Lecture 5: The Motion of the Planets

The heavens themselves, the planets, and this centre
Observe degree, priority, and place,
Insisture, course, proportion, season, form,
Office, and custom, in all line of order:

-- William Shakespeare, Troilus and Cressida, 1609

5.1 Direct and Retrograde Motion

(Discovering the Universe, 5th ed., §2-0)

• In addition to the stars, the Sun, and the Moon, there are several other objects in the sky which are easily visible at night.
   
  From the ancient perspective, a planet is a point of light in the sky that moves relative to the stars, much as the Sun and Moon do.
  
  The name comes from the Greek for "wanderer".

Photo Information

• With the naked eye, one can see five planets: Mercury, Venus, Mars, Jupiter, and Saturn.

Extra: the Sun, Moon, and planets are associated with ancient gods, and their number is the basis of our seven-day week.
 

• Like the Sun and the Moon, the planets all move near the ecliptic, never being more than a few degrees away.
  
  In the photo at the right, you can see (from top to bottom) Saturn, Venus, Jupiter, and Mercury in alignment with the recently set Sun.
   
  

• The planets move slowly enough that their positions change only slightly from night to night.
  
  They therefore rise in the east and set in the west as part of the sky's diurnal motion.
   
   
  
• Relative to the stars, however, the planets generally move from west to east , like the Sun and Moon.
  
  Their speeds vary, but Mercury is the fastest, followed by Venus, Mars, Jupiter, and then Saturn, the slowest.
  
  This motion is called direct motion.
  
  
• What distinguishes the planets from the Sun and Moon is that they will also sometimes reverse their motion, travelling from east to west relative to the stars.
  
  This reverse motion is known as retrograde motion.
  
  Retrograde motion can last from weeks (Mercury) to months (Saturn).
  
   
• The image below displays the actual retrograde motion of Jupiter (brighter) and Saturn (dimmer) over eleven months:

5.2 Geocentric Cosmology

(Discovering the Universe, 5th ed., §2-0)

• According to the laws of physics, there is no preference between saying that the Sun revolves around the Earth or the Earth revolves around the Sun.
  
  Each are equally true, although one perspective may be more useful than the other for a particular purpose, as we have seen.
   
  
  
• When it comes to the planets, however, the principles of science forces one to make a distinction between an Earth-based view and a Sun-based view.
   
  
  
• Given that the motion of the Earth cannot be perceived, it is natural to assume the planets revolve around it. This is known as a geocentric cosmology, and it was widely accepted until just a few hundred years ago.
   
  
  
• A good scientist, however, must try and make sense out of retrograde motion, which cannot be simply explained in an Earth-centered perspective.
   
   
  
• The Greek astronomer Hipparchus described a model for retrograde motion which placed each planet in motion around a circle called an epicycle, which in turn revolved around a circle centered on the Earth called a deferent.
  
  The former produces the retrograde motion, while the latter is primarily responsible for the direct motion.
  
  In the adjacent animation, the arrow points towards the stars we see behind the planet; watch where the arrow points as the planet moves around its epicycle.
  
   
  
• In the 1st C. A.D., Ptolemy, an astronomer at the Alexandria observatory in Egypt, took Hipparchus' model and fit it to the several centuries of observational data that was available to him.
  
  For each planet, Ptolemy determined the sizes of its deferent and epicycle, and the speed of revolution of its epicycle and the planet itself.
  
  By projecting his model forward in time, Ptolemy was then able to correctly predict where the planets would be located centuries into the future.
  
  Because of the success of his model, Ptolemy's treatise on the subject, which became known as the Almagest ("the Greatest"), was the bible of astronomers through the Middle Ages.
   
  
  
• After a millenium, however, the Ptolemaic model increasingly deviated from the planets' observed motions.
  
  Other astronomers tried to correct it by adding additional levels of epicycles, but the result was exceedingly complex.
  
  It was clear that something was not quite right with this model.

5.3 Heliocentric Cosmology

(Discovering the Universe, 5th ed., §2-1)

• An alternative to the geocentric cosmology was actually suggested a century before Hipparchus by Aristarchus (3rd C. B.C.).
   
  
• In the heliocentric cosmology, the planets orbit the Sun rather than the Earth.
  
  The Earth also orbits the Sun, but the Moon still orbits the Earth.
  
  Mercury and Venus (the inner planets) have smaller orbits than the Earth, while Mars, Jupiter, and Saturn (the outer planets) have larger orbits.
   
   
  
• The heliocentric cosmology also explains retrograde motion, by relying on the fact that the planets move at different speeds; in particular, inner planets move faster and outer planets move more slowly.
  
