Fusion & Stars (SL IB Physics)

Under the right conditions, four hydrogen nuclei can fuse to make a helium nucleus in a process known as the proton–proton cycle.    

Nuclei	Mass/ u
	1.0078
	4.0026

Show that 4 × 10^–12 J of energy is released as a result of the fusion of four hydrogen nuclei.

Fusion occurs naturally in the core of stars.

Explain why very high densities of matter and very high temperatures are needed to bring about and maintain nuclear fusion in stars.

While on the main sequence, the Sun maintains a constant luminosity of 3.86 × 10²⁶ W. It is predicted that the Sun will spend a total of 10¹⁰ years in this phase of its evolutionary cycle. 

Show that the Sun will convert a total mass of 2 × 10²⁹ kg of hydrogen into helium during its time on the main sequence. 

One day, the Sun will leave the main sequence and move on to the next stage of its evolutionary cycle.

Discuss what will happen to the Sun.

In your answer, you should outline

• the conditions that will initiate this change
• the nuclear processes that will occur 
• the physical changes that the Sun will undergo.

A star has a luminosity that is known to be 4.5 × 10²⁷ W. A scientist observing this star measures the intensity of the light received on Earth from the star as 2.6 nW m^–2.

Determine the distance of the star from Earth.

The scientist observing the star finds that the wavelength for which the maximum rate of emission occurs from the star is 430 nm.

The Hertzsprung–Russell diagram shows three main sequence stars, A, B and C. Star A is the one described in (a) and (b).

Determine the ratio of the radius of star A to the radius of star B.

Outline the final stages in the evolutionary cycle of star C.

The table summarises some information about four stars in the constellation Cygnus.

State which star has

The graph shows the rate of emission from two of the stars, Sadr and Aljanah, plotted against wavelength.

Without calculation, explain how the curves can be used to determine the surface temperature of the stars.

The surface temperatures of Aljanah and Sadr are known to be 4700 K and 7600 K respectively.

Determine whether this graph is consistent with Wien's displacement law.

Calculate the radius of Sadr in units of solar radius R_Sun. 

Radius of the Sun, R_Sun = 6.96 × 10⁸ m
Luminosity of the Sun, L_Sun = 3.83 × 10²⁶ W

State what is meant by the luminosity of a star.

In a comparison of two stars, A and B, the following data was collected

Surface temperature of star A = 25 000 K

Surface temperature of star B = 4300 K

The radius of star B was determined to be 1.1 × 10⁵ times larger than the radius of star A.

Calculate the ratio

Determine the wavelengths of light for which the maximum rate of emission occurs from stars A and B.

The graph shows the variation of rate of emission against wavelength for the Sun.

On the graph, sketch the variation of rate of emission against wavelength for stars A and B.

In the research into nuclear fusion, one of the most promising reactions is between deuterium and tritium nuclei  in a gaseous plasma.

Although deuterium can be extracted relatively easily from seawater, tritium is more difficult to produce. It can, however, be produced through the induced fission of lithium−6  nuclei. 

These reactions can be represented in the following nuclear equations:

The masses of the nuclei involved are given in the following table:

Calculate the mass of deuterium and lithium-6 required to produce a power output of 500 MW from a fusion plant for a year.

Assume an efficiency of one-third for the conversion of fusion energy into electrical energy. 

Over the coming decades, nuclear fusion reactors are expected to become commercially viable for the first time. Meanwhile, nuclear fission reactors have been in operation for decades already.

The aim of current fusion research programs is to achieve fusion 'ignition', which occurs when enough fusion reactions take place for the process to become self-sustaining.

Plasma is superheated matter. It is so hot that the electrons are stripped from their atoms, forming an ionised gas. 

The Sun is made up of gas and plasma and can be thought of as a giant fusion reactor. At its core where fusion takes place, the plasma is (mainly) protons with a temperature of about 1.5 × 10⁶ K.

Near the Sun's surface, however, protons have a mean kinetic energy of 0.75 eV, which is too low for fusion to take place.

Calculate the temperature of the Sun near its surface. Outline any assumptions you make.

By considering the distance of closest approach between two protons, explain why fusion does not occur near the Sun’s surface.

The energy produced by the Sun comes from a cycle of hydrogen fusion, during which the net effect is the fusion of 3 protons into a helium nucleus. One of the steps in the cycle is:

The amount of energy radiated away in this step is 5.49 MeV. 

The following data are available:

• Mass of nucleus = 2.01355 u
• Mass of proton = 1.00728 u  

Calculate the mass of the helium nucleus, in standard units

A Hertzsprung-Russell (HR) diagram shows how the luminosity L depends on the surface temperature T for a cluster of stars.

Massive stars in the later stages of their evolutionary cycle have a layered structure with different chemical elements in different layers.

Discuss the structure of these stars by referring to the nuclear reactions taking place.

One of the stars in the cluster is the variable star Alpha Cephei. It is known to be at a distance of 4.6 × 10¹⁷ m from the Earth.

Using measurements of parallax angle, astronomers can determine distances to all stars with a parallax angle larger than 2.4 × 10^–7 rad.

Deduce whether the distance to Alpha Cephei could be determined from parallax measurements.

The H-R diagram shows the logarithmic relationship between luminosity and temperature  for a sample of stars, where L_☉ and T_☉ represent the luminosity and temperature of the Sun respectively.

Two stars, X and Y, are shown with their corresponding radii, in terms of the radius of the Sun R_☉.

Deduce the colours of star X and star Y using the H-R diagram and the information in the table below.

The relationship between the mass M and luminosity L of the stars shown on the H-R diagram is given by

Using this mass-luminosity relation and the H-R diagram, determine the ratio 

In 2017, an ultra-cool star named Trappist-1 was discovered in our galaxy. It was found to have at least five of its own orbiting planets. 

Some of the data associated with Trappist-1 is shown in the table.   

The temperature T, in K, of a planet, its distance d from the star and the luminosity L of the star are related by the expression

[4]