AQA A-Level Physics: Astrophysics

An astronomical refracting telescope is set up in normal adjustment to view a distant star. It is built from two converging lenses: an objective of focal length $f_o = 1.20 \ \text{m}$fo​=1.20 m and an eyepiece of focal length $f_e = 25.0 \ \text{mm}$fe​=25.0 mm.

Describe and explain how this telescope produces a magnified image of a distant object when used in normal adjustment. Your answer should make clear:

• where the intermediate image of the objective is formed, and the role of each lens;
• what is meant by normal adjustment and the separation of the two lenses;
• how the angular magnification is defined and why a large $\frac{f_o}{f_e}$fe​fo​​ ratio is desirable.

(6 marks)

Astronomers studying a distant red supergiant record the following measurements.

Quantity	Symbol	Value
Apparent magnitude	$m$m	$+0.50$+0.50
Absolute magnitude	$M$M	$-5.0$−5.0
Peak wavelength of emission	$\lambda_{max}$λmax​	580 nm
Luminosity	$L$L	$8.00 \times 10^{27} \ \text{W}$8.00×1027 W

(a) Use the relationship $m - M = 5\log\left(\frac{d}{10}\right)$m−M=5log(10d​), where $d$d is in parsecs, to calculate the distance to the star in parsecs. (3 marks)

(b) Use Wien's displacement law, $\lambda_{max} T = 2.9 \times 10^{-3} \ \text{m K}$λmax​T=2.9×10−3 m K, to calculate the surface temperature of the star. (1 mark)

(c) Use Stefan's law, $L = 4\pi r^2 \sigma T^4$L=4πr2σT4 with $\sigma = 5.67 \times 10^{-8} \ \text{W m}^{-2}\,\text{K}^{-4}$σ=5.67×10−8 W m−2K−4, together with your answer to (b), to calculate the radius $r$r of the star. (2 marks)

Three stars are observed. For each one the wavelength at which its black-body emission peaks, $\lambda_{max}$λmax​, and its luminosity relative to the Sun are recorded. (The Sun's luminosity is $L_\odot$L⊙​.)

Star	$\lambda_{max}$λmax​ / nm	Luminosity
A	290	$0.010 \ L_\odot$0.010 L⊙​
B	970	$5000 \ L_\odot$5000 L⊙​
C	500	$1.0 \ L_\odot$1.0 L⊙​

The approximate temperature ranges of the spectral classes are: A $7500-11000$7500−11000 K; G $5000-6000$5000−6000 K; M below $3500$3500 K.

(a) Use Wien's displacement law, $\lambda_{max} T = 2.9 \times 10^{-3} \ \text{m K}$λmax​T=2.9×10−3 m K, to estimate the surface temperature of each star, and assign each to one of the spectral classes above. (3 marks)

(b) Using both the temperatures and the luminosities, identify which star is a white dwarf, which is a red giant, and which is a main-sequence star. Justify your choices. (2 marks)

A spectral line that has a rest (laboratory) wavelength of 500.0 nm is observed in the light from a distant galaxy at a wavelength of 530.0 nm.

Take $c = 3.00 \times 10^8 \ \text{m s}^{-1}$c=3.00×108 m s−1, $H_0 = 65 \ \text{km s}^{-1}\,\text{Mpc}^{-1}$H0​=65 km s−1Mpc−1 and $1 \ \text{pc} = 3.09 \times 10^{16} \ \text{m}$1 pc=3.09×1016 m.

(a) Use the Doppler relation $\frac{\Delta\lambda}{\lambda} = \frac{v}{c}$λΔλ​=cv​ to calculate the speed of recession of the galaxy. (2 marks)

(b) Use Hubble's law, $v = H_0 d$v=H0​d, to calculate the distance to the galaxy in Mpc. (1 mark)

(c) Use $H_0$H0​ to estimate the age of the universe in seconds, and convert your answer to years. (1 year $\approx 3.16 \times 10^7$≈3.16×107 s.) (2 marks)

Two reflecting telescopes are used to observe stars at a wavelength of $\lambda = 550 \ \text{nm}$λ=550 nm. Their objective (mirror) diameters are given below.

Telescope	Objective diameter $D$D
X	0.10 m
Y	2.0 m

(a) Using the Rayleigh criterion $\theta \approx \frac{\lambda}{D}$θ≈Dλ​, calculate the minimum angular separation that telescope Y can just resolve. (2 marks)

(b) The light-collecting power of a telescope is proportional to $D^2$D2. Calculate how many times greater the collecting power of Y is than that of X, and hence state one observational advantage of telescope Y. (2 marks)

The brightness of a star can be described using its apparent magnitude $m$m and its absolute magnitude $M$M.

(a) Define what is meant by the apparent magnitude and the absolute magnitude of a star. (2 marks)

(b) The star Sirius has an apparent magnitude of $m = -1.4$m=−1.4, while a fainter star has $m = +2.0$m=+2.0. State, with a reason, which star appears brighter in the night sky. (1 mark)