Download Week 3: Kepler`s Laws, Light and Matter

Week 3: Kepler’s Laws, Light and Matter
Hassen M. Yesuf ([email protected])
October 7, 2011
1 Lecture summary
• As we discussed last time, the apparent retrograde motion (a reversal in direction of motion) of the planets
is caused by the fact the Earth and the other planets revolve around the Sun at different velocities. The
Ptolemaic model of geocentric system, unsuccessfully tried to explain this motion by introducing a secondary
motion of a planet around the Earth in small circles (epicycles) whose center moves on the large circular
orbit around the Earth. Although Ptolemaic model could account for the retrograde motion, it made wrong
testable predictions and had to be abandoned. For instance, according to this model, Venus could only
have new and crescent phases but Galileo and others have observed Venus in many different phases. So the
Ptolemaic model had to be abandoned once and for all and the modern view of the Sun centered solar system
advocated by Copernicus was adopted. Furthermore, as improvement to the Copernican model, Kepler later
developed a planetary model that agreed with the observational data.
• Kepler’s three laws of planetary motion state that 1) a planet’s orbit is an ellipse whose one focus is the Sun
2) the planet sweeps out equal areas in equal times 3) the square of the planet’s orbit is proportional to the
the cube of its semi-major axis. So the first law tells us the distance of a planet varies during its orbit and the
second law tells us that the planet moves a greater distance when it is near perihelion (closet to the Sun) than
it does in the same amount of time near aphelion (farthest from the Sun). According to the third law, a more
distant planet orbits the Sun at slower speed and therefore takes longer time to go around the Sun once.
• A wave is a pattern of motion of that carries energy without carrying matter with it. Light is electromagnetic
wave generated by motions of electrical charges in atoms. Light is not quite like the waves we are familiar
in our everyday experience in that it does not have a medium it propagates through (in comparison a wave
on the pond moves through the water as water particles or molecules move up and down). Light waves are
vibrations of electric fields and magnetic fields caused by the motions of charged particles. We characterize
a wave by its frequency, wavelength and speed. The visible light we see is a small portion of the complete
electromagnetic spectrum. All electromagnetic waves, ranging from gamma rays rays to radio, move at
a constant speed. The higher the frequency or the shorter the wavelength, the more energetic the electromagnetic wave is. For instance, a blue light has a higher frequency and more energy than a red light does. For
similar reason, an X-ray is more energetic than a radio or a microwave. Similarly, contrary to our everyday
experience, light is not only a wave but is also a particle and therefore can be counted individually as it hits
an atom one at a time.
• All ordinary matter is composed of atoms. An atom has a tiny nucleus at the center with protons (+1 charge),
neutrons(0 charge) and a negatively charged electrons smeared out as cloud around the nucleus. Different
elements have different numbers of protons and neutrons. Therefore, they have different atomic number
(number of protons) and atomic mass (numbers of proton + neutrons ). Isotopes are variants of a particular
chemical element which have the same numbers of protons but differing numbers of neutrons. The property
of an atom depends on its electrical charge which fundamentally determines how an atom interacts with
electromagnetic fields (light). Atoms contain electrical potential energy that depends on the arrangement of
electrons around their nuclei. The available energy state of electron in an atom is quantized. This means that
an electron can only change its energy by moving up or down in a discrete way between particular energy
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levels. If an electron absorbs photon, it gains energy and moves to a higher energy level and if it emits
photon, it loses energy and descends to a lower level.
• Matter leaves its fingerprints when it interacts with light. We can learn a lot from this interactions about
what things are made of as each chemical element has its own fingerprint of energy levels. There are three
types of spectra we can learn from. Emission spectrum is produced when we observe a low density warm
gas cloud. In this warm gas collision are frequent to move electrons to higher energy levels and the electrons
emit photons when they come down. Absorption spectrum is produced when we observe a cold gas cloud
in front of a hot source of light. In this case, the electrons take away some energy from the light and
move up. They may lose this energy by emitting photons in random directions or any other mechanism but
obviously this does not cancel out the absorption we observe. On the other hand, in dense or opaque object
(such as a planet or a star), photons randomly bounce around multiple times, sharing their energy at each
encounter. By the time they escape to reach us, their energy matchs the kinetic energy of the object’s atoms
or molecules. Therefore, when we observe an opaque object we see a continuous thermal radiation spectrum
which depends on a temperature of the object.
