Friday, January 7, 2011
Direct and Retrograde Motion
- 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".
- 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:
Tests of the Heliocentric Model
- 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 andelongation: 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; hisDialogue 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.
Kepler's Laws
- 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 twopoints F1 and F2(each called a focus, and together thefoci) 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 distancea(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 Lawquantifies the observation that more distant orbits have longer periods:
a3 = P2
Here, the semimajor axisa 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.
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.
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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)longerthan 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.
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