Stars and Stellar Physics

Star formation begins when dense clumps within giant molecular clouds, called diffuse nebulae, begin to collapse as their gravity exceeds the pressure from the temperature of the gas. For much of the collapse the clump remains transparent and so can radiate away the energy of the collapse. Consequently the clump stays at a very constant temperature (until a protostar begins to form at the core, when the temperature does rise rapidly as the gas becomes opaque), and so this part of the collapse happens essentially in freefall. This means it can happen rather fast (by astronomical timescales!).

Consider one of the very dense clumps within a nebula. Assume it is spherical with an initial radius of $r_0$, and begins to collapse at time $t=0$, until it has shrunk to radius $r$ by some time $t$ later. If the initial density of this clump, $\rho_0$, is uniform everywhere in the sphere (called a homologous collapse) then we can describe its collapse with the following formulae:

$$\theta + \frac{1}{2}\sin 2\theta = \left(\frac{8\pi G\rho_0}{3}\right)^{1/2} t \quad \text{where} \quad \frac{r}{r_0} = \cos^2\theta$$

(a) The duration of the freefall is $t_{\mathrm{ff}}$, and by the end of that part of the collapse $r \ll r_0$. Using the above equations derive an expression for $t_{\mathrm{ff}}$ in its simplest form. What do you notice about your formula? [Note: $\theta$ is in radians]

(b) If the density of the clump is $5.0 \times 10^{-16} \mathrm{~kg~m^{-3}}$, calculate a value for $t_{\mathrm{ff}}$. Give your answer in years.

(c) Draw what you expect a graph of $r$ against $t$ to look like.

From the upper parts of the Sun's corona comes a stream of charged particles called the solar wind, meaning the Sun is slowly losing some of its mass (although the effect is negligible compared to the mass loss in nuclear fusion). The particles travel at supersonic speeds until the pressure from interstellar space causes their speed to drop to subsonic speeds instead -- this transition is called the termination shock, and has been explored by the two Voyager probes as they leave the solar system.

(a) By considering the solar wind to be radiated equally in all directions as a spherical shell, derive a formula for the rate of mass lost by the Sun, $\Delta M/\Delta t$, given the density, $\rho$, and speed, $v$, of the solar wind at a given distance $r$.

(b) At the orbit of the Earth, the solar wind is measured to have a density of 7 protons $\mathrm{cm^{-3}}$ and to be travelling at $500~\mathrm{km~s^{-1}}$. Calculate $\Delta M/\Delta t$, giving your answer in units of $\mathrm{M}_{\odot}~\mathrm{year^{-1}}$.

(c) As the solar wind moves further from the Sun its speed increases (at an ever decreasing rate), until it asymptotes at a speed, $u_{\infty}$, equal to the escape velocity, $v_{\mathrm{esc}}$, of the Sun. Given that $v_{\mathrm{esc}} = \sqrt{2GM/R}$ for an object with radius $R$ and mass $M$, calculate the escape velocity of the Sun. Give your answer in $\mathrm{km~s^{-1}}$.

(d) At the termination shock boundary the particles slow down from their (supersonic) $u_{\infty}$ to a much slower (subsonic) speed. Another way of defining this boundary is that it is where the pressure of the interstellar medium, $P_{\mathrm{ISM}}$, is equal to the kinetic energy density (i.e. KE per unit volume) of the gas. If $P_{\mathrm{ISM}}$ is estimated to be $10^{-13}~\mathrm{Pa}$, use your values of $u_{\infty}$ and $\Delta M/\Delta t$ to calculate an estimate for the distance to the termination shock. Give your answer in AU.

(e) Voyager 1 detected the termination shock at a distance of 94 AU, whilst Voyager 2 detected it at a distance of 84 AU. What does this tell us about the shape of the boundary?

Light from a star is split into a line spectrum of different colours. The line spectrum from the star is shown, along with the line spectra of some individual elements. Identify the elements present in the star.

A. Helium and hydrogen
B. Potassium and sodium and hydrogen
C. Hydrogen and sodium
D. Sodium and potassium

Main sequence stars fuse hydrogen atoms to form helium in their cores. About $90\%$ of the stars in the Universe, including the Sun, are main sequence stars. These stars can range from about a tenth of the mass of the Sun to up to 200 times as massive. The main source of energy in main sequence stars is from nuclear fusion. The mass of one hydrogen nucleus is $m_{\mathrm{H}} = 1.674 \times 10^{-27}~\mathrm{kg}$, and the mass of one helium nucleus is $m_{\mathrm{He}} = 6.649 \times 10^{-27}~\mathrm{kg}$.

(a) By considering the difference in mass between the net inputs and outputs in the proton-proton chain, how much energy is produced in this reaction? Hint: you will need to use Einstein's most famous equation.

(b) Calculate the number of hydrogen nuclei that must fuse to form helium every second to provide the measured solar luminosity, and the percentage of hydrogen mass per reaction that is converted into energy via fusion.

(c) The Sun is composed of about $71\%$ hydrogen, $27\%$ helium and some heavier elements. However, only $13\%$ of the hydrogen is available for hydrogen fusion in the core. The rest remains in layers of the Sun where the temperature is too low for fusion to occur. Use these figures and the answers to previous questions to calculate the lifetime in years of hydrogen fusion in the Sun (called its main sequence lifetime), assuming the Sun's luminosity remains constant.

