Download Astronomy 242: Review Questions #1 Distributed: February 10

Astronomy 242: Review Questions #1
Distributed: February 10, 2016
Review the questions below, and be prepared to discuss them in class on Feb. 12 and 17.
Modified versions of some of these questions will be used in the midterm exam on Feb. 19.
1. As seen from Earth, the Sun’s apparent diameter is α⊙ ≃ 0.57◦ , and the bolometric
flux we receive is Fbol,⊙ ≃ 1380 W m−2 . Using this information, calculate the Sun’s effective
temperature Teff,⊙ . Show that your answer is independent of the Sun’s actual distance.
2. A star is measured to have a parallax of π ′′ = 0.05 ± 0.01 seconds of arc, and a V-band
apparent magnitude of mV = 4.25 ± 0.05.
(a) Calculate this star’s V-band absolute magnitude MV , and estimate of the uncertainty.
(b) What is the greatest source of uncertainty in MV , π ′′ or mV ?
3. The bright star Canopus (αCar) has the following measured properties:
bolometric magnitude mbol = −0.72,
radial velocity vr = 20.5 km s−1 ,
proper motion µ = 4.8 × 10−15 rad s−1 ,
parallax π ′′ = 0.0104 ± 0.0005 arc-sec,
effective temperature T = 7350 K.
(a) What is the distance d to Canopus? Include an estimate of the uncertainty in d.
(b) How fast is Canopus moving with respect to the Sun?
(c) What is the absolute bolometric magnitude Mbol of Canopus?
(d) Given that the Sun has absolute bolometric magnitude Mbol,⊙ = 4.74, what is the
bolometric luminosity of Canopus in units of Lbol,⊙ ?
(e) Given that the Sun’s effective temperature is T⊙ = 5780 K, what is the radius of Canopus
in units of R⊙ ?
4. Hydrogen is the most common element in the universe, and makes up ∼ 75% of the mass
in almost all stars. Yet only stars with surface temperatures near T ≃ 10, 000 K exhibit
prominent spectral lines of hydrogen in visible light.
(a) Sketch a diagram of the energy levels of hydrogen, labeling each level by the quantum
number n. Then identify the transitions which emit or absorb visible photons.
(b) To ionize a hydrogen atom takes an energy of χ = 13.6 eV = 2.18 × 10−18 kg m2 s−2 . At
what temperature is the kinetic energy of a typical particle comparable to this energy?
(c) Visible hydrogen lines are weak in stars with surface temperatures T < 10, 000 K. Explain
this fact. The diagram you drew in part (a) may be helpful.
(d) These lines are also weak in stars with surface temperatures T > 10, 000 K. Explain this
fact.
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5. Fig. 1 shows a Hertzsprung-Russel diagram. It includes representative masses for stars
along the main sequence. The light diagonal lines are lines of constant radius in units of the
Sun’s radius, R⊙ .
(a) Using the information in this diagram, estimate the range of surface gravities g for stars
along the Main Sequence. Which end of the main sequence has the highest surface gravities?
(b) Typical white dwarf stars have masses Mwd ≃ 1M⊙ . How do the surface gravities of
white dwarf stars compare to those of main sequence stars?
(c) Giant and supergiant stars have masses Mg ≃ 1 to 30 M⊙ . How do their surface gravities
compare to those of stars on the main sequence?
(d) The surface gravity of a star can be estimated from its spectrum. Briefly explain (i) how
the spectrum of a high-g star differs from the spectrum of a low-g star of the same effective
temperature T , and (ii) the physical effect which accounts for this difference.
6. Main-sequence stars fall along a particular curve in a plot of (U − B) vs. (B − V ) colors,
as indicated by the solid line in Fig. 2.
(a) You observe a star with the spectral lines characteristic of a main-sequence B5 star.
Such stars typically have (B − V ) ≃ −0.2, yet this star is observed to have (B − V ) ≃ 0.7.
Explain.
Figure 1: HR diagram, indicating stellar radii and masses for stars along the main sequence.
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(b) Roughly what (U − B) color do you expect to observe for this star?
(c) Estimate the net extinction to this star in the V band.
7. The CO molecule has an astrophysically important set of energy levels associated with
rotation. Picture a CO molecule rotating end-over-end around its center of mass. Let req ≃
1.13 × 10−10 m be the equilibrium separation between the C and O atoms, and mC = 12mH
and mO = 16mH be the mass of the C and O atoms, respectively.
(a) Suppose the molecule is rotating with angular velocity ω. Express its angular momentum
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L and rotational energy E in terms of ω and the moment of inertia I = req
mC mO /(mC +mO ).
Figure 2: Color-color diagram. The labeled curve shows intrinsic colors of main-sequence
stars. The slanted arrow shows the effects of interstellar dust on observed colors. The full
length of the arrow represents AV = 3 magnitudes of extinction in the V band; tick marks
along the arrow show AV = 1 and AV = 2.
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Then eliminate ω to show that
L2
.
2I
(b) The molecule’s angular momentum L is quantized according to the rule
E=
L2 = J(J + 1)h̄2 ,
(1)
(2)
where J = 0, 1, 2, 3, 4, . . . is the rotational quantum number and h̄ ≃ 1.06 × 10−34 kg m2 s−1
is the reduced Planck constant. What is the energy ∆E of the photon emitted when a CO
molecule jumps from level J to level J − 1?
