E.5 Fusion and stars

The Hertzsprung–Russell diagram shows the Sun on the main sequence and four possible paths, A, B, C and D, that a star might follow after the hydrogen in its core is used up.

Which path represents the evolution of the Sun?

The graph shows the spectra of the light emitted by two stars, X and Y. Both stars may be treated as black bodies.

What is the ratio (surface temperature of X) / (surface temperature of Y)?

Star X has luminosity 4L and is at distance 2d from the Earth. Star Y has luminosity L and is at distance d.

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

A main-sequence star has a mass of 15 solar masses.

Which sequence describes the evolution of the star after its core hydrogen is used up?

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

In which region are stars whose outer layers expand and contract periodically, so that their luminosity varies?

Two stars X and Y have the same radius. The wavelength at which the spectrum of Y peaks is half the wavelength at which the spectrum of X peaks.

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

Stars P and Q are both on the main sequence. The mass of Q is greater than the mass of P.

Which row compares the surface temperature and the radius of Q with those of P?

Two stars, X and Y, have the same surface temperature. The luminosity of X is 100 times the luminosity of Y. What is \(\frac{R_X}{R_Y}\), the ratio of their radii?

What prevents a main-sequence star from collapsing under its own gravity?

Two deuterium nuclei fuse to form a helium-4 nucleus. Which statement explains why energy is released?

On an HR diagram, star P lies directly above star Q. Both stars have the same surface temperature, and the luminosity of P is 10⁴ times that of Q. What is \(\frac{R_P}{R_Q}\), the ratio of the radius of P to the radius of Q?

Hydrogen nuclei in the core of a main-sequence star fuse only because the core temperature is about 10⁷ K. What is the main reason such a high temperature is needed?

The apparent brightness of the Sun at the Earth's orbit (1.0 AU) is 1.36 kW m⁻². A solar probe is designed to operate at 0.50 AU from the Sun. What is the apparent brightness of the Sun at the probe?

A robotic survey telescope measures the parallax angle of a nearby star as 0.040 arcseconds. What is the distance to the star?

Two stars A and B have the same radius. The peak wavelength in the spectrum of A is half the peak wavelength in the spectrum of B. What is the value of \(L_A/L_B\)?

Which statement explains the dark absorption lines seen in the spectrum of a star?

Which statement describes the equilibrium of a main-sequence star?

A red giant and a white dwarf have the same luminosity. The surface temperature of the white dwarf is five times that of the red giant.

What is the ratio (radius of the red giant) / (radius of the white dwarf)?

The rate of fusion in the core of a main-sequence star briefly increases.

Which sequence describes how the star returns to equilibrium?

Three changes to the core of a star are proposed. Each change is made on its own, with everything else kept the same:

I. increasing the temperature of the core
II. increasing the density of the core
III. replacing the hydrogen nuclei by nuclei of larger proton number

Which changes would increase the rate of fusion in the core?

In a young star, deuterium fuses with protons: ²₁H + ¹₁H → ³₂He + γ.

The binding energy per nucleon is 1.11 MeV for ²₁H and 2.57 MeV for ³₂He.

What is the energy released in one reaction?

Observed from the Earth six months apart, a nearby star appears to move by a total of 0.032 arc-second against the very distant background stars.

Which row gives the distance to the star?

The Hertzsprung–Russell diagram shows the Sun and four stars W, X, Y and Z.

Which star has the largest radius?

On a Hertzsprung–Russell diagram, luminosity is plotted on a logarithmic scale against surface temperature on a logarithmic scale that decreases to the right. Lines of constant radius are straight lines on this diagram.

Which row describes the lines of constant radius?

For main-sequence stars, luminosity L is proportional to M^3.5, where M is the mass. Assume the fraction of a star's mass that undergoes fusion is the same for all main-sequence stars. A star has a mass 4 times that of the Sun. What is the ratio of its main-sequence lifetime to that of the Sun?

After a supernova, the collapsed core left behind has a mass of 2.5 solar masses. Take the Chandrasekhar limit as 1.4 solar masses and the Oppenheimer–Volkoff limit as about 3 solar masses. What is the remnant?

