Section 17: Configurations and apparent motions of the planets 1

Transcription

1 Section 17: Configurations and apparent motions of the planets 1 Configurations are arrangements of the planets that are special. Some configurations involve three objects in the solar system lining up: these are called "syzygies". Syzygy is a great word for Scrabble. An inferior planet is a planet that is closer to the sun than we are. The inferior planets to Earth are and. Inferior planets have the following configurations with respect to the sun: Greatest elongation West Earth Inferior conjunction Greatest elongation East Sun Superior conjunction An inferior planet can thus be in: greatest elongation -- when the angle between where the sun is and where the planet is in the sky is as big as possible. There is "greatest elongation east" and "greatest elongation west"; but it is probably easier to think about whether the planet will rise and set BEFORE the Sun or AFTER the Sun. In the diagram above, Earth's rotation and revolution is counterclockwise, so the inferior planet will rise before the sun at greatest elongation. From the picture above: conjunction -- when the planet is between Earth and the Sun. conjunction --when the planet is on the Earth - Sun line but beyond the Sun. The best time to observe an inferior planet is near because at that time the angle between the sun and the planet (in the sky, as seen from Earth) is greatest and so the planet is up when the sky is darker, or is up longer before sunrise or after sunset. The configuration diagram above can be used to draw a scale model of the inner solar system; one need merely observe the angle sun-earth-planet at maximum elongation, then one can construct the right triangle that is indicated in the figure. That right triangle forces the orbits of the two planets to be to scale. Tricks like this were used by Copernicus to set up his model of the solar system.

2 Section 17: Configurations and apparent motions of the planets 2 EXERCISE 17.1: At greatest elongation, Venus is about 46 degrees from the sun as seen from Earth. In the sketch below, draw a line indicating the direction to Venus at greatest elongation, and use the fact that the Earth-Venus line is tangent to Venus' orbit at greatest elongation, so the line from the sun to Venus must make a right angle with the line from Earth to Venus at greatest elongation, to select the circle that is closest in size to the orbit of Venus on this scale plot: Then, use your ruler to measure the distance from Earth to Sun on this sketch: and to measure the distance from the Earth to the orbit of Venus on this sketch: and thus find the distance to Venus in Astronomical Units:

3 Section 17: Configurations and apparent motions of the planets 3 A superior planet is a planet that is farther from the sun than Earth is. Mark on this diagram where the Earth is: Opposition Quadrature Quadrature A B conjunction From this picture: a superior planet has the following configurations with respect to the sun, as seen from Earth: -- when they appear together in the sky -- when the planet is in the opposite direction to the direction to the sun -- when the planet is 90 o from the sun in the sky. To build a scale model for superior planets in our solar system, one must observe the time it takes the planet to travel from conjunction to quadrature (which gives angle B) and the time it takes to travel from quadrature to opposition (which gives angle A). The synodic period of a planet is the time it takes to go from one configuration back around to the same configuration -- such as from opposition to the next opposition, for Mars, or inferior conjunction to the next inferior conjunction, for Venus. This can be very different than the sidereal period, the planet's year or the time it takes the planet to revolve once around the sun. For planets that have very long sidereal periods, like Pluto with a sidereal period of about 250 years, the synodic period is close to one Earth-year, because the planet doesn't move very far in its orbit while we go once around the sun. For planets closer to us, for Venus and Mars particularly, the synodic period is quite different than the sidereal period. A general formula for the synodic period, S, in terms of the sidereal period, P, is (1/S) = 1/P - 1 for an inferior planet, and 1/S = 1-1/P for a superior one. or using the absolute value, 1/S = 1-1/P for either case. EXERCISE What is the synodic period of Mars? EXERCISE What is the synodic period of Venus?

