October 26, 2002 | 9:00 pm | 26^{0} |
6^{0} |

Before this data is interpereted, it is necesary to explain this rotation of the stars. Our days are determined by the sun; one day is the time from sunrise to sunrise, or from sunset to sunset, or some similar measurement; the standard division between days is, arbbitrarily, in the middle of the night, but the length of time remains the same. But the position of the stars repeats over one true, 360^{0} rotation of the earth. The difference between these rotations is represented in the difference between synodic and sidereal time; synodic time is time based on the apparant cycle of the sun, and sidereal time is time based on the apparent cycle of, well, basically everything else outside of our solar system. The difference between these times is roughly four minutes per day for earth, although it is often very different for other planets. Here are a series of pictures that help to explain how all of this ties together:

This illustrates the difference between sidereal and solar time. The earth always points in the same direction, towards the same stars, at a constant sidereal time, and it always points at the same angle towards the sun at a constant synodic time. But the angle to a given star changes, albeit predictably, at a constant synodic time, and the angle to the sun changes at a given sidereal time.

View of Earth's pole, at a constant synotic time of day

Our default unit of time is synodic time, so at a constant time, the sun is at a constant angle with the earth. This means that, at a given standard time of day, the stars that we are facing will appear to rotate. Synodic and sidereal time match up once per year, so the stars will complete one revolution per year.

Distances between objects in the sky are measured in terms of angles. Also, the star that I measured makes one 360^{0} circle around the north star in one year; it circles the north star and there are 360^{0} in a circle, and it returns to its starting point after exactly one year. I used these two facts to calculate the fraction of a year that passed, and then compared that to the number of days between the measurements to determine, in days, the length of a year. I only used the first and last measurements, because they are the most widely spaced and therefore the ones with, hopefully, the least relative error.

(90 - a) = arctan(1/6)

(90 - a) = 9.46^{0}

a = 80.54^{0}

a + a + theta = 180^{0}

2(80.54^{0}) + theta = 180^{0}

theta = 18.92^{0}

theta = 19^{0} (I could only effectively measure to two significant figures)

So the star went 19^{0} around the north star in the time of the measurement. The measurements that I used here were three weeks, or 21 days, apart. Calculating the length of a year for both of these numbers,

(X days)/360^{0} = (21 days)/19^{0}

X = 398 days

(X weeks)/360^{0} = (3 weeks)/19^{0}

X = 57 weeks

Well, as compared to the now-standard 365 days and 52 weeks, these measurements aren't perfect. The error on them is (1 - (365/398)) = 8%. The likely reason for this is the small block of time over which I watched this star. Given the measuring mechanisms at my disposal, I could only really measure accurately to the degree, and being off by even one degree would throw off the answer in the other direction. Take, for example, the example that the horizontal measurement was 5^{0} (instead of 6^{0}):

(90 - a) = arctan(1/5)

a = 78.69^{0}

a + a + theta = 180^{0}

2(78.69) + theta = 180^{0}

theta = 22.62^{0}

(X days)/360^{0} = (21 days)/23^{0}

X = 328 days

Most likely, the actual horizontal measurement was somewhere between 5^{0} and 6^{0}; the exact amount was just below the level of accuracy in measurement. The best solution to this would be to spread the measurement over a greater period of time, such as three or four months; unfortunately, we didn't have this much time to do this lab, so that option was not feasable in this situation. But if the length of time for the measurement, and hence the distance of the star, were to change, one would need to take into account that the stars are not coplanar. This math assumes that the degrees across the sky correspond to linear measurements; they, in fact, do not. They correspond to non-Cartesian, spherical distances. I haven't yet learned spherical geometry, however, so I wouldn't know how to apply it effectively; in any case, the measurements here are small enough that the resulting error is probably negligible. It is like negating the arc distance difference in the recent proof that { a_{gravity} = (V^{2})/R_{circle} } in our physics class, for example. Overall, this lab illustrated a reasonable measurement of the length of an Earth year.