A star is plotted in the bottom-left region of a Hertzsprung–Russell diagram. What can be deduced about the star?

Fusion in the core of a massive star produces successively heavier elements, but stops once the core is mainly iron. Why?

A space telescope orbits the Sun in a circular orbit of radius 1.5 AU. Observations made six months apart show that a distant star has a parallax angle of p arcseconds. The parallax angle is half the total angular shift. What is the distance to the star in parsecs?

Star A has a black-body spectrum that peaks at 300 nm. Star B has a black-body spectrum that peaks at 900 nm. The two stars have the same radius. What is \(\frac{L_A}{L_B}\), the ratio of the luminosity of A to the luminosity of B?

An interstellar gas cloud of mass M, radius R and temperature T is close to the Jeans criterion for collapse. Which single change makes gravitational collapse more likely?

After a supernova, the remnant core of a massive star has a mass of 1.8 M_☉. No fusion takes place in the remnant. What is the remnant, and what prevents it from collapsing further?

Stars X and Y have the same luminosity. The parallax angle of X is four times the parallax angle of Y. What is \(\frac{b_X}{b_Y}\), the ratio of the apparent brightness of X to the apparent brightness of Y?

Stars X and Y have the same surface temperature. The radius of X is 3 times the radius of Y. X is 2 times as far from Earth as Y. What is \(\frac{b_X}{b_Y}\), the ratio of the apparent brightness of X to that of Y?

The hydrogen Balmer absorption lines are produced when electrons in hydrogen atoms in the n = 2 level absorb photons. These lines are weak in the spectra of the hottest O-type stars (T > 30 000 K). What is the best explanation?

Star P has its peak emission at 400 nm. Star Q has its peak emission at 800 nm. The two stars have the same radius. What is \(\frac{L_P}{L_Q}\), the ratio of the luminosity of P to that of Q?

Which statement about the CNO cycle in stars is correct?

After a supernova, the collapsed core left behind by a massive star has a mass of 1.8 solar masses. Take the Chandrasekhar limit as about 1.4 solar masses and the Oppenheimer–Volkoff limit as about 3 solar masses. Which row correctly identifies the remnant and what prevents its further collapse?

A main-sequence star has a mass four times that of the Sun. Assume the mass–luminosity relation L ∝ M^3.5. What is the ratio of this star's main-sequence lifetime to the Sun's main-sequence lifetime?

The binding energy per nucleon is 1.11 MeV for hydrogen-2, 2.83 MeV for hydrogen-3 and 7.07 MeV for helium-4. In the fusion reaction hydrogen-2 + hydrogen-3 → helium-4 + neutron, what is the energy released?

A star has the same surface temperature as the Sun and a luminosity of 100 times that of the Sun. What is its radius compared with the Sun's radius?

Why must the core of a star reach a very high temperature before hydrogen fusion can occur?

The core of a contracting protostar is modelled as a fixed mass of ideal gas. As the protostar contracts, the radius of the core halves and the temperature of the core becomes three times larger. The number of particles in the core does not change.

By what factor does the pressure in the core change?

A star has luminosity L and its apparent brightness at the Earth is b. One parsec is equal to P metres.

What parallax angle, in arc-seconds, would be measured for this star from the Earth?

A proposed space telescope would orbit the Sun at a radius of 5.2 AU. It measures the parallax angle of a star, using the radius of its own orbit as the baseline, as 0.13 arc-second.

What is the distance to the star?

Star A has twice the surface temperature of star B. The luminosity of A is one quarter of the luminosity of B. What is \(\frac{R_A}{R_B}\)?

Over a stage of its evolution, a star's luminosity stays constant while the wavelength at which its spectrum peaks increases by 20%. By what factor does its radius change?

For main-sequence stars, luminosity is proportional to M^3.5, where M is the mass of the star. The Sun's main-sequence lifetime is about 1×10¹⁰ years. Assume the same fraction of each star's mass is fused. What is the approximate main-sequence lifetime of a star of mass 4 M_☉?

