A LevelPhysicsCIETopic Questions25. Astronomy & Cosmology25.2 Stellar Radii

Syllabus Edition

First teaching 2023

First exams 2025

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Stellar Radii (CIE A Level Physics)

Topic Questions

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1a
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State Wien’s displacement law.

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1b
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The colour of a star indicates its surface temperature. In a sample of stars, the main colours observed are

white            blue            orange            yellow            red        

blue-white            yellow-white   

List the colours from the coolest to the hottest.

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1c
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Fig. 1.1 shows how the rate of emission for three stars, A, B and C varies with wavelength.

25-1-2c-e-wien-graph-three-stars-cie-ial-sq

Fig. 1.1

Using Fig. 1.1, identify 

 
(i)
the hottest star,
[1]
(ii)
the coolest star,
[1]
(iii)
the brightest star.
[1]

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1d
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Fig. 1.2 shows the variation of wavelength with intensity for four different stars. 

On Fig. 1.2, identify the visible, ultraviolet and infrared regions of the graph.

8-2-3a-q-sl-sq-easy-phy
Fig. 1.2

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2a
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The Stefan-Boltzmann law states that the power output of a star depends on two factors. 

Place a tick (✓) next to the two correct factors.

Mass

  

Radius

  

Surface temperature

  

Core temperature

  
  

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2b
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The Sun emits radiation with a peak emission occurring at a wavelength of 5.2 × 10−7 m.

Show that the Sun has a surface temperature of about 6000 K.

Wien's displacement constant = 2.898 × 10−3 m K

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2c
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The radiation received from the Sun at the top of the atmosphere is 1.37 kW m−2. 

Show that the Sun's luminosity is about 4 × 1026 W.

Distance from the Sun to the Earth = 1.49 × 1011 m

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2d
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Use the Stefan-Boltzmann law to determine the radius of the Sun.

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3a
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In Table 1.1 below, place a tick (✓) next to the statement(s) that is/are associated with Wien's Law.

the lower the surface temperature of a star, the shorter the wavelength it emits at the peak intensity

  

the relationship between the highest intensity wavelength and surface temperature of a star is inversely proportional

  

the higher the surface temperature of a star, the greater the intensity of the radiation emitted at each wavelength

  
the relationship between the lowest intensity wavelength and surface temperature of a star is inversely proportional  

Table 1.1

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3b
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Fig. 1.1 shows the black body radiation curves of two stars with surface temperatures 750°C and 1000°C respectively.

8-2-2b-q-sl-sq-easy-phy

Fig. 1.1

On Fig. 1.1, sketch a curve to show how the peak intensity of radiation varies with wavelength.

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3c
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An astronomer observes a star which emits radiation with a peak emission wavelength of 644 × 10−9 m. 

Show that the surface temperature of the star is 4500 K.

Wien's displacement constant = 2.898 × 10−3 m K

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3d
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The radius of the star is 5.6 × 109 m.

The Stefan-Boltzmann law can be written as:

L space equals space sigma A T to the power of 4

(i)
State the meaning of L, A and T.
[3]
(ii)
Calculate the value of A. State the correct unit.
[3]
(iii)
Calculate the value of L. State the correct unit.
[3]

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1a
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In recent years, astronomers have discovered sources of fast radio bursts (FRBs) in other galaxies. Studies suggest that these sources may be a type of standard candle.

An FRB source emits intense bursts of radio waves, each burst lasting for a fraction of a second. The closest FRB source is in a massive spiral galaxy 4.60 × 1024 m from the Earth.

A detector of area 1.00 × 10−4 m2 on the surface of the Earth received bursts of radio waves. In one burst, 9.40 × 10−23 J of energy was received in a time of 1.15 ms.

(i)
Show that the luminosity of the source is about 2 × 1035 W.
[4]
(ii)
When FRB sources were first discovered, some observers suggested that the bursts might be extraterrestrial communications.
 
Suggest why this is unlikely.
 
luminosity of the Sun = 3.8 × 1026 W
[2]

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1b
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It has been suggested that FRBs are produced by a type of star known as a magnetar.

Magnetars are extremely dense, hot remnants of stars which produce intensely powerful magnetic fields. These magnetic fields power the emission of high-energy radiation, particularly X-rays and gamma rays.

Fig. 1.1 shows how the radiated intensity from a magnetar varies with photon energy.

25-1-1b-h-25-1-h-magnetar-wien-curve-photon-energy-cie-ial-sq

Fig. 1.1

Using Fig. 1.1, determine the surface temperature of the magnetar.

Wien's displacement constant = 2.898 × 10−3 m K

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1c
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Some literature about magnetars states they are similar in size to a large city.