(c) What is the wavelength of the photon emitted due to the J = 1 → 0 transition?
(d) In a molecular cloud, CO in the J = 0 state can be excited to the J = 1 state by collisions
with H2 molecules. A CO molecule typically stays the J = 1 state for trad ≃ 1.4 × 107 s
before spontaneously radiating a photon and returning to the J = 0 state. If the typical
time between collisions tcoll is longer than trad , most of the CO will be in the J = 0 state and
no line emission is produced. Show that this implies there is a critical density ncrit , below
which a molecular cloud will not produce CO emission lines. Assuming the cross-section
of a CO molecule is comparable to the cross-section of a H2 molecule, estimate the critical
density for a cold molecular cloud with a temperature of T = 10 K.
8. Draw a diagram of the Milky Way. Include all the major components, indicate the
Sun’s position, and provide a scale-bar. Label the population of each component. Finally,
summarize the key properties of the two stellar populations.
9. Sketch HR diagrams for an open cluster (Pop. I) and a globular cluster (Pop. II).
(a) What is the key factor which accounts for the observed variation in HR diagrams among
different open clusters?
(b) Likewise, what is the key factor which accounts for the observed variation in HR diagrams
among different globular clusters?
10. Near the mid-plane of the Milky Way’s disk, the vertical acceleration g is given by
g(z) ≃ −2πGρ0 z ,
(3)
where z is the distance above the mid-plane, and ρ0 is the mass density. (This approximation
is valid as long as z is much less than the vertical scale height of the disk, zd .)
(a) Assume that the vertical motion of a star above and below the mid-plane is independent
of its other orbital motions, and that the star in question always stays well within the range
z = ±zd . Using eq. (1), write down a simple differential equation for the vertical position of
the star, z(t).
(b) This differential equation has solutions of the form
z(t) = z1 sin(µ0 t) ,
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(4)
where z1 is the maximum distance of the star from the mid-plane, and 2π/µ0 is the vertical
period of the star’s motion. Find µ0 in terms of G and ρ0 .
(c) Evaluate µ0 , using the mass density ρ0 ≃ 0.15 M⊙ pc−3 we find in the Sun’s neighborhood.
(d) If it takes the Sun ∼ 225 Myr to complete one orbit around the center of the galaxy, how
many vertical oscillations does it complete in the same amount of time?
11. Summarize the key steps required to determine distances to (a) nearby stars, (b) star
clusters in the Milky Way, and (c) nearby galaxies (ie, those close enough that we can see
individual stars).
12. You observe a sample of Cepheid variable stars in a nearby galaxy. Plotting the average
apparent K-band magnitude of each one against the period of pulsation yields Fig. 3. The
straight line, a least-squares fit to the data, has the equation mK = 16.40 − 3.53 log(P/day).
(a) Does it seem reasonable to assume that all these stars are at approximately the same
distance? Base your argument on the data presented in this plot, regardless of whatever else
you may know about galaxies.
(b) You make similar observations for δ Cephei, a Cepheid variable star only d = 273 pc
away. δ Cephei has a period P = 5.37 day, and an average apparent magnitude in the Kband of mK = 2.34. What is the average absolute magnitude in the K-band of δ Cephei?
(c) Assuming that δ Cephei obeys the same period-luminosity relation as the sample of
Cephieds plotted in Fig. 3, what is the distance to this galaxy?
Figure 3: Plot of average K-band apparent magnitude as a function of period for a sample
of Cepheid variable stars in a nearby galaxy. The straight line is a least-squares fit.
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13. Sketch Hubble’s classification system for luminous galaxies, and briefly describe the
criteria used classify galaxies. How do key properties like ISM makeup and star formation
change along the Hubble sequence?
14. Two years ago, an undergraduate class at University College London discovered Supernova 2014J in the nearby spiral galaxy M82 (Fig. 4). This type of supernova results from the
thermonuclear explosion of a white dwarf star, and such explosions reach a peak absolute
V-band magnitude MV ≃ −19.3. At maximum luminosity, these supernovae have colors
similar to Vega, with (B − V ) ≃ 0.0.
(a) What (B − V ) color does this supernova have at peak brightness? Is this what you’d
expect?
(b) The galaxy M82 is at a distance d = 3.5 ± 0.3 Mpc. What apparent V-band magnitude
mV would you expect this supernova to have at peak brightness? Is this what we observed?
(c) The colors, apparent magnitudes, and even the position of this supernova within its host
galaxy all point to a simple interpretation of the observations which explains the results in
parts (a) and (b). What is it?
(d) At maximum luminosity, this type of supernova has an absolute bolometric magnitude
Mbol ≃ −19.5 and a surface temperature T ≃ 10, 000 K. Assuming it radiates like a blackbody, estimate the radius of the supernova when it reaches maximum luminosity, in units of
the Sun’s radius.
Figure 4: Left: image of Supernova 2014J and its host galaxy M82. The dashes identify
the supernova. Right: apparent magnitude of Supernova 2014J plotted against date of
observation. Blue, green, and red represent B, V, and R magnitudes, respectively.
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