  As a result, the Earth will regularly overtake and pass the outer planets, and the inner planets will do the same to the Earth.
  
  Like one car passing another on the highway, the second car will appear to "move backward".
  
  This is another example of parallax.
  
   
  
• Although the heliocentric cosmology had a simpler geometry than the geocentric cosmology, it never achieved acceptance in ancient Greece because it required that the Earth move.
  
  The heliocentric cosmology was forgotten for almost 2000 years, until the 16th century, when the Polish astronomer Nicolaus Copernicus rediscovered it.
   
  
  
• Copernicus performed his own calculations using this model (assuming circular orbits), and found that he could describe the planets' observed motions to an accuracy similar to that of the geocentric model.
  
  He was also able to make very accurate predictions of the planets' relative distances from Sun (see the table below).
   
  
  
• Although Copernicus was convinced that his heliocentric model was a well-founded improvement to astronomy, he delayed publication of his results until near the end of his life, presumably because he was concerned that his work was so radical that it would be rejected or lead to censure.
  
  Finally, in 1543, Copernicus' book On the Revolutions of the Celestial Spheres appeared, shortly before he died.
  
  The book was widely read in Europe, and gained enough support that it seriously threatened the geocentric model, which was virtually an article of faith in the Catholic Church.
  
  In 1616 the Church banned Copernicus' book (and the ban was not lifted until the end of the 18th century!).

5.4 Planetary Configurations

(Discovering the Universe, 5th ed., §2-1)

• For millenia, observational astronomers have described the positions of planets and other celestial bodies using several special configurations, which can be easily understood in terms of the Copernican model.
   
  
  
• The configuration between two celestial bodies can be described using the angle between them (measured along the ecliptic), which is called their elongation.
  
  Question: what other type of celestial angular measurement is elongation similar to?
   
   
  
• When two or more celestial bodies pass each other along the ecliptic, they have an elongation of 0°, and they are said to be in conjunction.
  
  At the right, the Sun and the red planet are in conjunction.
  
  Question: what is the phase of the Moon when it is in conjunction with the Sun?
   
  

Photo Information

The image at the right shows a "triple" conjunction that occurred on April 23, 1998, between the Moon,Venus, and Jupiter.

• When two celestial bodies are at right angles in the sky, they have an elongation of 90°, and they are said to be in quadrature.
  
  At the right, the Sun near the western horizon and the red planet near the meridian are in quadrature.
  
  Question: what is the phase of the Moon when it is in quadrature with the Sun?
   
   
  
• When two celestial bodies are directly opposite each other, they have an elongation of 180°, and they are said to be in opposition.
  
  At the right, the Sun near the western horizon and the red planet near the eastern horizon are in opposition.
  
  Question: what is the phase of the Moon when it is in opposition to the Sun?
   
  
  
• Mercury and Venus differ from the other planets in that they never appear far from the Sun.
  
  Mercury is at most 23° away from the Sun, while Venus is at most 46° away; this is their maximum elongation.
  
  Maximum elongation of a planet is not readily explained using the geocentric model, but it arises naturally out of the heliocentric model, simply by assuming that the orbits of Mercury and Venus lie inside the Earth's orbit.
   
   
  
• Maximum elongation of an inner planet from the Sun can be seen from the geometry of the picture at the right; it is determined by the tangent line from the Earth to the planet's orbit.
  
  From the rotation of the Earth we can determine the relative directions of the planet and the Sun, allowing us to distinguish the two sides of the planet's orbit as eastern and western.
  
  When an inner planet is at maximum eastern elongation it will only be visible shortly after sunset (an "evening star"); when it is at maximum western elongation it will only be visible shortly before sunrise (a "morning star").
   
  
  
• Note that an inner planet can never be in opposition to (180° away from) the Sun, or even in quadrature (90° away).
   
  
  
• For the planet and the Sun to be in conjunction, they must be along the same line of sight from the Earth.
  
  From the picture above we can see that there are two different ways in which an inner planet can be in conjunction with the Sun, one in between the Sun and the Earth (called inferior conjunction) and the other on the opposite side of the Sun from the Earth (called superior conjunction).
  
  Question: as observed from Earth, what is the phase of the inner planet at each of the four positions in the picture?
   
   
  
• Unlike Mercury and Venus, Mars, Jupiter, and Saturn can appear in opposition to the Sun (180° away).
  
  In the heliocentric model, their orbits must therefore lie outside the Earth's orbit.
   
  
  
• An outer planet can be in opposition with the Sun, in quadrature with it (both eastern and western quadrature), and in conjunction with it (but only one way, corresponding to an inner planet's superior conjunction).
  
