E.5 Fusion and stars

A main-sequence star keeps a steady size for billions of years.

Which row identifies the outward and the inward effects that are in equilibrium?

The Sun radiates energy at a rate L_☉. This energy is released by nuclear fusion in its core.

At what rate does the mass of the Sun decrease because of the energy it radiates? (c is the speed of light.)

A star has a parallax angle of 0.25 arc-second. (1 pc = 3.09 × 10¹⁶ m, 1 ly = 9.46 × 10¹⁵ m)

What is the distance to the star in light years?

The spectrum of a star consists of a continuous spectrum crossed by dark absorption lines.

Which row gives the property of the star that can be determined from each feature of the spectrum?

Some stars are unstable: their radius, surface temperature and luminosity rise and fall regularly. The Hertzsprung–Russell diagram shows the main sequence and four regions, A, B, C and D.

Which region is the instability strip?

A star is 1.24 × 10⁶ AU from the Sun. What is this distance in parsecs? (1 AU = 1.50 × 10¹¹ m, 1 ly = 9.46 × 10¹⁵ m, 1 pc = 3.09 × 10¹⁶ m)

Which row describes the axes of a Hertzsprung–Russell (HR) diagram?

An astrometry satellite measures the parallax angle of a red dwarf star as 0.025 arcsec. What is the distance to the star?

Why can the stellar parallax method only be used for relatively nearby stars?

Dark absorption lines appear in the spectrum of a star. How are they formed?

Which of the following defines the parsec?

The tungsten filament of an old-style incandescent lamp operates at a temperature of 2900 K. Treating the filament as a black body, at what wavelength is its emitted intensity a maximum?

A star similar to the Sun exhausts the hydrogen in its core and becomes a red giant. Which element is produced by fusion in its core during the red giant stage?

Why can fusion in a stellar core not continue to release energy once iron has formed?

On a Hertzsprung–Russell diagram, luminosity increases upwards and surface temperature decreases from left to right. How does the position of a Sun-like star change as it leaves the main sequence to become a red giant?

Two stars X and Y have the same surface temperature. The radius of X is three times the radius of Y. What is \(\frac{L_X}{L_Y}\), the ratio of the luminosity of X to the luminosity of Y?

Which of the following is the apparent brightness of a star?

An astronomer wants to determine the radius of a distant star using the Stefan–Boltzmann law. Which set of measurements is sufficient?

An astronomy app shows a Hertzsprung–Russell (HR) diagram. What is plotted along the horizontal axis, and in which direction does it change?

A solar-farm engineer notes that the energy collected by her panels was originally released in the core of the Sun. Which process releases this energy in the Sun's core today?

A red giant and the Sun are compared. The red giant lies above and to the right of the Sun on an HR diagram. Compared with the Sun, the red giant is:

Engineers at a tokamak research facility heat a deuterium–tritium plasma to about 10⁸ K. What is the main reason such a high temperature is needed for fusion to occur?

A main-sequence star keeps a constant radius for billions of years. Which pair of effects is balanced in the star?

A deuterium nucleus and a tritium nucleus fuse to form helium-4 and a neutron, releasing about 17.6 MeV. Where does this energy come from?

Fusion of hydrogen requires two nuclei to come within about 10⁻¹⁵ m of one another.

Why does this need both a very high temperature and a very high density?

A star lies at the bottom left of a Hertzsprung–Russell diagram: it has a high surface temperature but a very low luminosity.

What type of star is it, and what does its position imply about its radius?

Two main-sequence stars form at the same time: one of about one solar mass and one of about twenty solar masses.

Which row states which star leaves the main sequence first and the final state of the 20-solar-mass star?

In the core of a main-sequence star, hydrogen is fused into helium. A fraction f of the mass of the hydrogen that is fused is converted into energy, and all of this energy eventually leaves the star as radiation. The luminosity of the star is L.

What mass of hydrogen is fused each second? (c is the speed of light.)

A protostar is a cloud of gas that is contracting under its own gravity. Nuclear fusion has not yet started in its core.

Which row describes the protostar as it contracts?

The overall result of hydrogen fusion in the core of the Sun is

4 ¹₁p → ⁴₂He + 2e⁺ + 2ν

The mass of a proton is m_p, the mass of a helium-4 nucleus is m_α and the mass of a positron is m_e. The neutrinos have negligible mass. What is the energy released in this reaction? Ignore any later annihilation of the positrons. (c is the speed of light.)