For main-sequence stars, luminosity L is related to mass M by L ∝ M^3.5. The Sun's main-sequence lifetime is T_☉. What is the best estimate of the main-sequence lifetime of a star of mass 2M_☉?

An experimental fusion reactor uses the reaction ²H + ³H → ⁴He + n, which releases 17.6 MeV. The reactants have negligible momentum. Which is the best estimate of the kinetic energy of the neutron?

A main-sequence star has twice the mass of the Sun. For main-sequence stars, \(L \propto M^{3.5}\). Assume the main-sequence lifetime is proportional to the available fuel divided by the luminosity. What is \(\frac{\text{lifetime of the star}}{\text{lifetime of the Sun}}\)?

Star X has twice the luminosity of star Y. The parallax angle of X is 0.050 arcseconds and that of Y is 0.020 arcseconds. What is the value of the ratio of the apparent brightness of X to the apparent brightness of Y?

An astronomer studies six stars that move together through space as a group. All six have identical spectra, so she proposes that they have the same luminosity L. For each star a space telescope measures the parallax angle p and a calibrated photometer measures the apparent brightness b. If the luminosity is the same for every star, then b = Lp²/(4πK²), where p is in arc-seconds and K = 3.09 × 10¹⁶ m is the number of metres in one parsec.

The graph shows b against p² with a line of best fit through the origin. (1 pc = 3.26 ly)

An astronomer determines the luminosity L and the surface temperature T of six white dwarfs of similar mass. The luminosities come from measurements of apparent brightness and parallax, and the temperatures from the stellar spectra. She suspects that the six stars have the same radius, and plots lg(L/L_☉) against lg(T/K). The graph shows the data with a line of best fit.

(L_☉ = 3.85 × 10²⁶ W, R_☉ = 6.96 × 10⁸ m, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

A nearby star lies in the plane of the Earth's orbit. Over six months an astronomer measures its angular position x relative to a reference point fixed by very distant background stars; the time t is measured in months from the moment when x is greatest. The motion is modelled by x = x₀ + p cos(2πt/12), where x₀ is a constant and p is the parallax angle. Each value of x is uncertain by ±0.8 × 10⁻³ arc-second.

The graph shows x against cos(2πt/12) with the line of best fit (solid) and the lines of maximum and minimum gradient (dashed).

A spectrometer on a telescope records the continuous spectrum of a nearby star. The relative intensity is measured at 13 wavelengths λ between 350 nm and 950 nm, and a smooth best-fit curve is drawn through the points. The broad, flat top of the curve makes the peak hard to locate precisely. Interferometer measurements give the radius of the star as (0.84 ± 0.03) R_☉.

The graph shows the data and the curve. (R_☉ = 6.96 × 10⁸ m; σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

At noon on a clear day a student points a thin copper disc, blackened on its front face, directly at the Sun. The disc has a mass of 25.0 g and a diameter of 4.00 cm, and its back face is insulated. A thermocouple fixed to the disc records its temperature θ every minute, starting when the disc is at the temperature of the surrounding air. The graph shows θ against time t with a curve of best fit.

(Specific heat capacity of copper = 385 J kg⁻¹ K⁻¹; Earth–Sun distance = 1.50 × 10¹¹ m; R_☉ = 6.96 × 10⁸ m; σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

An ultraviolet space telescope observes a star whose continuous spectrum peaks at a wavelength of 350 nm. The radius of the star is 2.1 times the radius of the Sun.
Radius of Sun = 6.96×10⁸ m; luminosity of Sun = 3.83×10²⁶ W; surface temperature of Sun = 5.8×10³ K; Wien constant = 2.9×10⁻³ m K; Stefan–Boltzmann constant σ = 5.67×10⁻⁸ W m⁻² K⁻⁴.

A space astrometry mission orbits close to Earth. It measures the parallax angle of a faint red dwarf as 4.2×10⁻³ arcseconds.
1 pc = 3.086×10¹⁶ m.