  Question: as observed from Earth, what is the phase of the outer planet at each of the four positions in the picture?
   
  
  
• It can be shown that an outer planet is least illuminated at quadrature.
  
  Question: when will an outer planet be brightest?

5.5 Orbital Period

(Discovering the Universe, 5th ed., §2-1)

• The time it takes for a planet to complete one orbit is called the orbital period of revolution, often simplified to "orbital period" or just "period".
  
  As with the Moon, we must distinguish between star-relative and sun-relative positions when determining the period:
  
  The sidereal period of a planet refers to the time it takes for the planet to return to the same position with respect to the stars, e.g. from one position on its orbit back to the same position.
  
  The synodic period of a planet refers to the time it takes for the planet to return to the same position with respect to the Sun, e.g. from inferior conjunction to inferior conjunction, or from opposition to opposition.
  

As can be seen in the table at the right, the farther a planet is from the Sun, the longer is its sidereal period. The synodic period doesn't have a simple behavior, however; it initially increases, and then decreases until it is slightly larger than one year. This complicated behavior is due to the relative motion of both the planet and the Earth.

Planet Average Distance Sidereal Period Synodic Period Mercury 0.3871 AU 0.2408 y = 87.97 d 115.88 d Venus 0.7233 AU 0.6152 y = 224.70 d 583.92 d Earth 1.0000 AU 1.0000 y = 365.26 d ---- Mars 1.5237 AU 1.8809 y = 686.98 d 779.94 d Jupiter 5.2028 AU 11.862 y 398.9 d Saturn 9.5388 AU 29.458 y 378.1 d Uranus 19.1914 AU 84.01 y 369.7 d Neptune 30.0611 AU 164.79 y 367.5 d Pluto 39.5294 AU 248.5 y 366.7 d

• For an inner planet, the sidereal period is shorter than the synodic period, because when the planet returns to its original position the Earth has moved in its orbit, so the planet must travel further to catch up to the Earth.
  
  In the animation at the right, Venus actually completes two sidereal periods (225 d) before it finally catches up with the Earth after the synodic period (584 d).
   
   
  
• For an outer planet, the sidereal period is (usually)longer than the synodic period, because when the Earth returns to its original position (one year) the planet has only moved slightly in its orbit, and the Earth doesn't have to travel very far to catch up to the planet.
  
  Mars is an exception to this because it is so close to the Earth; after one year it has already traveled more than half an orbit, so the Earth has to complete two orbits before it can finally catch up to Mars.
  

5.6 Tests of the Heliocentric Model

(Discovering the Universe, 5th ed., §2-2, §2-4)

• Tycho Brahe (1546 - 1601), a Danish nobleman, was renowned for his development of astronomical instruments and his use of them to make measurements of the positions of stars and planets.
  
  His data were the most accurate available prior to the introduction of the telescope into astronomy, shortly after his death.
  
  Amongst other discoveries, he made accurate measurements of a supernova in 1572, and showed that it was in the realm of the stars, which was believed to be unchanging.
  
   
  
• Tycho noted that the Copernican model predicts that stars should appear to shift their position as the Earth moves around the Sun, due to parallax:
  
  
  
  Tycho attempted to measure this parallax, but he was unable to do so, and therefore concluded that the premise that the Earth moves around the Sun was wrong.
  
  Actually, it was Tycho's confidence in his own measurements which was ill-founded!
  
  As we have already seen, the stars do exhibit parallax, but because they are so far away, a telescope is required to observe it!
  
  However, this is still a good example of the scientific method in action: a prediction is made, and then tested by subsequent observations.

• The telescope was invented by a Dutch optician late in the 16th century.
  
  Galileo Galilei (1561-1642), a professor of mathematics at the University of Padua, heard about the telescope in 1609.
  
  Recognizing the telescope's possibilities, Galileo immediately built one of his own, based only the sketchy details he had heard.
  
  Galileo then improved the design of the telescope to the point where it could be used for astronomy.
  
  Galileo quickly made several important astronomical discoveries, which were published in 1610 in his book The Starry Messenger.
   
  
• One of Galileo's observations was that Venus exhibited phases similar to the Moon's:
  
  
  
  Galileo noticed that Venus' phases were related to its angular diameter and elongation: it is smaller (farther away from us) at the gibbous phase and larger (closer to us) at the crescent phase, with the extremes occurring at small elongations.
  
  These observations were strong confirmations of the heliocentric model.
  
   
  
• Galileo also saw the four large moons of Jupiter, now called the Galilean satellites.
  
  The Galilean satellites were obviously orbiting Jupiter, which was contrary to a basic assumption of the geocentric model, viz. everything in the heavens orbited the Earth.
   
