ASTR 1210 (O'Connell) Study Guide

4. MOTIONS IN THE SKY

Star trails over the Himalayas in a
2-hour exposure (Anton Jankovoy).

A. Motions of Bright Objects

If you're patient for about 20 minutes, you can easily detect the diurnal rotation of the whole night sky by comparing the position of the stars to a fixed foreground object (e.g. a tree). If you have a digital camera capable of taking exposures longer than a few minutes, you can readily take images showing star trails, like the one at the top of the page.
Celestial motions may be complicated, but they are repeatable over periods of months to years, and their cyclic nature was quickly recognized by many cultures. It was this feature that encouraged people to search for a deeper understanding of them.
The cycles also naturally became the basis of practical calendar systems. For instance, Western calendars are based on 7-day weeks (one day for each of the bright moving lights in the sky: the Sun, Moon, and five planets) and 30-day months (one lunar cycle).

Thought experiment

Imagine that we couldn't easily see the cyclic sky phenomena---that, for instance, the Earth were perpetually shrouded in fog-like clouds. In these circumstances, it would have been difficult or impossible to recognize the regularity in the sky cycles, and astronomy might well never have developed. That would have inhibited the development of fields such as physics as well and perhaps all of science. What kind of world might we live in today if that one piece of the scientific enterprise had been missing? In a somewhat different context, the famous science fiction story "Nightfall" by Isaac Asimov describes the terrible fate of a society whose planet experiences night only once in a few thousand years.

B. Explanation of Motions

• The Greeks built mathematical models of the sky based on the concepts in plane and spherical geometry which they had developed. These reduce a bewildering array of complex phenomena to a much simpler set of mathematical concepts.

• Although it is fashionable today to criticize "reductionism" in science, there would, in fact, have been very little progress in understanding nature (or in converting that understanding to useful technologies) without the tremendous simplifying insights of reductionism.

• Intrinsic motions of the objects themselves with respect to one another

• The motion of the observer, or the platform on which he/she is standing---in this case, the Earth
  Because the motions of objects in the sky can be induced by movement of the observer, we call them apparent motions.

• THE EARTH WE LIVE ON IS A ROUND, TILTED, SPINNING, MOVING PLATFORM.

• Although it is simple to state, it is very difficult for most people to visualize this situation. You instinctively feel yourself to be "upright" and at rest on a flat, stationary Earth, but your instinct here is seriously misleading. This is one of the main reasons it took humans nearly 2000 years to fully accept this explanation, which became the first major discovery of the scientific age (by Copernicus, see Study Guide 6).

C. Effects of Earth's Shape and Spin

• Earth is a sphere, with a diameter of 7900 miles.

• Half of Earth is always in sunlight; half is always in shadow. We experience night on the shadow side (away from the Sun).

• Daylight = sunlight scattered by the Earth's atmosphere whenever the Sun is above the horizon. The daylight glare overwhelms the light of the planets and stars, so we cannot see them (though they are, of course, always there). The glare rapidly declines as the Sun dips below the horizon ("dusk"). "Civil twilight," the point at which sunlight is just barely perceptible, occurs when the Sun is 6 degrees below the horizon.

• Earth spins on its axis with respect to the stars once in 23 hours, 56 minutes (one "sidereal day").
  This causes the apparent diurnal rotation of the whole sky. (The universe is not moving around the Earth once per day.)
  If the Earth were assumed not to be rotating, then the diurnal movement of the sky would necessarily imply that it was at the center of the universe. This was the first assumption discarded by Copernicus on his way to rejection of the ancient "geocentric" cosmologies.

• The spin is counterclockwise (eastward) as seen from above Earth's N pole; this means that the apparent rotation of the sky seen from Earth's surface is westward
  • The left-hand panel of diagram above shows Earth viewed from above its North Pole. The Earth rotates counterclockwise in this diagram, carrying observers with it.
    • The positions where observers on the equator are experiencing sunrise, noon, sunset, and midnight are marked. Mean local times of day corresponding to these points are 6 AM, 12 noon, 6 PM, and 12 midnight, respectively.

  • The right hand panel shows the local "horizon plane," which is the plane "tangent" to Earth at your location. You can see (in principle) objects above the plane but not below.
    • Note that half of the entire celestial sphere is always above your horizon plane. However, the horizon plane (and visible hemisphere) is different at each location on the Earth's surface.

  • The horizon plane sweeps across the sky as Earth spins. Astronomical objects appear to move in the opposite direction (shown by the arrows in the figure). Objects rise over the eastern horizon and set toward the western horizon.

• By combining the concept of the horizon plane with the definition of local time in the left-hand panel, you can visualize what parts of the sky are observable at any given time.

D. Effects of Earth's Motion in Orbit

• Earth is a planet moving in orbit around the Sun
  • Its orbit is nearly circular (distance to the Sun varies only 3.4%), with a mean radius of 150,000,000 km or 93,000,000 miles. The mean radius is defined to be the Astronomical Unit (AU).