A space-based astrometry telescope measures the stellar parallax angle of a nearby star as 0.0125 arc-seconds.

A robotic space telescope observes a star whose parallax angle is 0.125 arcseconds.

The diagram shows the Earth at two positions, E₁ and E₂, six months apart in its orbit of radius 1 AU about the Sun S. The parallax angle p of the nearby star Q, marked on the diagram, is 0.0400 arc-second.

(1 AU = 1.50 × 10¹¹ m, 1 pc = 3.26 ly, L_☉ = 3.85 × 10²⁶ W)

The Hertzsprung–Russell diagram shows the predicted evolutionary track of the Sun. S marks the Sun today, P the top of its red-giant stage and W the white dwarf that it eventually becomes.

(L_☉ = 3.85 × 10²⁶ W, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, 1 AU = 1.50 × 10¹¹ m)

When the hydrogen in the core of a star of two solar masses has been used up, the core contracts until helium fusion begins at a temperature of about 10⁸ K. Hydrogen fusion needs only about 1.5 × 10⁷ K. Helium fusion occurs in two steps:

3 ⁴₂He → ¹²₆C     and then     ¹²₆C + ⁴₂He → ¹⁶₈O

Atomic masses: ¹H 1.007 825 u, ⁴He 4.002 602 u, ¹²C 12.000 000 u, ¹⁶O 15.994 915 u. 1 u = 931.5 MeV c⁻² = 1.66 × 10⁻²⁷ kg, 1 MeV = 1.60 × 10⁻¹³ J, L_☉ = 3.85 × 10²⁶ W, solar mass = 1.99 × 10³⁰ kg, 1 year = 3.16 × 10⁷ s.

The Hertzsprung–Russell diagram shows the Sun and three other stars S, V and W. The dashed lines are lines of constant radius. Stars in the shaded band have a luminosity that varies periodically.

(R_☉ = 6.96 × 10⁸ m, L_☉ = 3.85 × 10²⁶ W, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, 1 AU = 1.50 × 10¹¹ m)

Two stars A and B formed at the same time in the same cluster. Star A has a mass of 20 solar masses and a main-sequence luminosity of 4.0 × 10⁴ L_☉. Star B has a mass of 1.0 solar mass and a luminosity of 1.0 L_☉; its main-sequence lifetime is about 1.0 × 10¹⁰ years.

(Binding energy per nucleon: ²⁸Si 8.448 MeV, ⁵⁶Ni 8.643 MeV, ⁵⁶Fe 8.790 MeV — close to the maximum of the binding-energy curve.)

In the nineteenth century physicists debated what could power the Sun. The Sun has mass 1.99 × 10³⁰ kg, radius 6.96 × 10⁸ m and luminosity 3.85 × 10²⁶ W. Geological and radioactive dating show that it has been shining for about 4.6 × 10⁹ years at roughly its present luminosity.

(G = 6.67 × 10⁻¹¹ N m² kg⁻², c = 3.00 × 10⁸ m s⁻¹, 1 year = 3.16 × 10⁷ s; atomic masses: ¹H 1.007 825 u, ⁴He 4.002 602 u)

A star of initial mass 20 solar masses is on the main sequence. It later evolves off the main sequence, and its life ends in a supernova that leaves a collapsed core of mass 1.7 solar masses.
Take the Oppenheimer–Volkoff limit to be about 2.5 solar masses.

The table gives data for four stars. The Sun has a surface temperature of 5800 K.
Star — L / L_Sun — Surface temperature / K
P — 0.010 — 12 000
Q — 1.0 — 5800
R — 1.0×10⁴ — 3500
S — 3.0×10⁴ — 30 000

Models of the Sun's future predict that it will become a red giant with radius 150 R_Sun and surface temperature 3500 K. Later, it will become a white dwarf with radius 0.012 R_Sun and surface temperature 25 000 K. The Sun's present surface temperature is 5800 K.