  
  
• In 1616, when Copernicus' book was banned, Galileo was instructed by the Vatican that he could only discuss the heliocentric model as a "mathematical supposition" because anything else would "restrict God's omnipotence".
  
  Nevertheless, in 1632 Galileo published Dialogue Concerning the Two Chief World Systems--Ptolemaic and Copernican, which was such a masterpiece of exposition of the heliocentric model that readers ignored the ordained conclusion.
  
  Galileo was then brought before the Inquisition and forced to publicly recant; his Dialogue was banned, and he spent the last eight years of his life under house arrest.
  
  The ban on Galileo's Dialogue wasn't lifted until 1822, and the Vatican's censure of Galileo himself wasn't removed until 1992!
  
  Extra: you can find out much more about Galileo at PBS/Nova's website, Galileo's Battle for the Heavens.

5.7 Kepler's Laws

(Discovering the Universe, 5th ed., §2-3)

• Although the heliocentric model worked just as well as the geocentric model, to make it work over a millenium Copernicus still had to add epicycles.
  
   
  
• The German astronomer Johannes Kepler (1571-1630) had a different idea, however.
  
  Kepler didn't believe planetary orbits were necessarily circles, but could instead be other closed curves, such as the ellipse or oval.
  
   
  
• Recall that a circle is defined as the set of all points that are a constant distance r (the radius) from the center C.
  
   
  
• An ellipse is a generalization of a circle, involving two points F₁ and F₂(each called a focus, and together the foci) and two distances r₁ and r₂, whose sum is a constant:
  
  r₁ + r₂ =2a
  
  When the foci coincide (coming together at the center), the result is a circle with a radius a.
  
• The constant 2a is equal to the length of the longer or "major" axis, so a is called the semimajor axis.
  
  The semimajor axis therefore describes the overall size of the ellipse.
  
  It can be shown that a is the average distance of the ellipse from one focus.
   
   
  
• The constant c describes how far each focus is from the center, which determines how elongated the ellipse is (for a given value of a).
  
  However, it is more useful to use the eccentricity:
  
  e = c/a
  
  Because c is always less than a, the value of e varies between 0 and 1.
  
  When e = 0, c = 0, the foci coincide, and we have a circle.
  
  When e =1, the foci approach the opposite ends of the ellipse; the result is so elongated that, from one focus, both the center and the other focus are infinitely far away, forming a curve called a parabola.
  
   
  
• Kepler came to work with Tycho in 1600, and the latter's astronomical records provided Kepler with the data he needed to test his hypothesis.
  
  After many years of laborious calculations, Kepler was able to demonstrate what is now known as Kepler's First Law:
  
  Planetary orbits are ellipses, with the Sun at one focus.
   
  
• Because the Sun is off-center, we can describe two special positions on a planet's orbit, both on the major axis:
  
  The perihelion is the point of closest approach to the Sun; it is a distance a(1 - e) from the Sun.
  
  The aphelion is the point where the planet is farthest from the Sun; it is a distance a(1 + e) from the Sun.
   
  

The orbital inclination is usually quite small, except for Pluto.

Question: where did we see orbital inclination previously?

Question: why doesn't a planet usually disappear behind the Sun when they are in conjunction?
 

• Kepler also noticed another characteristic of planetary motion: planets move fastest at perihelion, and slowest at aphelion.
  
  Kepler was able to quantify these varying speeds in what is known as Kepler's Second Law:
  
  Planets sweep out equal areas in equal times.
   
  
  
• Kepler published his First and Second Laws in 1609 in a book entitled New Astronomy.
  
  
• Ten years later, in 1619, Kepler discovered and published an additional relationship.
  
  Kepler's Third Law quantifies the observation that more distant orbits have longer periods:
  
  a³ = P²
  
  Here, the semimajor axis a is measured in A.U. and the orbital period P is measured in years.
  
  The graph at the right shows log P vs. log a; the data falls along a straight line, with a slope of 3/2.
   
  
• Kepler also noticed that the Galilean satellites obeyed the Third Law, as can be seen by the same 3/2 slope in the graph at the right.
  
  This implied that Kepler's Third Law was a general principle.
   
  
  
• Galileo himself refused to accept Kepler's ideas, clinging to the notion that planetary orbits must be circular, though his reasons were based on his studies of motion rather than on tradition.
  

The retrograde motion, Venus phase, and Galilean satellite animations, and star charts, were produced on a Macintosh with the Voyager II program, and are ©1988-93 Carina Software, 830 Williams St., San Leandro, CA 94577, (510) 352-7328. Used under license.

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