A1.1 Lenses and Optical Telescopes 


The principal focus F, of a lens is the point through which rays parallel to the principle axis meet after passing through the lens


The focal length f, is the distance between the optical centre of the lens and the principle focus.


A refracting telescope is in normal adjustment when the final image is formed at infinity. The eye will then be most relaxed and

the objective and eyepiece lenses are separated by the sum of their focal lengths.



Angular magnification, M   =    (angle subtended by the FINAL IMAGE at the eye)          =  b / a   =  fo / fe

                                                       (angle subtended by the OBJECT at the unaided eye)


Cassegrain reflecting telescope has a concave (parabolic) objective and a convex secondary mirror.


Chromatic aberration is due to longer (red) wavelengths of light being refracted through smaller angles by a lens than shorter (blue) wavelengths.


Spherical aberration is due to rays further from the axis of a spherical mirror meeting closer to the mirror than rays nearer to the axis. The cure is

          to make the main concave mirror parabolic.

Reflectors vs. Refractors




Chromatic Aberration

Affects both objective and eyepiece lenses.

Less problematic, affects eyepiece only.

Spherical Aberration

Affects all spherical lenses.

Eliminated in main mirror by using a parabolic mirror

Light gathering power

Reflectors can be made with much larger apertures, so potentially much greater light grasp

Resolving power

Reflectors can be made with much larger apertures, so potentially higher resolving powers



Reflectors only need one surface to be ground accurately. For lenses its two surfaces and the glass must be perfect throughout.

Disadvantage of reflectors

The secondary mirror not only absorbs some of the incoming light, but generates a diffraction pattern too 

Collecting power of a telescope is proportional to D2  (diameter squared)

The resolving power of a telescope is the smallest angular separation that can be observed, and is given by  θ = λ / D.

where λ is the wavelength of light and D is the diameter of the telescope objective lens or mirror,  θ is in RADIANS.

The smaller θ is then the greater is the resolving power. Clearly a large diameter telescope objective is one way to achieve this.

Another measure of resolving power is to use Rayleigh's Criterion between diffraction patterns which states that 2 objects are

 just resolved when the bright central maximum of one object coincides with the first minimum of the other object.

See the diffraction pattern from a circular aperture:



A Charge Couple Device (CCD) is a silicon chip divided into picture elements or pixels.


Quantum efficiency = no. of incident photons producing a signal

                                             total no. of incident photons




Quantum Efficiency

About 1%

About 80%


0.0003 radians

New ‘retina’ sensors can easily exceed this figure

Variable time exposure

Long exposure photography means a very feint image (not visible with eye) can be captured with a CCD.

Disadvantage of CCD

High resolution CCDs are expensive


A1.2 Non Optical Telescopes

The resolving power of a radio telescope is also given by   q = l / D,

since λ is LARGE, diameter D needs to be LARGE to get a good resolving power.

The dish does not have to be as perfect as a mirror for a light telescope.  As long as the surface is within about 1/20 wavelength, then the focusing will be unaffected by imperfections.  Also the reflector does not have to solid.  Fine wire mesh will do, since radio waves will not pass through a gap less than one wavelength.

          UV is strongly absorbed by the atmosphere so space telescopes are best (eg. SOHO)

          Used to determine chemical composition, densities, and temperature of stars and the interstellar medium
X-rays are absorbed by Earth’s atmosphere so again space telescopes are best (eg. satellite CXO)
X-rays are expected from sources with very high temperatures: eg.neutron stars, black holes

A 1.3 Classification of stars by brightness

Apparent magnitude m   = brightness as observed from Earth

Hipparcos (or Hipparchus) Greek Astronomer around 100BC classified visible stars into 6 brightness categories

or magnitudes. The lower or more negative, the brighter is the star (eg. Sun is - 26). A non-quantitative brightness scale.

Best (feintest) naked eye visual magnitude = + 6 , with long exposure cameras this can be increased considerably.

with a camera and telescope it can be as high as +31 (Hubble telescope)  

A revised, quantitative form of the Hipparcos scale is in use today.