In the Sun, hydrogen nuclei fuse only in the core, the region within about a quarter of the Sun’s radius from its centre. About 70 % of the mass of the Sun’s outer layers is also hydrogen.

Which statement best explains why no fusion takes place in the outer layers?

The Hertzsprung–Russell diagram shows four main-sequence stars P, Q, R and S.

Which row identifies the star with the greatest mass and the star that will stay on the main sequence for the longest time?

A red giant and a red supergiant have the same surface temperature.

Which row compares the red supergiant with the red giant?

An astronomer determines the radius R of a star by using L = 4πR²σT⁴.

Which row identifies the observations of the star from which T and L are determined?

The hydrogen in the core of a main-sequence star is used up and fusion in the core stops. Hydrogen continues to fuse in a shell around the core.

Which row describes what happens next to the core and to the outer layers of the star?

Stars X and Y are both main-sequence stars. Their continuous spectra are most intense at the same wavelength. The apparent brightness of X is 1/400 of the apparent brightness of Y.

Which row gives (surface temperature of X)/(surface temperature of Y) and (distance to X)/(distance to Y)?

Seen from Earth, a star has a parallax angle of 0.010 arcsec. A proposed spacecraft would orbit the Sun in a circular orbit of radius 1.5 AU. It would observe the same star from two diametrically opposite points of its orbit. The star lies perpendicular to the line joining these points. What parallax angle would the spacecraft measure?

An ultraviolet space telescope finds that the spectrum of a star peaks at a wavelength of 290 nm. Wien's displacement constant is 2.9×10⁻³ m K. What is the surface temperature of the star?

Stars X and Y have the same luminosity. The peak wavelength in the spectrum of X is 400 nm and the peak wavelength in the spectrum of Y is 800 nm. What is the ratio \(\frac{\text{radius of Y}}{\text{radius of X}}\)?

Two stars have the same luminosity. Star X has a parallax angle of 0.04 arcsec and star Y has a parallax angle of 0.01 arcsec. What is the ratio \(\frac{\text{apparent brightness of X}}{\text{apparent brightness of Y}}\)?

The parallax angle of a star is measured as (0.020 ± 0.002) arcsec. The star's apparent brightness has negligible uncertainty. What is the percentage uncertainty in the luminosity calculated for the star?

Two stars A and B have the same radius. The peak wavelength in the spectrum of A is three times the peak wavelength in the spectrum of B. What is the ratio \(\frac{L_A}{L_B}\) of their luminosities?

The surface temperature of a black body is doubled. What happens to the wavelength at which its emission peaks, and to the power it emits per unit area?

A main-sequence star has a mass of 20 solar masses. Consider these statements.
I. Fusion in its core eventually produces iron.
II. It will end its life in a supernova.
III. It will spend longer on the main sequence than the Sun.
Which statements are correct?

Star A has twice the surface temperature of star B and half the radius of star B. What is \(\frac{L_A}{L_B}\)?

Stars P and Q have the same luminosity. P is four times as far from Earth as Q. What is \(\frac{b_P}{b_Q}\), the ratio of their apparent brightnesses?

An engineer designs solar panels for a rover on Mars. Earth orbits the Sun at 1.0 AU and Mars at 1.5 AU. What is the ratio \(\frac{\text{intensity of sunlight at Mars}}{\text{intensity of sunlight at Earth}}\)?

A red giant and a blue main-sequence star have the same luminosity. The surface temperature of the blue star is four times the surface temperature of the red giant. What is \(\frac{R_{\text{giant}}}{R_{\text{blue}}}\)?

Two stars A and B have the same surface temperature and the same apparent brightness when viewed from Earth. Star A is twice as far from Earth as star B. What is \(\frac{R_A}{R_B}\)?

A space telescope is upgraded so that the diameter of its mirror is doubled. It then observes the same star as before. Which row correctly describes the apparent brightness of the star and the power collected by the telescope?

A space telescope finds that two stars X and Y have the same surface temperature. The luminosity of X is 100 times the luminosity of Y. What is \(\frac{\text{radius of X}}{\text{radius of Y}}\)?

Which equation correctly summarises the overall result of the proton–proton chain in the Sun?

Engineers building fusion reactors on Earth need plasma temperatures of order 10⁸ K. Why is a very high temperature needed for nuclear fusion, in the Sun's core or in a reactor?