• We have learned about two radiation laws. According to Stefan-Boltzmann law, each square meter of a hotter
object’s surface emits more light at all wavelengths. For instance, hot stars emit light at UV wavelengths
that cooler stars does not emit at all and they also emits more infrared than cool stars. From Wien’s law, we
learned that hot objects emit photons with higher energy therefore their spectra peaks at shorter wavelength.
For instance, a very hot star may have its spectrum peak at UV wavelength and a cold star may have its peak
in infrared.
• A Doppler effect tells us how fast an object is moving toward us or away from us. Spectral lines are shifted to
shorter wavelength (a blueshift) in object moving toward us and to longer wavelength (a redshift) in objects
moving away from us.
2 Review/Comprehension Questions
1. The Earth and the other planets revolve in elliptical orbits centered around the Sun. Is this statement
correct ?
Ans: No, even though the planets revolve in elliptical orbits, the Sun is not at the center but at the focus of
the ellipse. One of the focus is empty.
2. If your classmate tells you that he/she saw a crescent Jupiter recently with his/her powerful binoculars
and is now convinced that Earth is not the center of the solar system, will you believe him/her?
Ans: Your classmate is testing the depth of your understanding, sound smart by explaining to him/her planets
farther from the earth cannot go through the crescent and “new planet” phase because obviously for that to
happen the observer (the Earth) has to be farther than the planet to only see a small fraction of the illuminated
half of the planet facing the Sun.
3. Earth is farthest from the Sun in July and closest to the Sun in January. During which Northern Hemisphere season is Earth moving fastest in its orbit?
Ans: Winter because according to Kepler’s second law the Earth moves fastest when it is closet to the Sun.
Remember, it has to sweep out equal area in equal time, so when it closest to the Sun the “pie piece” it covers
is shorter in radius but thicker (longer arc length). So the Earth must travel faster to cover this arc.
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4. According to Kepler’s third law (p2 = a3 ), how does a planet’s mass affect its orbit around the Sun?
Ans: Kepler’s third law makes no allowance for planetary mass, and the planets mass is very small compared
to The Sun’s mass and therefore the planet’s mass has virtually no effect on its orbit and the orbit is solely
determined by mass of the Sun. Similarly, the eccentricity of the orbit also has no effect on the period.
5. According to Kepler’s third law which planets in the solar system orbit the Sun faster than the Earth ?
Ans: Mercury and Venus because they√are closest to the Sun. Remember, velocity = circumference of orbit/period, from Kepler’s law, v ∝ 1/ a. In other words, the closest the planet to the Sun, the faster it is
orbiting
6. Why is the sky blue ?
Ans: The light from the Sun is scatters by gas molecules in the atmosphere. These molecules scatter the
higher-energy (shorter wavelength) blue portion of the sunlight more than they scatter the lower-energy red
portion of the sunlight. It is for a similar reason that the moon looks red during an lunar eclipse.
7.
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Al and 26 Al are two isotopes of Aluminum, how are these two isotopes different?
Ans: They have different numbers of neutron (but have the same numbers of proton)
8. From the fact that O stars are hotter than A stars in which stellar type do you expect see stronger absorption spectral features?
Ans: A star. Remember that an absorption spectrum is produced when a light passes through cooler gas
before reaching the observer. The atmosphere of a star act as a cooler blanket around the hotter interior of
the star so that typical stellar spectra are absorption spectra on top of their blackbody spectra but this feature
is relatively stronger in A stars since they are relatively cooler than O and B stars.
9. Star forming clouds like the Orion Nebula has pinkish-red color called HII region, what kind of spectrum
would you expect if you observe this region ?
Ans: Emission spectrum because the newly formed stars are hot they ionize the gas surrounding them and
they electrons them recombine with Hydrogen atoms and give off photon as they cascade down the energy
level.
10. Andromeda galaxy is not participating in the Hubble expansion (as viewed from the Earth) and so instead
of moving away from our Galaxy, it is moving toward it and may collide with it. How is the spectrum of
Andromeda different from the other galaxies around us participating in Hubble flow?
Ans: The spectrum of Andromeda is blueshifted as it is moving toward us and the spectra of other galaxies
is redshifted as they are moving away from us.
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