In the 1850s a British astronomer, Pogson, devised the following rule:

'for 2 stars a magnitude difference of 5 magnitudes represented a difference in brightness of 100X'

A change in 1 magnitude is in fact a change in brightness of 2.512 times, (1001/5 = 2.512). 

To compare the brightness of 2 objects we need to: 

·         calculate the difference in apparent magnitudes = n

·         then work out the ratio of the brightnesses = 2.512n.

Absolute magnitude M  = apparent magnitude at a distance of 10pc (parsecs)

1 pc = 3.26 light years  (1 lightyear (ly) is the distance travelled by light in 1 year

1 Astronomical Unit (AU)  is the distance between the Earth and the Sun.


 m  -  M  = 5 log (d / 10),   where d is the distance to the star, in parsecs, from Earth


A 1.4 Classification of stars by temperature


A black body is a perfect emitter and absorber of electromagnetic radiation. We assume that a star is a black body emitter


Study the intensity / wavelength variation for stars of different temperature:

Wien’s Law:   the wavelength at which most radiation is emitted is inversely proportional to the absolute temperature,  λmaxT = constant = 0.0029mK.


Stefan’s Law:  the luminosity (power output) of a star is proportional to its surface area, and to the fourth power of its absolute temperature, P = σAT4.

Limitations of the inverse square law.


Stellar spectral classes: O(hottest, blue)  B  A  F  G  K  M(coolest, red):


Balmer absorption lines for hydrogen (which occur when hydrogen atoms in the star get excited to the n=2 level)


the intensity of the Balmer absorption lines depends on the temperature of the star, so the temperature can be measured.


A 1.5 Hertzsprung-Russell (HR) Diagram

HR Diagram - classification and evolution of stars:

A plot of absolute magnitude ( -10 to +15) against temperature (50,000K to 2,500K) or spectral class OBAFGKM 

              O    B            A             F             G             K             M



 NB the HR diagram does not indicate time - the Sun actually spends most of its life (13 billion yrs) on the main sequence:


File:Solar Life Cycle.svg

A more massive star will use up its hydrogen much faster and go through the life cycle at a faster rate.


A 1.6 Supernovae, neutron stars, black holes

Supernova - a very luminous star explosion, occurs towards the end of a very massive stars life.


neutron stars - the remains of a massive star consisting entirely of neutrons- extremely dense, escape velocity close to light speed.


black hole - remains of very massive star, absorbs all light that hits it, escape velocity > than speed of light. Supermassive black holes at the centre of galaxies.

Gamma ray bursts - due to the collapse of supergiant stars to form neutron stars or black holes.

type 1a supernovae used as standard candles to determine distances.

Controversy concerning accelerating universe and dark energy.

The event horizon of a black hole is the boundary beyond which light cannot escape.


The radius of the event horizon is called the Swartzschild radius Rs, given by  Rs = 2GM / c2


A1.7 Cosmology


Doppler Shift    Df is will be + for recession, - for approach:



Evidence for the Big Bang theory is the observed red-shift of galaxies, and also the cosmic microwave background. Also the relative abundance of hydrogen and helium.


Hubble’s Law:    distance away of galaxy is directly proportional to recessional velocity,   v = Hd,    H = Hubble constant  = 65 ± 10 km s-1 Mpc-1


1/ H = age of the universe.


An open universe will expand for ever, but a closed universe will eventually collapse into a “Big Crunch”. The answer lies in knowing the density of the universe.


Quasars are the most distant, the most luminous and the most red-shifted bodies discovered so far. Many are intense radio sources.

Formation of quasars from active supermassive black holes.

A 1.8 Exoplanets

A planet orbiting a star other than the Sun

Detection methods: transit method - typical light curve for one orbit of planet.

Radial velocity method - doppler spectroscopy detects periodic shifts in the wavelength of light from the host star.

Detection difficulties due to long time scale between readings and variations too small to detect.


Tony Mead 3/7/17