4 Section 17: Configurations and apparent motions of the planets 4 EXERCISE 17.4: It takes Mars about 212 days to go from quadrature through opposition to quadrature again; the synodic period of Mars is 780 days, so it takes Mars about =568 days to go from quadrature through conjunction to quadrature. This means that the angle from quadrature to opposition to quadrature is 212/780 of 360 degrees, or about 98 degrees, and therefore the angle from quadrature to opposition is half that, or about degrees. On this sketch, draw lines from the sun in the direction of the two positions of quadrature, and use that to determine where on this sketch Mars should be located: Use your ruler to measure the distance from the Sun to the Earth in this sketch: and the distance from the Sun to the orbit of Mars in this sketch: and thus find the distance from the Sun to Mars in Astronomical Units:

5 Section 17: Configurations and apparent motions of the planets 5 Conjunctions that don't involve the Sun: When we say "Mars is at conjunction" we mean "Mars is in conjunction with the Sun". Planets can also be in conjunction with each other or with the moon. When we just say "Venus is at conjunction" we mean "with the sun". Otherwise, we can say "Venus is in conjunction with Mars", when they are along the same line of sight as seen from Earth: When two planets are in conjunction with each other they will be close together in the sky; but they need not be in exactly the same place. This is true because their orbits are not all exactly along the ecliptic, but rather, each orbit has some small inclination. Thus: This sketch shows about 60 days of motion of Mars and Jupiter before and after a conjunction. Mars' apparent motion is faster than Jupiter's, because Mars is closer to the sun (moves faster) and Mars is closer to us. In the conjunction pictured here, the closest that Mars and Jupiter get is a little less than one degree. If you want to see what the relative positions of these two planets were before or after the conjunction, divide each "track" into 6 equal segments; then each segment represents 10 days of apparent motion.

6 Section 17: Configurations and apparent motions of the planets 6 The problem of translating "what we see" into "what is truly happening" is particularly challenging when it comes to figuring out how the planets move in the solar system. We will be talking in lecture about some of the historical steps that led to our modern understanding of the arrangement of the planets in the solar system. By studying configurations, and using them to reconstruct some of the arguments that let Copernicus, Galileo and Kepler figure out how the planets orbit the sun, we can understand better how this tremendous puzzle was solved. By noticing when planets reach certain configurations, we can place them most easily and accurately in a scale model of the solar system. For example: If a superior planet is at opposition, then there is only one place in its orbit that it can be. The same is true for a superior planet at conjunction. For a superior planet at quadrature or for an inferior planet at greatest elongation there are two possible positions the planet may be in. To choose between these possibilities we can use the Earth-clock:

7 Section 17: Configurations and apparent motions of the planets 7 Example: Suppose Mars transits at 6 am on a certain day. Then where is it in this diagram? At A or at B? Remember that Earth rotates in the same sense as it revolves around the sun. Mark the times on the little "Earth Clock" and indicate where Mars must be: A B EXERCISE 17.5: Venus is at maximum elongation in the evening sky. Where is it in this diagram -- at A or at B?? A B EXERCISE 17.6: Mars is at quadrature; it rises near midnight. Where is it in this diagram? Earth Sun

8 Section 17: Configurations and apparent motions of the planets 8 Earth Mercury Sun Venus A Mars E Sun Earth Sun Mercury Earth EXERCISE Write a,b,c,d,e or "none" after each description below to indicate which picture fits, if any. The observer is located on Earth. B Conjunction of a superior planet with an inferior one. Earth Venus Opposition of a superior planet Greatest elongation of an inferior planet Sun Planet transits at midnight C Mars Quadrature of a superior planet Planet transits at noon Conjunction of inferior planet with the sun, inferior conjunction Sun Earth Mars Conjunction between two inferior planets D