A planetarium display places stars on an HR diagram. A white dwarf has a surface temperature of about 20 000 K and a luminosity of about 10⁻³ times that of the Sun. Where is it plotted?

The Sun has kept an almost constant radius for billions of years. Which statement best explains this stability?

Fusion in the Sun converts about 4.3×10⁹ kg of mass into energy every second. Which value is closest to the Sun's luminosity? (c = 3.0×10⁸ m s⁻¹)

In a fusion experiment, the temperature of a deuterium plasma is raised from 50 MK to 200 MK. By what factor does the root-mean-square speed of the deuterons increase?

A space telescope observes a star whose core has just run out of hydrogen. What happens to the core immediately afterwards?

In laser inertial confinement fusion, powerful lasers squeeze a tiny fuel pellet to a very small volume. At a given temperature, why does this compression increase the rate of fusion reactions?

In a main-sequence star, the rate of fusion in the core increases slightly. Which sequence describes how the star returns to equilibrium?

Main-sequence stars of greater mass have higher core temperatures. Which statement best explains this?

Two stars have the same luminosity, but star X has twice the surface temperature of star Y.

What is (radius of Y)/(radius of X)?

The luminosity of star X is four times that of star Y. The parallax angle of X is half that of Y.

What is (apparent brightness of X)/(apparent brightness of Y)?

Two stars X and Y, which behave as black bodies, have the same radius. The surface temperature of X is 4500 K and that of Y is 6000 K.

Each graph shows the variation with wavelength λ of the power P emitted per unit wavelength by X (solid line) and by Y (dashed line). Which graph is correct?

The Hertzsprung–Russell diagram shows four stars P, Q, R and S.

Which row identifies the star with the largest radius and the star with the smallest radius?

A proposed space telescope would orbit the Sun in a circle of radius 5.0 AU. It would measure the parallax angle of a star from two positions on opposite sides of its orbit, in the same way as parallax angles are measured from the Earth.

What is (parallax angle measured from the telescope)/(parallax angle measured from the Earth) for the same star?

The Hertzsprung–Russell diagram has logarithmic scales on both axes. Four straight lines, A, B, C and D, are drawn through the position of the Sun.

Which line passes through the positions of all stars that have the same radius as the Sun?

The apparent position of a nearby star is measured against very distant background stars on two dates six months apart, when the Earth is on opposite sides of its orbit. Between the two dates the star appears to move through a total angle of 0.0500 arc-second.

What is the distance to the star? (1 pc = 3.26 ly)

The hydrogen absorption lines in the visible spectrum (Balmer lines) are weak in the hottest stars (surface temperature above about 30 000 K), even though these stars are mostly hydrogen. What is the best explanation?

In the future, the Sun's luminosity will increase by a factor of 1000 while its surface temperature halves. By approximately what factor will its radius increase?

As a star evolves off the main sequence, its luminosity increases by a factor of 100 and its surface temperature halves. By what factor does its radius increase?

At the temperature of the Sun's core, the mean kinetic energy of a proton is a few hundred times smaller than the energy needed to overcome the electrostatic repulsion of another proton. Why does fusion still happen in the Sun?

A simple model gives the pressure needed at the centre of a star in equilibrium as proportional to M²/R⁴, where M is the mass and R the radius of the star. Star Y has twice the mass and twice the radius of star X. What is the ratio (central pressure of Y)/(central pressure of X)?

A student uses a star catalogue produced by a space-based astrometry mission to test whether five stars that appear close together in the sky belong to the same star cluster. The catalogue gives the parallax angle p of each star in milliarcseconds (1 mas = 1 × 10⁻³ arc-second).

(1 pc = 3.26 ly)

A space telescope in orbit around the Sun with the Earth measures the position of a nearby star against very distant galaxies at regular intervals for 1.5 years. The steady drift of the star caused by its own motion through space has been subtracted. The star lies in the plane of the Earth’s orbit, so the remaining offset Δx of its apparent position is along a straight line.

The graph shows Δx, in milliarcseconds (1 mas = 10⁻³ arc-second), against time t. Each measurement has an uncertainty of ±1.5 mas. (1 pc = 3.26 ly)

To model how the apparent brightness of a star depends on its distance, a student measures the intensity I of the light from a small filament lamp with a light sensor at different distances d from the filament, in a darkened room. Each distance is measured with a metre rule (uncertainty ±0.5 cm).

The student suggests that I = P/(4πd²), where P is the power emitted by the lamp as light. The graph shows log(I / W m⁻²) against log(d / m) for six of the readings.

