I wrote up a mix of 2 and 3 from the work sheet on LST. Hope you don't mind
Abstract
Palomar is an important observatory for the Caltech astrophysics community. Because of this, we seek to constrain the visibility of the star Tau Ceti from the Palomar Observatory over the course of the year. We determine first whether its declination allows it to be seen and then when it can be observed. We also examine the elevation from month to month. We find that Tau Ceti is observable from Palomar during certain times of the year.
Introduction
Tau Ceti is a star located in the constellation Cetus at RA = 1:44 (25.0167 degrees) and Dec = -15.9375 degrees in the southern hemisphere of the celestial sphere. The Palomar Observatory is located at 33.358 degrees north and 116.864 degrees west. It is important to understand how and where Tau Ceti can be observed from Palomar in order to study the star. To do this, we determine the elevation of Tau Ceti when it is on the meridian passing through Palomar as well as the times that it is observable optically.
Methods
First we determine what the maximum elevation of Tau Ceti is from Palomar. The elevation of a star is the angle it forms with a line connecting the centre of the earth to the observer's local horizon. We calculate this using the equation:
Where a is elevation, delta is declination, phi is the local latitude, and H is the hour angle (the difference in LST from the meridian at the time of viewing). When the star is on the meridian, it is clear that H = 0. We then get:
Since one local sidereal day is a 360 degree revolution of the earth, or the time it takes for the meridian to re-align with the object in question, it is obvious that this is the elevation any time Tau Ceti is on the meridian and also that this is the maximum elevation of Tau Ceti as observed from Palomar.
Next we notice that if we set a = 0, or say the star is at the horizon, we can find the hour angle the time the star rises. This is given by:
Dividing this by 15, we get H = 6.72 hours, which we round down to 6.5 to account for the fact that stars at the horizon are invisible to the ground based observer.
Conclusions
Plotting the elevation as a function of time for the 20th of each month of the year, we get the following graph. We choose the 20th of each month for convenience's sake as the vernal equinox is on 20 March.
This shows, as expected from considering the problem, that at LST = 1:44, or when Tau Ceti is aligned with the meridian, the elevation is always 40.71. The graph also shows the approximate times at which Tau Ceti is aligned with the meridian in UT for planning of observations. Now plotting the time (UT) of Tau Ceti's maximum elevation against the passing months, we get the following graph:
As expected, the change in time is a linear relationship to the time of year. The error bars of this graph mark the total time that Tau Ceti is above the horizon. Regions plotted in red are the times before and after sunrise and sunset, respectively. These are the times that Tau Ceti is actually visible from Palomar because interference from the sun is absent. Predictably, this range is from July to January, with maximum visibility in the October-November region where it is dark the entire time that Tau Ceti is above the horizon. This agrees with observation data that indicates Cetus is best visible in late autumn.
Acknowledgments
We thank Professor Johnson and Jackie for providing us with this problem to solve. Also, Wikipedia for the image describing horizontal coordinates and Professor Harold Geller for his notes on calculating the altitude of a star. We also thank Microsoft Excel for cooperating long enough to produce these graphs.
Lovely graphs! Can you come up with a way to figure out the time at which Tau Ceti is at the meridian throughout the year without using fancy formulas, just spatial reasoning? Same for elevation - can you figure out the elevation when it is at the meridian (i.e. at its highest elevation) just using the latitude of Palomar and the declination of the star, and no fancy formula? The fancy formulas are wonderful and very useful for computing, but it's good to be able to bootstrap it too. Post-apocalyptic astronomy skills :P (but actually also very useful on a day-to-day basis so you don't need a computer every time you want to figure out if you can observe something).