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

Week 3: Kepler’s Laws, Light and Matter
Hassen M. Yesuf (hyesuf@ucsc.edu)
January 30, 2013
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 luminosity of the sun is, L⊙ = 3.8 × 1033 erg/s and its radius is R⊙ = 7 × 1010 cm. What is the effective
temperature, T eff,⊙ , of the surface of sun ? At what wavelength does the sun radiate its energy ? use σ = 5.67×10−5
erg cm −2 s −1 K −4
Ans: Using the formula, L = 4πR2 σT 4 for a blackbody radiation
the temperature of the sun, T⊙ =
using Wien’s law, λmax =
2.9×106 nm
T (inK)
L⊙
4πR2⊙ σ
1/4
= 5744 K
= 504.9 nm
Therefore, above wavelength falls in green region. But why does the sun appear yellow then? Because the
sun emits a continuous radiation at wavelength shortward and longward of 504.9 nm but the emission longward
(redward) of this wavelength dominates. As the result the sun appears yellow to on averge to our eyes.
2. What is the frequency and energy of green light with λ = 504.9 nm light? use h = 6.63 ×10−34 m2kg/s
Ans: Using c = λ ν, ⇒ ν =
3×108 m/s
504.9×10−9 m
= 5.94 × 1016 Hz
Energy of light is given by E = hν = 3.93 × 10−17 Joule = 9.4 × 10−18 calories. Note that from the above
equation high frequency light have more energy than low frequency light. For instance blue light has more energy
than red light.
3. The rest wavelength of vega of Hα, n = 3 → 2 transition, is at 656.285 nm but when you analyize the
spectrum of vega you found that this transition has an observed wavelength of 656.255 nm. What is the redshift of
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vega and its radial velocity ?
Ans: using 1 + z = λobs /λres = 656.255/656.285 = 0.999954
Therefore z = −4.6 × 10−5 . Since the redshift is negative vega is moving toward you. But at what speed?
vrad = cz = 3 × 105 km/s × −4.6 × 10−5 = 13.8 km/s
4. 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.
5. 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.
6. 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.
7. What is the diffraction limit, θ, of the 2.4 m Hubble Space Telescope for a visible light with λ = 500 nm?
Ans: θ = 2.5 × 105 × λ/D = 2.5 × 105 ×
500×10−9 m
2.4 m
= 0.05 arcsecond.
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