Lecture 5
September 8, 2000
· Note positions of Moon on SC1 chart.
Today we apply knowledge of planetary orbits, including Kepler’s Laws, to some important problems in astronomy.
What time is it? If I ask you that, you will give me Central Daylight Time. How does that relate to the astronomical concepts we have discussed? Is it equivalent to the Hour Angle of the Sun, i.e. the number of hours until the Sun transits the meridian (morning) or the number of hours after it has transited (afternoon)? The answer is approximately, but not exactly. There are a number of fine points involved.
Time Zones Until about 150 years ago all time was probably local time, i.e. determined by the hour angle of the Sun (position of the Sun in the sky as locally observed). There was no need for watches to be synchronized in Iowa City and Des Moines, since there was no way of rapidly communicating or traveling between the two places.
But, Iowa City and Des Moines will have different local solar time. The longitude of Iowa City is 91 degrees 34 minutes, while that of Des Moines is 93 degrees 56minutes, so there is almost 2.5 degrees of longitude difference. This corresponds to 10 minutes of time difference, so it is not totally trivial.
The railroads (and probably the associated technology of the telegraph) represented the technological change that made a difference. A train could travel fast enough across Iowa to be able to let notice that clocks in Iowa City were different by 10 minutes. Since it would have been a nightmare to have every town along the Burlington railroad have its own time, there was a push to introduce Time Zones; big areas where the time was defined to be the same.
For the Central Time Zone, the time is defined as the Mean Solar Time for a longitude of 90°, i.e 1.5 degrees to the east of Iowa City. At the time the Sun (or any other astronomical object) transits at a longitude 1.5 degrees to the east, it will be 1.5 degrees to the east of the meridian here in Iowa City. Thus a sundial in Iowa City would always be early relative to standard time by 6 minutes. This is true in the mean, or on the average.
Throughout this course I have talked about means and averages. A mean solar day is 86400 seconds. In the previous paragraph I described average differences between a sundial and a clock on standard time. What is going on?
Recall what makes the difference between the sidereal day and the solar day. The solar day is what normal people think of as the day. The sidereal day is the more fundamental physical quantity (the time it takes the Earth to turn on its axis relative to the stars).
There are two reasons why the amount the Sun moves to the east is not the same each day. This means that there will be a difference between standard time, which is based on the mean motion of the Sun (one in fact speaks of the mean Sun) and actual solar time.
· The ecliptic is inclined 23.°5 with respect to the celestial equator. This means the daily solar motion (along the celestial equator) varies through the year.
>>>> Look at SC1 chart.
Kepler’s 2nd Law: The orbit of the Earth is, precisely speaking, an ellipse. Its eccentricity is 0.017. This means the greatest distance (aphelion) is 3.4% greater than its closest distance (perihelion). Kepler’s 2nd Law then says that the angular rate at which the Earth sweeps around the Sun in its orbit will be 3.4% greater. Thus true (or apparent) solar day is a certain fraction (or multiple ) of the mean solar day, and the differences between clocks based on the two can accumulate.
The difference between the apparent solar time and mean solar time (standard time) is referred to as the equation of time.
>>>>>> Plot of the equation of time
This difference can get large, as big as 17 minutes in early November, or 15 minutes in February. It is also the reason why, although the shortest day of the year is December 22, the date of earliest sunset is in the first week of December.
You could check out all of these facts by measuring the time of local apparent noon with a Gnomon or vertical stick, and comparing it with standard time over a period of several weeks.
Right now, sundials are about 2 minutes ahead of clocks. That will increase to 10 minutes by the end of the month.
Eclipses are striking natural phenomena. Total Solar Eclipses might be the most awe-inspiring natural phenomenon that doesn’t kill the observer. Stories of famous eclipses in history abound.
· Eclipses correspond to shadows cast by one astronomical object (either the Moon or the Earth) on another astronomical object (either the Moon or the Earth).
· They are a consequence of the fact that the orbit of the Moon is nearly in the plane of the ecliptic.
· The fact that the orbit of the Moon is not exactly in the plane of the ecliptic is responsible for the fact that eclipses are rare phenomena, and so scared the bejabbers out of people in antiquity, when their nature was not understood.
(This extended to the Athenians in the time of the
great war with Sparta, and the disasterous expedition to Sicily of the general
Nicias).
Lunar Eclipses occur when the Moon moves into the shadow
cast by the Earth into space. Everyone on the night side of the Earth then sees
the Moon move into darkness.
>>>>> Diagram showing lunar eclipse.
Solar Eclipses when the Moon casts a shadow on the Earth.
>>>>> Diagram of solar
eclipse
Because the Moon is a relatively small
astronomical body, the shadow it casts on the Earth is small, and a total
eclipse is seen only in a narrow band across the Earth.
At times of solar eclipse, the Moon blocks
out the light from the disk of the Sun (termed the photosphere) and you see
the faint, structured light of the upper atmosphere of the Sun, the corona.
Solar eclipses occur because of the
remarkable coincidence that the angular
size of the Sun and Moon as viewed
from Earth are the same, even though they are a factor of 400 different in
physical diameter.
Question for audience: what is the explanation for an annular eclipse in which the Moon doesn’t quite cover up the disk of the Sun, but a ring
(annulus in Latin) of light shines around the Moon. Is the Sun bigger in size
at that time? Does the Moon shrink like
a prune?
The orbit of the Moon is inclined with respect to the plane of the ecliptic by 5°. The orbit intersects the plane of the ecliptic at two points called nodes (the ascending node and the descending node). The line in the plane of the ecliptic which connects them is called, reasonable enough, the line of nodes.
This inclination of 5° is actually easily visible if you make careful, naked eye observations of the Moon and its position relative to the stars.
This inclination is very important with regard to eclipses; it means that they do not usually occur.
>>>>>> Transparency with Moon’s orbit.
Because the Moon is up out of the plane of the ecliptic most of the time its shadow passes above or below the Earth at times which are favorable for a solar eclipse, and IT is above or below the shadow of the Earth at times which are favorable for a lunar eclipse.
For an eclipse to occur, the Moon must be in the right position (New Moon for a solar eclipse, Full Moon for a lunar eclipse) and the Moon must be near one of the nodes. An equivalent statement is that the line of nodes must be pointing towards the Sun.