Stellar Radii (Cambridge (CIE) A Level Physics)

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]

The Sun has a surface temperature of 5800 K. The wavelength 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]

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

Table 1.1

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]

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

Fig. 1.2

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

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

Determine the surface temperature of Sadr.

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

The wavelength λ_max 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.

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

Fig. 1.1

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

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]

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.

Fig. 1.1

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

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

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

Assess the accuracy of this statement.

Fig. 1.1 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_☉.

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

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

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

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     

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

(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]