A medical thermal-imaging camera detects radiation from human skin at a temperature of 307 K. Astronomers use the same black-body physics to study a red supergiant. The supergiant's spectrum peaks at 828 nm and its luminosity is 1.0×10⁵ L_☉.
Wien constant = 2.9×10⁻³ m K; σ = 5.67×10⁻⁸ W m⁻² K⁻⁴; L_☉ = 3.83×10²⁶ W.

The table gives data for two stars, P and Q.

Star P: parallax angle 0.0500 arcsec; apparent brightness 2.4×10⁻⁹ W m⁻²; peak wavelength 290 nm
Star Q: parallax angle 0.0125 arcsec; apparent brightness 1.5×10⁻¹⁰ W m⁻²; peak wavelength 725 nm

1 pc = 3.086×10¹⁶ m; Wien constant = 2.9×10⁻³ m K.

A white dwarf has a surface temperature of 2.5×10⁴ K and a luminosity of 0.026 L_☉.
Wien constant = 2.9×10⁻³ m K; σ = 5.67×10⁻⁸ W m⁻² K⁻⁴; L_☉ = 3.83×10²⁶ W.

A spacecraft orbiting Mars is powered by solar panels. The panels have a total area of 15 m² and an efficiency of 22%, and they always face the Sun. The luminosity of the Sun is 3.83 × 10²⁶ W. The mean distance of Mars from the Sun is 2.28 × 10¹¹ m.

A student analyses data for two stars from an online star catalogue.

Star P: peak wavelength 290 nm; apparent brightness 4.0 × 10⁻¹⁰ W m⁻²; parallax 0.020 arc-seconds.
Star Q: peak wavelength 725 nm; apparent brightness 9.0 × 10⁻¹⁰ W m⁻²; parallax 0.050 arc-seconds.

Use Wien's constant = 2.9 × 10⁻³ m K, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, 1 pc = 3.09 × 10¹⁶ m, L_☉ = 3.83 × 10²⁶ W and R_☉ = 6.96 × 10⁸ m.

A university's robotic telescope records the spectrum of a star. The spectrum shows a continuous background crossed by dark lines. One strong dark line, at 656 nm, is due to hydrogen. (h = 6.63 × 10⁻³⁴ J s, c = 3.00 × 10⁸ m s⁻¹, e = 1.60 × 10⁻¹⁹ C)

Sirius is the brightest star in the night sky. Its apparent brightness is 1.2 × 10⁻⁷ W m⁻², its distance from Earth is 2.64 pc and its surface temperature is 9900 K. (1 pc = 3.09 × 10¹⁶ m, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴)

An astrobiology team is deciding whether to search for life on planets orbiting a main-sequence star of mass 12 solar masses. The Sun's main-sequence lifetime is 1.0×10¹⁰ years. Assume L ∝ M^3.5 for main-sequence stars. Assume also that every main-sequence star fuses the same fraction of its mass before leaving the main sequence.

Astronomers estimate the age of the Galaxy from the coolest white dwarfs they can find. A newly formed white dwarf has a surface temperature of 25 000 K. Its radius stays constant as it cools.

The spectrum of a star has its peak intensity at a wavelength of 483 nm. Use Wien's constant 2.90×10⁻³ m K, σ = 5.67×10⁻⁸ W m⁻² K⁻⁴, solar luminosity 3.83×10²⁶ W and solar radius 6.96×10⁸ m.

A proposed fusion reaction in a stellar core is deuterium + deuterium → helium-3 + neutron. The nuclear masses are: deuterium 2.013553 u, helium-3 3.014932 u, neutron 1.008665 u. Use 1 u = 931.5 MeV c⁻².

A red supergiant has surface temperature 3500 K and luminosity 1.20×10³¹ W. Use σ = 5.67×10⁻⁸ W m⁻² K⁻⁴ and solar radius 6.96×10⁸ m.

A fishing-fleet navigation survey records a star with a parallax angle of 0.050 arcseconds and an apparent brightness of 3.4×10⁻¹⁰ W m⁻². Use 1 pc = 3.09×10¹⁶ m, solar luminosity 3.83×10²⁶ W.