  • Its orbit lies in a plane called the "ecliptic" plane. Seen face-on, the orbit is technically an ellipse but deviates only slightly from a circle. Seen edge-on, the orbit is a thin line.

  • Earth orbits the Sun in 365.25 days (one year) moving at ~66,000 mph. Its motion is counterclockwise as seen from above the N pole

  

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• Stars visible at night are those "opposite" the Sun. See figures above. The night side of Earth is that opposite the Sun. So, in May, the constellation Scorpio will be prominent in the night sky, while in November, it lies in the direction of the Sun and therefore is not visible because of the atmospheric glare.
  • Warning! the drawings in this guide, and most others you will see in this course either in the text or in class, are grossly distorted and not to scale! Don't take them literally. In contrast to the drawing above of the Earth's orbit, the real orbit is 100 times the diameter of the Sun; the Earth is 100 times smaller than the Sun; the stars are vastly distant from the Earth's orbit; and the stars in a given constellation are not necessarily near one another in space.

  • Obviously, no one could produce or sensibly view a figure like this drawn to actual scale.

  • We are viewing Earth's orbit here from an inclination such that it looks highly elliptical, whereas it is nearly circular seen face-on.

• The Earth's motion around the Sun is counterclockwise in the drawings above (drawn from a viewpoint that is north of the Earth's orbital plane).
  • This orbital motion produces an apparent eastward "reflex motion" of the Sun in the sky against the stellar reference frame as seen from the Earth.

  • From the drawings, we find that the Sun in November would be seen in projection in the direction of Libra, while in December, 30 days later, the Earth has moved such that the Sun is seen in projection against Scorpio, which is approximately 30 degrees east of Libra. Therefore, the average "reflex motion" of the Sun with respect to the stars as seen from Earth is 1 degree per day eastward.

• The ecliptic path:
  • The Sun's annual path seen from the moving Earth against the stars is called the "ecliptic path." This is simply the projection of the ecliptic plane on the celestial sphere.

  • The "Zodiac" is the set of constellations through which the ecliptic path passes. Those are the only constellations which have been illustrated in the drawings above. Of course, there are 76 other constellations not shown.

  • The only astronomical significance of the 12 Zodiacal constellations, therefore, is accidental: they happen to lie on the projection of the plane of the Earth's orbit. They are not the largest, brightest, or most prominent constellations (in fact a number are quite inconspicuous).

  • Practice exam question: what Zodiacal constellation will be best seen at midnight in August?

• "Solar" vs. "sidereal" days: The daily motion of the Earth in its orbit means that the Earth must spin a little more than once on its axis to bring the Sun back to the point of local "noon" (i.e. halfway between rise and set). The extra amount is 4 minutes, on average.
  • This accounts for the difference between the sidereal spin period of the Earth (23 hours, 56 min) and the average elapsed time between successive noons (24 hours, the "solar day"). Our ordinary clocks, of course, are set such that the time between successive noons is defined to be 24 hours. But the stars come back to the same position in the sky as seen from Earth 4 minutes earlier each day. This means, for instance, that the stars you see at midnight in a given month have the same configuration as they did at 10 PM one month earlier or 8 PM two months earlier.

  • See the effect of Earth's motion during one sidereal day here.

E. Tilt of Earth's Axis

• The polar rotation axis of the Earth is not perpendicular to its orbital plane. It is tilted by 23.5 degrees from the vertical, or, equivalently, it is tilted 66.5 degrees out of the plane. See figure below (again, exaggerated for clarity):
  
• The rotation axis is fixed in direction with respect to the stars, not with respect to the Sun. This means that, as Earth orbits the Sun, the axis continues to point in the same direction in 3D space. This implies that the north pole is sometimes "tilted toward" the Sun, sometimes "tilted away." In the illustration above, the North Pole is shown at its maximum tilt away from the Sun (assuming the North Pole is at the top).
  The fixed axis direction of the Earth during a year is illustrated in the drawing in Section F below.

• This, in turn, implies that the Sun, as viewed from the Earth, will appear to move north and south of the celestial equator through the year by a maximum of + or - 23.5 degrees. In the drawing above the Earth is situated such that the Sun will be seen at its most southerly position during the year.
  The total amplitude of the Sun's N/S swing across the celestial equator is 2 x 23.5 = 47 degrees.

• Equivalently, the ecliptic path is inclined 23.5 degrees to the celestial equator. The following diagram shows the ecliptic and the equator on the celestial sphere (see Guide 3 for the definition of the celestial sphere).
  
• The ecliptic crosses the celestial equator at two points called equinoxes.
  • When the Sun is at an equinox, night and day are each 12 hours long at all latitudes. The "Vernal Equinox" occurs around March 21, while the "Autumnal Equinox" occurs around September 21.

  • The Sun is at its greatest distance from the equator (23.5 degrees) at the solstices ("sun stationary"), which are approximately June 21 and Dec 21. At these times, one hemisphere experiences its longest day, the other its shortest.

