Chapter 2, The Hertzsprung–Russell Diagram Video Solutions, Stars and Stellar Processes

Problem 1

Estimate the color index $B-V$ for a star assumed to be a blackbody with a surface temperature of $18,000 \mathrm{~K}$. In making this estimate, assume the appropriate color filters to have a $\delta$-function (that is, very sharply peaked) response of the actual response for the filter.

Narayan Hari

Numerade Educator

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Problem 2

In the Sirius binary system, which has a parallax of 0.38 " as observed from Earth, the companion Sirius B has apparent visual magnitude 8.44 and its spectrum suggests an effective temperature of $\sim 30,000 \mathrm{~K}$. In addition, the semimajor axis for the primary orbit and companion orbit are observed to be 6.54 and $13.26 \mathrm{AU}$, respectively, and the period for the binary is found to be 50.13 years.
(a) What are the absolute visual magnitude and absolute bolometric magnitude if the bolometric correction is $B C=$ -3.3 ?
(b) What is the luminosity of Sirius B based on this bolometric magnitude?
(c) What is the radius of Sirius B, assuming it to be a spherical blackbody?
(d) What is the total mass of the binary and the individual mass of the companion Sirius B?
(e) Assuming Sirius B to be spherical, what is its average density?
(f) What region of an HR diagram does Sirius B occupy? What kind of star is it, based on the information deduced in this exercise?

Andrew Eddins

Emory University

01:49

Problem 3

A certain star has an absolute bolometric magnitude of -5 but its spectrum suggests that the surface temperature is only about $3000 \mathrm{~K}$. Assuming the star to radiate as a spherical blackbody, estimate the luminosity and the radius of this star in comparison with the Sun. How much of the inner Solar System would this star encompass if placed at the position of the Sun? Assuming the mass of this star to be no more than 10 solar masses, estimate its maximum average density. What region of the HR diagram does this star occupy, based on the information just deduced?***

Farhanul Hasan

Numerade Educator

04:26

Problem 4

Deneb (the tail of Cygnus the Swan) has apparent visual magnitude 1.25 and is of spectral-luminosity class A2Ia. Use spectroscopic parallax to estimate its distance from Earth.

Urjeet Deshmukh

Numerade Educator

01:08

Problem 5

Assume that apparent visual magnitude $\sim 28$ is the limit at which such systems can reliably determine light variability and spectrum in order to use the Cepheid variable method (the spectrum is necessary to ensure that the star has been identified correctly as a Cepheid variable). Estimate the maximum distance that space-based or ground-based adaptive effects systems can determine by using Cepheids. Assume the limit for standard ground-based telescopes to be apparent visual magnitude $\sim 23$. What is the maximum distance in that case?

Narayan Hari

Numerade Educator

04:12

Problem 6

Suppose that the main sequence of a cluster of stars at unknown distance is observed and is found to be approximately described by a linear relation $m \simeq 5.8 \cdot \mathrm{CI}+15$, where $m$ is the apparent magnitude and $\mathrm{CI}$ is the $B-V$ color index. Assuming the main sequence of stars in the cluster to be similar to the main sequence observed for stars near the Sun, estimate the distance to the cluster.

Sarah Mccrumb

Numerade Educator

02:34

Problem 7

The Hubble Space Telescope has been used to study Cepheid variables in the galaxy M100. Assume the period-luminosity relation to be parameterized by [see Eq. (1.35)]
$$M_{\mathrm{v}}=-2.76 \log P-1.4$$
with the period $P$ expressed in days. What is the distance to M100 indicated by an observed Cepheid having a period of 51 days and average apparent visual magnitude of 24.9? In estimating the distance, assume that interstellar absorption between us and M100 has dimmed the Cepheid light by 0.15 magnitudes.

Farhanul Hasan

Numerade Educator

00:57

Problem 8

A cluster of stars contains a type-II Cepheid variable with apparent magnitude 20 and a period of 10 days. How far away is the cluster?

Farhanul Hasan

Numerade Educator

02:48

Problem 9

In the solar spectrum there are strong calcium absorption lines (the $\mathrm{Ca} \mathrm{II} \mathrm{H}$ and $\mathrm{K}$ lines), corresponding to excitations from the ground state of singly ionized calcium. These lines are hundreds of times stronger than the hydrogen Balmer series absorption lines, even though calcium is about a million times less abundant than hydrogen on the Sun. Estimate the expected strength of the calcium $\mathrm{K}$ line and Balmer series lines at the solar surface temperature assuming that the partition functions for $\mathrm{Ca}$ I and Ca II are 1.32 and 2.30 , respectively, the electron pressure is $\log P_{\mathrm{e}}=1.5$ dyn $\mathrm{cm}^{-2}$, the ionization energy of calcium is $6.11 \mathrm{eV}$, the $\mathrm{K}$ line corresponds to a transition from the Ca II ground state to an excited state at $3.12 \mathrm{eV}$ with statistical factors $g_1=2$ for the Ca II ground state and $g_2=4$ for the excited state, and that the ratio of calcium abundance relative to hydrogen abundance in the solar surface is $2.2 \times 10^{-6}$. Hint: You can solve this problem using the Saha equations to determine the population of ionic species and the Boltzmann formula to determine the relative population of ground and excited states within each ionic species.