9 Section 17: Configurations and apparent motions of the planets 9 EXERCISE On the attached sketch (to scale) of the orbits of the inner planets, fill in the approximate relative positions of the five naked-eye planets, using the clues below. a. Mercury was visible in the early morning sky about six weeks ago; now, it is invisible. Six weeks from now it will be visible in the evening, shortly after sunset. Approximately where is Mercury? b. Venus is a very bright "morning star"; it rises about 3 hours before the sun. c. Mars is on the Meridian one hour before midnight. d. Jupiter transits around 6 P.M. e. Saturn rises around 11P.M.. We have found approximately where in the solar system the 5 naked-eye planets are located. If we know when a planet transits, then we can place it more precisely using the following facts: The difference in time between when an object transits and when the sun transits gives the angle between the Earth-Sun line and the Earth-Object line (convert to degrees using 15 o = 1 hour). The sun transits at noon, i.e. 12 hours before (or after) midnight. f. Improve your position for Saturn using this method and the information that Saturn transits at 4:30 A.M. g. Why doesn't Saturn transit exactly 6 hours after it rises? Is Saturn N or S of the Celestial Equator on this occasion? Explain! Lee Anne Willson printed 12:13, September 7, 1997

10 Section 17: Configurations and apparent motions of the planets 10 In this diagram (for Exercise 17.8) the Sun and the Earth are indicated, and the two arrows mark the direction of revolution of the planets -- counterclockwise. Is the rotation of the Earth in this diagram clockwise or counterclockwise? Lee Anne Willson printed 12:13, September 7, 1997

11 Section 17: Configurations and apparent motions of the planets 11 The most complicated motions that occur in the sky are the apparent motions of the planets, with their retrograde loops. If we observe Mars over a few months near opposition, then we will see it move, first Eastward (like the annual motion of the sun, "prograde"); then Westward for a while ("retrograde motion"); then Eastward again. In this "snapshot" of several months' apparent motion of Mars against the background stars the square at the left represents Mars at the end of the year; the smaller square 'tick-marks" are about ten days apart. The following diagram illustrates how this motion can occur, according to our current, heliocentric, view of the solar system: This illustration shows nearly a year of Earth's and Mars's motion. The dashed line shows where they were when Mars was at opposition. The other lines show the direction that an observer on Earth must look to see Mars. Lee Anne Willson printed 12:13, September 7, 1997

12 Section 17: Configurations and apparent motions of the planets 12 In the plot on the previous page, for an observer standing on Earth at midnight, which direction is East? Thus which sequences of lines show eastward motion? Highlight them on the plot. At the beginning, Mars appears to be moving with respect to the stars -- and as that is also the direction that it is truly moving, that is called direct motion. Near opposition, however, Mars will appear to an observer on Earth to be moving, opposite to the direction it is truly moving; that is called retrograde motion. EXERCISE Here is a view of Mars' motion, relative to Earth, for several months in ; also a view of the sun's apparent motion around the Earth, for the same interval. The little boxes mark 40 day intervals, and both objects are moving mostly counter-clockwise. Such views could have been sketched by pre-telescopic astronomers, using where the planet appeared relative to the stars to find the direction to each point and the apparent magnitude as an indication of the distance (closer = brighter, all else being equal). It shows how complicated the motions are if one takes a geocentric point of view! What was the configuration of Mars relative to the sun at the time of the loop? Hint: You may find it helpful to number the positions of the Earth and of Mars, starting from the present and going backwards. Then you can use these numbers to find the relative positions of Earth, Sun and Mars when Mars made its little loop. Lee Anne Willson printed 12:13, September 7, 1997

13 Section 17: Configurations and apparent motions of the planets 13 EXERCISE 17.10: How did Ptolemy explain the little loop in Mars' apparent motion? (explain/sketch) How did Copernicus explain the little loop in Mars' apparent motion? Explain, in your own words, including a sketch.! Lee Anne Willson printed 12:13, September 7, 1997

14 Section 17: Configurations and apparent motions of the planets 14 EXERCISE Here is a similar diagram for Venus (the hint above works here, too): What is the configuration when Venus shows retrograde motion? EXERCISE It would have been very difficult for ancient astronomers to deduce this motion, because the face that Venus shows to Earth is sometimes larger and sometimes smaller, sometimes fully illuminated and sometimes completely dark. Thus Venus appears brightest not when it is closest but when it is near greatest elongation. Venus at several phases as seen from Earth is shown below. Indicate on the configuration plot where each phase occurs. Lee Anne Willson printed 12:13, September 7, 1997