Assess the accuracy of this statement.

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2a
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Fig. 1.1 shows the logarithmic relationship between luminosity L over L subscript ☉ and temperature T over T subscript ☉ 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☉.

25-1-3a-h-25-1-h-luminosity-temperature-main-sequence-graph-cie-ial-sq

Fig. 1.1 

 

Deduce the colours of star X and star Y using the information in Fig. 1.1 and Table 1.1.

 

Table 1.1

star colour surface temperature / T☉
blue > 5.5
blue-white 1.7 − 5.5
white 1.3 − 1.7
yellow-white 1.0 − 1.3
yellow 0.8 − 1.0
orange 0.6 − 0.8
red < 0.6

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2b
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The relationship between the mass M and luminosity L of the stars shown on Fig. 1.1 is given by

L space proportional to space M to the power of space 3.5 end exponent

Using this mass-luminosity relation and information from Fig. 1.1, determine the ratio 

 

fraction numerator d e n s i t y space o f space s t a r space X over denominator d e n s i t y space o f space s t a r space Y end fraction

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2c
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In 2017, an ultra-cool star in our galaxy Trappist-1 was discovered with at least five of its own orbiting planets. 

Astronomers have studied it extensively to determine if there is a possibility of finding life on some of the planets orbiting Trappist-1.
 

Table 1.1 below shows some data.

Table 1.1     

  Trappist-1 Sun
Luminosity L / L☉ 5.26 × 10−4 1
Surface temperature T / T☉ 0.4 1
Radius of star / R☉ R 1
Distance between Earth and Sun / m − 1.5 × 1011
Distance between planets and Trappist-1 / m 1.6 × 109
to 9.0 × 109		 −

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

fraction numerator T to the power of 4 d squared over denominator L end fraction space equals space 419 cross times 10 cubed

(i)
Show that the radius R of Trappist-1 is smaller than the radius of the Sun. 
[2]
(ii)
The average temperature of the Earth is about 290 K.
 
Using data from Table 1.1, comment on the possibility of life existing on some of the planets orbiting Trappist-1.

[4]

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1a
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A star has a luminosity that is known to be 4.8 × 1029 W. A scientist observing this star finds that the radiant flux intensity of light received on Earth from the star is 2.6 nW m–2.

(i)
Name the term used to describe an astronomical object that has known luminosity.

[1]

(ii)
Determine the distance of the star from Earth.




distance = ....................................... m [2]

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1b
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The Sun has a surface temperature of 5800 K. The wavelength lambda subscript m a x end subscript of light for which the maximum rate of emission occurs from the Sun is 500 nm. 

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

(i)
Show that the surface temperature of the star in (a) is approximately 6700 K. Explain your reasoning.

[2]

(ii)
Use the information in (a) and (b)(i) to determine the radius of the star.




radius = .................................... m [2]

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2a
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Table 1.1 summarises some information about four stars in the constellation Cygnus.

Table 1.1

Name Colour Luminosity / LSun
Aljanah orange 62
Deneb blue-white 196 000
Fawaris blue 155
Sadr yellow-white 33 000

Identify the star in Table 1.1 with

 
(i)
the highest surface temperature,
[1]
(ii)
the lowest surface temperature,
[1]
(iii)
the largest radius.
[1]

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2b
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Fig. 1.2 shows the rate of emission from two of the stars, Sadr and Aljanah, plotted against wavelength.

25-1-2b-m-25-1-peak-wavelength-wien-graph-cie-ial-sq

Fig. 1.2

Explain how the curves in Fig. 1.2 can be used to determine the surface temperature of the stars.

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2c
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The surface temperature of Aljanah is known to be 4700 K.

Determine the surface temperature of Sadr.

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2d
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Calculate the radius of Sadr in units of solar radius RSun. 

Radius of the sun, RSun = 6.96 × 108 m
Luminosity of the Sun, LSun = 3.83 × 1026 W

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3a
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State what is meant by the luminosity of a star.

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3b
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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

fraction numerator l u m i n o s i t y space o f space s t a r space B over denominator l u m i n o s i t y space o f space s t a r space A end fraction

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3c
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The wavelength λmax of light for which the maximum rate of emission occurs from the Sun is 5.00 × 10–7 m.

The surface temperature of the Sun is 5770 K.

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

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3d
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Fig. 1.1 shows the variation of rate of emission against wavelength for the Sun.

25-1-3d-m-25-1-wien-displacement-graph-sun-star-a-b-cie-ial-sq

Fig. 1.1

On Fig. 1.1, sketch the variation of rate of emission against wavelength for stars A and B.

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