James Kiss

Numerade Educator

02:48

Problem 10

Absorption in the surface of a star involves both discrete lines, such as the Balmer absorption series for hydrogen, and continuum absorption. An important source of continuum absorption is boundfree transitions where an electron is excited from a bound state to an unbound state (ionization). Since the final state contains an essentially free electron, bound-free transitions produce continuous absorption for wavelengths shorter than the threshold wavelength corresponding to the energy difference between the bound state and the ionization threshold. Show that Balmer bound-free continuum absorption (corresponding to ionization from the $n=2$ hydrogen level at $10.2 \mathrm{eV}$ ) and Lyman bound-free continuum absorption (corresponding to ionization from the $n=1$ ground state of hydrogen) both occur at UV wavelengths where the Sun outputs little light. Show that Paschen bound- free absorption (corresponding to ionization from the $n=3$ level of hydrogen) occurs primarily in the visible spectrum where the solar light output peaks and therefore is expected to dominate the contribution of neutral hydrogen to continuum absorption at visible wavelengths. (However, note from Problem 2.11 that this is not the dominant overall contribution to continuum absorption for the Sun.)

James Kiss

Numerade Educator

03:13

Problem 11

Negative ions are generally difficult to produce and are of less consequence in astrophysics than are positive ions. One negative ion that is important in astrophysics is $\mathrm{H}^{-}$, which is formed by addition of an electron to neutral hydrogen and has a binding energy of only $0.7 \mathrm{eV}$. Negative hydrogen ions have very low abundance in the solar surface but they are thought to be the dominant source of continuum absorption at visible wavelengths. Work through the following considerations to see why this is so.
(a) Show that under conditions expected in the solar surface the abundance of $\mathrm{H}^{-}$ions relative to neutral hydrogen atoms is only about $3 \times 10^{-8}$.
(b) Show that bound-free absorption involving ionization of $\mathrm{H}^{-}$ occurs at wavelengths shorter than about $17,000 \mathrm{~nm}$ and therefore overlaps completely the visible spectrum of the Sun. Thus, essentially every $\mathrm{H}^{-}$ion in the solar surface is in a state that can contribute to visible continuum absorption.
(c) Neutral hydrogen is by far the most abundant species in the solar surface. In Problem 2.10 it was demonstrated that the dominant neutral hydrogen contribution to continuum absorption is expected to originate in ionization of electrons from the $n=3$ atomic hydrogen level (Paschen continuum bound- free transitions). Compare the relative abundance of $\mathrm{H}^{-}$ ions in the solar surface with the relative abundance of neutral hydrogen atoms in the $n=3$ level (the only ones that can contribute significantly to continuum absorption), and show that - even though neutral hydrogen atoms are of order 10 million times more abundant than negative hydrogen ions negative hydrogen ions are of order 100 times more abundant than neutral hydrogen in the $n=3$ level and therefore negative hydrogen ions are expected to dominate neutral hydrogen in producing continuum solar absorption. Since separate problems in this chapter demonstrate that other sources of visible continuum absorption (for example, from helium and metals) are less important, it is expected that ionization of $\mathrm{H}^{-}$ions is the dominant source of visible continuum absorption for the Sun.

Hubert Agamasu

Numerade Educator

06:49

Problem 12

Neutral helium has an ionization energy of $24 \mathrm{eV}$. Show that, although helium is the second most abundant element in the Sun, ionization from either the ground or first excited state at $20 \mathrm{eV}$ is not expected to be a significant source of continuum absorption for the Sun. Order of magnitude estimates are sufficient for this problem and the effect of statistical factors may be ignored by setting them all to one.***

Linda Winkler

Numerade Educator

01:56

Problem 13

Estimate the possible contribution of metals (elements with atomic number greater than that of helium, which have a total abundance of several percent in the Sun) to solar continuum absorption. Take as a representative example iron, which has an ionization energy of 7.9 $\mathrm{eV}$ and a sufficiently high density of bound states that one may assume as a rough approximation that for any energy below the ionization threshold there is a nearby bound state. The relative abundance of iron in the solar surface is about $10^{-4}$ that of hydrogen and, for order of magnitude estimation, you may assume that the statistical factor ratios are of order 1 .
(a) Consider continuum absorption at a wavelength of $4000 \AA$. Show, by considering the expected abundance of iron in a level appropriate to give absorption at that wavelength in the solar surface, that neutral iron is expected to be much less important than $\mathrm{H}^{-}$ions in contributing to continuum absorption at $4000 \AA$ (see the results of Problem 2.11 and don't forget the effect of ionization on neutral iron abundance).
(b) Repeat the considerations of part (a) but for a wavelength of $2000 \AA$ Å. Show that at these short wavelengths iron is expected to dominate both $\mathrm{H}^{-}$and neutral hydrogen in its contribution to continuum absorption through bound-free transitions.

Chai Santi

Numerade Educator

02:52

Problem 14

For stellar atmospheres the free-electron pressures $P_{\mathrm{e}}$ typically range from of order $1 \mathrm{dyn} \mathrm{cm}^{-2}$ for the coolest stars to of order 1000 dyn $\mathrm{cm}^{-2}$ for the hottest stars. Estimate the range of free-electron number densities for stellar atmospheres from the hottest to coolest main sequence stars.

Dominador Tan

Numerade Educator

06:55

Problem 15

Use the condition $n_{\mathrm{T}}=n_0+n_{+}+n_{++}$to derive Eqs. (2.7) for the fractions of ionized species. $* * *$

Katherine Mccandless

Numerade Educator

02:33

Problem 16

Suppose a pure atomic hydrogen gas is found to be $70 \%$ ionized at a temperature of $14,000 \mathrm{~K}$. What is the corresponding electron number density, assuming the ionization to be caused entirely by thermal effects?

Nishant Kumar

Numerade Educator