Saturday, October 24, 2009

Timeline of the Julian calendar reforms

239 BC Ptolemy III issues the Decree of Canopus

63 BC Caesar elected Pontifex Maximus

49 BC Civil War

48 BC Caesar meets Cleopatra

46 BC Forum of Caesar

46 BC Julian calendar

44 BC Assassination of Julius Caesar

42 BC Apotheosis of Julius Caesar

30 BC Cleopatra's death

26 BC Alexandrian calendar

After examining this timeline one has to ask if the Julian calendar reforms were incomplete due to the assassination of Julius Caesar?

Thursday, October 22, 2009

Are the number of days per month rational?

During the last week I have been studying the computation of the Julian Day Number that astronomers use to keep track of time. To do this one needs to know how to convert from month and day to day of year. The number of days in a month is quite irregular and one wonders how this came about. Most of the features of our present calendar are due to the reforms of Julius Caesar in 46 BC. Caesar turned to an Egyptian astronomer, Sosigenes of Alexandria, for assistance in correcting the errors in the Roman calendar at that time. The lengths of the months and leap day date from that time. At first the length of the months seem quite arbitrary but if one considers the period from March to the following February the pattern if more regular. Placing the leap day at the end of this period is consistent with December being the "10th month."

The problem of designing a calendar with twelve month and 365 days is how to distribute the odd 5 days. The pattern seems to be consistent with using multiples of 30 7/12 rather than the more obvious 30 5/12 as seen in the calculation below. An irregularity is that the sequence is shifted by one month.

In order to do this one needs to be able to perform integer division and this can be done quite easily using multiplication tables and Sosigenes would have been quite capable of doing this. This gives us the sums of the days of the months. The formula for computing the sums turns out to be rather simple and allows us to derive a formula for January through December counting January as the first month. This simplifies converting month and day to day of year with the inclusion of a leap day in leap years.

Sunday, October 4, 2009

Space Elevator Thermal Cycling

As a diversion from ancient history we might look to the future for a change.

A space elevator is a mechanism which claims to provide easy access to space. It is basically a cable with a counterweight that rotates with the Earth as it turns about its axis. The cable is heated by sunlight which varies as the elevator rotates. So there will be daily variations in the temperature of the cable also know as thermal cycling. Why study themperature variations? Most materials expand as they are heated and since the space elevator extends beyond geosynchronous orbit, 36,000 km above the Earth's surface, the change in length can be considerable.

It has been suggested that carbon nanotubes might be strong enough to create a cable that is self supporting. The carbon nanotubes are similar in structure chemically to graphite. Since the planes of carbon atoms form tubes, we would expect the density of a cable to be less than that of graphite. But one would expect the thermal properties per unit mass to be about the same.

So to approximate the thermal properties of a cable, we will assume it is made of graphite and behaves like a black body. The following calculation shows the daily temperature variation of the cable as it rotates about the Earth at the time of the equinoxes when the Sun is above the Equator. The cable absorbs energy from the sunlight which strikes it and radiates heat at a rate depending on its temperature. The method is similar to that used for simple climate models of the Earth.

The images below are a slightly condensed version of the program used to do the calculations. A simplifying assumption was that a section of the cable had a uniform temperature throughout. (For a better view of the images double click on them.)

The calculations above indicate the daily changes at the time of the equnoxes. There are two minimums because when the cable aligns with the direction of the Sun it is essentially in its own shadow and only experiences cooling. The shifts in times of the minimum and maximum temperatures for thicker cables can be attributed to thermal inertia.

The daily variations for different times of the year have to take into consideration changes in the angle of the Sun relative to the Equator. There is surprisingly little change though in the thermal cycles throughout the year. The reason is that the projection of sunlight onto the surface of the cable doesn't change that much. It is on the order of 10% as can be seen from the necessary change below. ι_S is the inclination of the rotational axis of the Earth, 23.5°. The first formula gives the declination of the Sun in terms of the angle φ which is the angle of the Sun in the ecliptic plane. θ is the angle of the cable relative to the Sun in the plane of cable's rotation.