Chapter 10, Life and Times on the Main Sequence Video Solutions, Stars and Stellar Processes

Problem 1

Spherical symmetry has been assumed for our basic discussion of hydrostatic equilibrium and for the Standard Solar Model. Rotation could flatten stars, thus invalidating this assumption. Show that rotation of a star should cause negligible deviation from spherical symmetry if the rotational frequency is much less than the inverse of the hydrodynamical timescale for the star. Would you expect such effects to be large or small for the Sun? What about for Saturn, viewed as a sphere of gas in hydrostatic equilibrium? Are your findings consistent with the observation that the oblateness (difference in equatorial and polar diameters divided by the average diameter) of the Sun is of order $10^{-5}$, while that of Saturn is about 0.1 ?

Vipender Rao

Numerade Educator

01:33

Problem 2

Mass loss is important for some stars in some phases of their lives. The Standard Solar Model does not assume any mass loss in the evolution of the Sun. The solar wind has an average velocity at Earth of about $400 \mathrm{~km} \mathrm{~s}^{-1}$ and a density of about 7 protons per cubic centimeter. Use this to estimate the rate of mass loss from the Sun. Will this loss be significant over the approximately $10^{10}$ year main sequence lifetime of the Sun?

Zachary Warner

Numerade Educator

04:02

Problem 3

Count the number of neutrinos emitted for each $4 \mathrm{H} \rightarrow{ }^4 \mathrm{He}$ in the PP chains. Use this, the average energy release in the PP chains, and the solar photon luminosity to estimate the neutrino luminosity of

Ren Jie Tuieng

Numerade Educator

02:11

Problem 4

Assuming a spherical star of uniform density $\bar{\rho}$, show that the central pressure may be (very crudely) approximated by $$
P \simeq \text { constant } \times \frac{M^2}{R^4}
$$
where $M$ is the mass and $R$ is the radius. Calculate the central pressure of the Sun in this approximation and compare with the results of the Standard Solar Model.

Dominador Tan

Numerade Educator

02:08

Problem 5

The most massive of the four bright stars in the Trapezium open cluster of the Orion Nebula is Theta Orionis C. Its spectrumluminosity class is $\mathrm{O} 6 \mathrm{~V}$, its visual apparent magnitude is $m_{\mathrm{v}}=5.13$, and its absolute visual magnitude is $M_{\mathrm{v}}=-3.2$. It is estimated to have a radius of $8 R_{\odot}$, a mass of $40 M_{\odot}$, and its luminosity is $251,000 L_{\odot}$ (much of this at UV wavelengths, since Theta Orionis C has the distinction of having the hottest surface temperature of any star visible to the naked eye). What is the parallax of Theta Orionis C? What is its surface temperature? Given that Theta Orionis C is thought presently to be about 1 million years old, how much longer will it spend on the main sequence?

Khoobchandra Agrawal

Numerade Educator

00:33

Problem 6

Assume a completely ionized ZAMS star from Pop I and neglect the contribution of metals to the mean molecular weight relative to that of hydrogen and helium. Show that the mean molecular weight $\mu$ defined in Eq. (3,19) can then be approximated as
$$
\mu \simeq \frac{4}{5 X+3} \simeq 0.6
$$
where Eq. (3.28) has been used to estimate $X$. Compare with the results of Fig. 10.7 for the mean molecular weight of the Sun computed in the Standard Solar Model.

Farhanul Hasan

Numerade Educator

03:39

Problem 7

Estimate the main sequence lifetimes for stars of spectral class O5, A0, K5, and M5. Hint: See Table 2.2 for relevant data.

Zachary Warner

Numerade Educator

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Problem 8

(a) Suppose that the central energy-producing regions of the Sun were unchanged but that (magically) the layers outside this region were made transparent so that they did not impede transport of photons. Assuming that it were still a blackbody radiator, in what region of the spectrum would the observed photons from the Sun peak?
(b) In reality the photon energy diffuses outward through the outer layers of the Sun. Estimate a diffusion time to the surface assuming a random walk with opacities characteristic of Thomson scattering. Compare this diffusion timescale with the Kelvin-Helmholtz timescale, which is a measure of how rapidly a star can convert gravitational energy into emitted photons. You should find that the diffusion timescale and the Kelvin-Helmholtz timescale are in fact very different. Can you explain what the fallacy is in assuming that the diffusion timescale estimated in this way and the Kelvin-Helmholtz timescales should be comparable?***

Andrew Eddins

Emory University

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Problem 9

Use data from Fig. 7.2 and Table 10.1 to estimate the mean free path for photons in the center of the Sun and at $90 \%$ of the solar radius. From Fig. 7.2 estimate an average internal opacity (excluding the surface region) and from that opacity calculate a time for photons to diffuse to the surface by a random walk.***

Andrew Eddins

Emory University

01:57

Problem 10

Derive a formula for the ratio of energy densities carried by the photons and by the gas in a star, assuming an ideal gas and photons with a Planck distribution. Estimate this ratio for the Sun using a representative interior temperature. From Table 2.2 most main sequence stars have an average density $\rho \sim 1 \mathrm{gcm}^{-3}$. Assuming this density, and that one can continue to treat the radiation and gas independently as above, at what temperature would the energy density of the radiation become equal to that of the gas?

Salamat Ali

Numerade Educator

01:31

Problem 11

In Example 6.1 the mean life for loss of protons in the first step of the PP chains was estimated to be about $6 \times 10^9$ yr. Make a similar estimate for the mean life of the deuterium formed in the first step and consumed in the second step in PP-I, and for the mean life of the ${ }^3 \mathrm{He}$ formed in the second step and consumed in the third step of PPI. Assume for the center of the Sun that
$$
T_6=15 \quad \rho=150 \mathrm{~g} \mathrm{~cm}^{-3} \quad Y_{\mathrm{p}}=0.4 \quad Y_{3 \mathrm{He}} \sim 4 \times 10^{-6}
$$
(the ${ }^3 \mathrm{He}$ abundance is rather uncertain). Use Eig 6.1 to estimate rates that you will need.***

Chai Santi

Numerade Educator

00:03

Problem 12

Show that if the radial coordinate $d r$ is parameterized in terms of the optical depth $d \tau$ (see Box 10.1), the equation for hydrostatic equilibrium can be expressed as $d P / d \tau=g / \kappa$, where $g$ is gravitational acceleration and $\kappa$ is radiative opacity.***

John Johnson

Numerade Educator

06:37

Problem 13

Stars have magnetic fields but the energy of those fields has generally been neglected in discussing the equilibrium configuration for stars. Justify this for the Sun by estimating an upper limit for the average magnetic field to be given by that in sunspots, which you may take to be about $0.1 \mathrm{~T}$ (the average field is certainly less than this), and computing the ratio of the energy density of the magnetic field to that of the gravitational field.

Mahnoor Amin

Numerade Educator

04:37

Problem 14

The electron number density is about $6.4 \times 10^{25} \mathrm{~cm}^{-3}$ at the center of the Sun and $5 \times 10^{24} \mathrm{~cm}^{-3}$ at $30 \%$ of the solar radius. Use results from the Standard Solar Model to calculate the ratio of the actual electron density to the critical density given by Eq. (?51) at these two radii.*

Keshav Singh

Numerade Educator