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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 ’s apparent diameter is α 0.57◦,andthebolometric 2 ⊙ ≃ flux we receive is Fbol, 1380 W m− .Usingthisinformation,calculatetheSun’seffective ⊙ ≃ Teff, . Show that your answer is independent of the Sun’s actual distance. ⊙

2. A is measured to have a parallax of π′′ =0.05 0.01 seconds of arc, and a V-band ± apparent of m =4.25 0.05. V ± (a) Calculate this star’s V-band MV, and estimate of the uncertainty.

(b) What is the greatest source of uncertainty in MV, π′′ or mV?

3. The bright star (αCar) has the following measured properties: bolometric magnitude mbol = 0.72, 1− vr =20.5kms− , 15 1 µ =4.8 10− rad s− , × parallax π′′ =0.0104 0.0005 arc-sec, ± effective temperature T =7350K. (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 of Canopus in units of Lbol, ? ⊙ (e) Given that the Sun’s effective temperature is T =5780K,whatistheradiusofCanopus ⊙ in units of R ? ⊙

4. Hydrogen is the most common element in the universe, and makes up 75% of the mass in almost all . Yet only stars with surface 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 whichemitorabsorbvisiblephotons. 18 2 2 (b) To ionize a hydrogen atom takes an energy of χ =13.6eV=2.18 10− kg m s− .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.

1 5. Fig. 1 shows a Hertzsprung-Russel diagram. It includes representative masses for stars along the . 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 g for stars along the Main Sequence. Which end of the main sequence has the highest surface gravities?

(b) Typical dwarf stars have masses Mwd 1M .Howdothesurfacegravitiesof ⊙ stars compare to those of main sequence≃ stars?

(c) Giant and supergiant stars have masses Mg 1to30M .Howdotheirsurfacegravities ≃ ⊙ compare to those of stars on the main sequence? (d) The surface 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)thephysicaleffectwhichaccountsforthisdifference.

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.

2 (b) Roughly what (U B)colordoyouexpecttoobserveforthisstar? − (c) Estimate the net 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 10 ≃ 1.13 10− mbetheequilibriumseparationbetweentheCandOatoms,andm =12m × C H and mO =16mH be the mass of the C and O atoms, respectively. (a) Suppose the molecule is rotating with angular velocity ω.Expressitsangularmomentum 2 L and rotational energy E in terms of ω and the moment of inertia I = reqmCmO/(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 =1andAV =2.

3 Then eliminate ω to show that L2 E = . (1) 2I (b) The molecule’s angular momentum L is quantized according to the rule

L2 = J(J +1)¯h2 , (2)

34 2 1 where J = 0, 1, 2, 3, 4, ...is the rotational quantum number andh ¯ 1.06 10− kg m s− 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 0transition? → (d) In a , CO in the J = 0 state can be excited to the J = 1 state by collisions 7 with H2 molecules. A CO molecule typically stays the J =1statefortrad 1.4 10 s before spontaneously radiating a photon and returning to the J =0state.Ifthetypical≃ × time between collisions tcoll is longer than trad,mostoftheCOwillbeintheJ =0stateand 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 =10K.

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 (Pop. I) and a (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) 2πGρ z, (3) ≃− 0 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(µ0t) , (4)

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. 3 (c) Evaluate µ0,usingthemassdensityρ0 0.15 M pc− we find in the Sun’s neighborhood. ≃ ⊙ (d) If it takes the Sun 225 Myr to complete one orbit around the center of the , 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 (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 m =16.40 3.53 log(P/day). K − (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 only d =273pc away. δ Cephei has a period P =5.37 day, and an average in the K- band 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.

5 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 change along the Hubble sequence?

14. Two years ago, an undergraduate class at University College London discovered Super- 2014J in the nearby spiral galaxy M82 (Fig. 4). This type of results from the thermonuclear explosion of a white , and such explosions reach a peak absolute V-band magnitude MV 19.3. At maximum luminosity, these supernovae have colors similar to , with (B≃−V ) 0.0. − ≃ (a) What (B V )colordoesthissupernovahaveatpeakbrightness?Isthiswhatyou’d expect? − (b) The galaxy M82 is at a distance d =3.5 0.3Mpc. WhatapparentV-bandmagnitude ± 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 19.5andasurfacetemperatureT 10, 000 K. Assuming it radiates like a black- body,≃− 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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