(d) For main sequence stars the luminosity scales with the mass as $L \propto M^{3.5}$. Consider a binary star system with two stars separated by 9.1 au and an orbital period of 3.83 years. If the stars are unequal in mass, with one being three times heavier than the other, and assuming similar structures to the Sun, estimate the main sequence lifetime of the larger star.

Which well-known star dropped in brightness by 40% between October 2019 and April 2020, leading to speculation it may be about to go supernova?
A. Aldebaran
B. Antares
C. Arcturus
D. Betelgeuse

In the early 1900s, Henrietta Leavitt made several observations of variable stars from the Harvard College Observatory that led her to propose that the period of variation was related to the intrinsic brightness of the star. In particular, for a class of variable stars known as Cepheids, there was a strong power law relating luminosity to period, and hence a straight line on a log-log graph. Since magnitudes are $\propto \log L$, this can be described by the empirical relation

$$\langle\mathcal{M}\rangle = -2.43(\log P - 1) - 4.05$$

where $P$ is the period measured in days and $\langle\mathcal{M}\rangle$ is the mean absolute magnitude of the star, defined as the apparent magnitude measured from a distance of 10 pc. The apparent and absolute magnitudes are related as

$$m - \mathcal{M} = 5 \log d - 5$$

where $d$ is the distance measured in pc.

In the past, distances to stars could only be determined using something called the parallax method, where the angular shift of a star relative to the background stars was measured at two points in the Earth's orbit six months apart. The parallax angle is the same as the angle subtended by 1 au at the distance of the star. In Henrietta's time, parallax measurements only reached about 100 pc, however her method with Cepheid variables extended this range to more than $10^{6}$ pc, enabling much better determination of distances across the Milky Way and into nearby galaxies.

The variable star RS Puppis (see Figure 1) is one of the biggest and brightest known Cepheids in the Milky Way galaxy and has one of the longest periods for this class of star at 41.5 days.

(a) Measure the mean apparent magnitude from Figure 1, and find the distance to the star, given that absorption of the light by interstellar dust means the star appears 1.42 magnitudes fainter.

(b) In 2018 the Gaia satellite measured a parallax angle of 0.5844 milliarcseconds (where there are 3600 arcseconds in a degree). What does this suggest is the distance to the star?

The Crab Nebula is the expanding remnant of a supernova that was observed by Chinese astronomers about 1000 years ago. Although faint in the visible, it is one of the brightest objects in the sky both in the radio and in X-ray. Figure 4 shows it as observed with the Very Large Array (a radio telescope) operating at a frequency of 3.0 GHz. The whole image has an angular size of 530 by 530 arcseconds (where there are 3600 arcseconds in a degree).

For long wavelengths (such as those found in the radio), Planck's Law for black-body radiation simplifies to the Rayleigh--Jeans Law,

$$B_{\nu} = \frac{2 \nu^{2} k_{B} T}{c^{2}},$$

where $B$ is the emitted intensity at frequency $\nu$ per unit frequency per steradian (a unit of solid angle, for which there are $4\pi$ steradians in a sphere), $k_B$ is the Boltzmann constant, $T$ is the temperature of the source, and $c$ is the speed of light. Typically what is measured is the emitted intensity per unit frequency, $I_\nu$, which is related to $B_\nu$ as

$$I_{\nu} = B_{\nu}\, \Omega,$$

where $\Omega$ is the solid angle in steradians subtended by the emitting source as seen by the detector. A small angular ellipse subtends approximately $\Omega = \pi a b$ steradians, where $a$ is the semi-major axis and $b$ is the semi-minor axis of the ellipse, both specified in radians.

(a) The total measured intensity in Figure 4 is $I_{\nu} = 7.43 \times 10^{-24}$ W m$^{-2}$ Hz$^{-1}$. By taking suitable measurements from Figure 4, calculate the temperature of the nebula.

(b) The speed of the expansion of the nebula along its longest axis is about 1500 km s$^{-1}$, although it was ejected at a higher speed during the supernova. Assuming an average deceleration of $15~\mu$m s$^{-2}$, taking the distance to the nebula as 2.0 kpc, and given the image was taken in 2017, estimate the year of the supernova.

The recent supernova SN 2023ixf was one of the closest to Earth of the past decade, reaching an apparent magnitude of 10.8 despite being in the galaxy M101, about 21 million light years away. What was the absolute magnitude of this supernova?

A. $-18.2$
B. $-19.5$
C. $-20.8$
D. $-101$

T Coronae Borealis (T CrB), nicknamed the Blaze Star, is a binary star system about 920 pc away in the constellation Corona Borealis. It is famous for being a 'recurrent nova' - that is, an object whose brightness suddenly increases and then dims again periodically. The last eruption was in 1946, and there are signs the next one will happen either at the end of 2024 or the beginning of 2025. When it does, its apparent magnitude will increase from 10.8 to around 2.5. By approximately what factor will its brightness change?
A. 4
B. 50
C. $2 \times 10^3$
D. $2 \times 10^8$