  • The diagram below shows the position of the Sun on the celestial sphere with respect to the equator plotted against date. The longest and shortest days in the northern hemisphere occur at the June and December solstices, respectively.

  

F. Astronomical Effects on the Weather

• The Sun's north/south distance from the celestial equator determines:
  1. The number of hours of daylight at a given latitude and

  2. The angle at which sunlight strikes Earth's surface at a given latitude. More energy is incident per unit area per second on the surface the closer are the rays to perpendicular to the surface.

• The Sun's distance from the celestial equator thus determines insolation, or the amount of solar energy deposited on a given area of the Earth's surface during 24 hours at a given latitude.

• This differential heating is responsible for the seasons. The "official" beginning dates of spring, summer, autumn, and winter correspond to the Vernal Equinox, Summer Solstice, Autumnal Equinox, and Winter Solstice, respectively.

• The diagram below shows how the hours of daylight (represented by the length of the white lines on the illuminated half of the Earth) and the solar angle of incidence at different latitudes correlate with the date. A close-up of the situation is shown here.
  
• The change in the geographic shadow distribution caused by the tilt is quite dramatic (even though the shadow always covers exactly one half of the Earth's surface). Here are two images of the way the shadow is distributed at two different times of year. The Earth's surface moves eastward (toward the right in the diagrams) through the shadow as the Earth rotates. You can immediately tell from the image which latitudes are receiving more sunlight in a 24 hour period. Click on the thumbnails for an expanded view.

  2 PM EDT August 1	2 PM EST December 1

• Therefore, the seasons are caused by the tilt of the Earth's pole to its orbital plane, not its distance from the Sun.
  If the seasons were a distance effect, winter, for instance, would occur simultaneously in both the southern and northern hemispheres. But instead, the seasons differ by 6 months in the two hemispheres.
  There is a small annual change in the distance of the Earth and the Sun caused by the finite ellipticity of Earth's orbit. This amounts to only + or - 0.017 AU. The change in distance has a real, but small, effect on the seasons. The Earth is nearest the Sun in January, one of the coldest months in the northern hemisphere. The seasonal changes therefore are milder in the northern hemisphere and larger in the southern hemisphere than they would have been if Earth's orbit were perfectly circular.

• Local Weather Extremes:
  The change in insolation at the mid-latitudes (like Charlottesville's) would not by itself cause the extremes of weather we experience here (e.g. frigid winter temperatures). Instead, it is mixing by wind currents of air masses from different latitudes that produces the extremes. In the case of winter, the north polar regions experience perpetual night starting in September and lasting until March. The resultant drastic atmospheric cooling creates the icy air masses that can produce very cold weather in the US if they move over us.

• The Milankovitch Effect
  The polar tilt and the properties of the Earth's orbit change slightly with time. Because of the gravitational effects of the other bodies in the solar system, they all exhibit secular (i.e. long-term) variations. Small changes in the polar tilt, the size of Earth's orbit, or the timing of closest approach to the Sun can have cumulative effects which drive major changes in climate over long time periods. One of these small effects is precession, discussed in Study Guide 5.
  There is good geophysical evidence for a correlation between such astronomical changes and the climate on Earth, particularly the ice ages. This is known as the "Milankovitch Effect."

Bennett textbook: Ch. 2.1, 2.2
Study Guide 4
Optional exercise (this is a PUZZLAH for Wednesday, January 30):
Measure the angular diameter of the Sun as follows. You can do this on any day (clear or partly cloudy) when you can see the disk of the Sun. Don't look directly at the sun. Instead put your hand (palm out & fingers together) in front of your eyes at arm's length. Close one eye. Then, carefully fold down fingers, keeping the Sun's light covered until you can't remove any more fingers without letting sunlight pass. Remember that your index finger will subtend about 1 degree in width when held at arm's length.
• How wide is the Sun in degrees?

Work alone or in a small group, as you wish, and be ready with your answers on Wednesday, Jan. 30..

Bennett textbook: CH 2.3 (lunar phases, eclipses), 3.1 (ancient astronomy)
Study Guide 5
Aztec Calendar Stone
Optional reading: eclipses and Stonehenge
Optional reference on Mayan astronomy: Skywatchers of Ancient Mexico by Anthony F. Aveni (Univ. of Texas Press, 1980/97).

Interactive Earth & Moon viewer (includes software & add'l links)
JPL Tutorial on Earth motions and astronomical reference systems
Ice Ages (Wikipedia)
Lecture on the Milankovitch Effect (J. Frogel, OSU)
Milankovitch Cycles (Wikipedia)

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Last modified February 2013 by rwo

Text copyright © 1998-2013 Robert W. O'Connell. All rights reserved. Zodiac and Earth tilt diagrams copyright © by Nick Strobel. Equator/ecliptic diagram copyright © Edmund Scientific Corp. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 1210 at the University of Virginia.