Radiometers on satellites measure the solar constant S, the power per unit area received from the Sun above the atmosphere at the mean Earth–Sun distance of 1.50 × 10¹¹ m. The table gives the annual mean values of S from an eight-year satellite record.

In the core of the Sun, the overall effect of fusion is to convert four hydrogen-1 atoms (mass 1.007825 u each) into one helium-4 atom (mass 4.002602 u). (1 u = 931.5 MeV c⁻², 1 MeV = 1.60 × 10⁻¹³ J)

Before analysing the spectra of stars, a student uses a computer simulation of black-body radiation. For each temperature T the simulation draws the spectrum, and the student places a cursor on the peak to read the peak wavelength λ_max. Each reading has an uncertainty of ±10 nm.

The student tests the hypothesis that λ_max ∝ Tⁿ with n = −1. The graph shows log(λ_max / nm) against log(T / K), with error bars, for six of the temperatures.

A student uses a published computer model of the core of a Sun-like star. The model gives the power ε released by hydrogen fusion per kilogram of core material. In Table 1 the density is fixed at 1.5 × 10⁵ kg m⁻³ and the temperature T is varied. In Table 2 the temperature is fixed at 1.5 × 10⁷ K and the density ρ is varied. The student tests the hypothesis that ε = kρTⁿ, where k and n are constants.

The graph shows log(ε/W kg⁻¹) against log(T/K) for five of the temperatures in Table 1. The values calculated by the model have no random uncertainty.

A student analyses the continuous spectrum of a star recorded by a spectrometer on a telescope. The spectrometer gives the relative intensity of the light at fourteen wavelengths, each with an uncertainty of ±0.03. The graph shows the results.

The luminosity of the star is known to be 45 L_☉ with an uncertainty of ±6 %. (Wien’s displacement constant = 2.9 × 10⁻³ m K, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, L_☉ = 3.83 × 10²⁶ W, R_☉ = 6.96 × 10⁸ m)

A student uses a published computer model of a protostar of mass 1 M_☉ that is contracting under its own gravity. The table gives the core temperature T of the protostar at seven values of its radius R. Each value of T has an uncertainty of ±5 %.

The student tests the hypothesis that T = kRⁿ, where k and n are constants. The graph shows log(T/K) against log(R/R_☉) for six of the stages, with error bars.

Two methods of finding the radius of a star are compared. For six stars, an optical interferometer measures the angular diameter, from which the radius R_int is found with an uncertainty of ±3 %. A stellar database also gives the luminosity L of each star (uncertainty ±4 %) and its surface temperature T, found from its spectrum (uncertainty ±2 %). A student uses L and T to calculate a second value of the radius, R_LT, and tests the hypothesis that the two methods agree, so that R_LT = R_int.

The graph shows R_LT against R_int, with error bars, for five of the stars. (σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, L_☉ = 3.83 × 10²⁶ W, R_☉ = 6.96 × 10⁸ m)

A student uses a database of nearby white dwarfs to test whether white dwarfs of similar mass have the same radius. For six white dwarfs with masses close to 0.6 solar masses, the database gives the luminosity L (uncertainty ±5 %) and the surface temperature T (uncertainty ±2 %).

(σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, L_☉ = 3.83 × 10²⁶ W, radius of the Earth = 6.37 × 10⁶ m)

A student uses a published star catalogue to study eight main-sequence stars in one open cluster. The catalogue gives the surface temperature T of each star (uncertainty ±2 %) and its luminosity L (uncertainty ±6 %) in terms of the luminosity of the Sun, L_☉. The student tests the hypothesis that, for main-sequence stars, L = kTⁿ, where k and n are constants.

The graph shows log(L/L_☉) against log(T/K), with error bars, for seven of the stars.

A student uses a published table of computer models of main-sequence stars. For each model the table gives the mass M of the star, its mean luminosity L while on the main sequence and its main-sequence lifetime t. The model lifetimes have an uncertainty of ±10 %.

The student tests the hypothesis that t = k(M/L)ⁿ, where k is a constant and n = 1. The graph shows log(t/10⁹ years) against log(M/L) for six of the seven models, with error bars. (M_☉ = 1.99 × 10³⁰ kg, L_☉ = 3.83 × 10²⁶ W, c = 3.00 × 10⁸ m s⁻¹, 1 year = 3.16 × 10⁷ s)