Download Second Year Problem sheet 3 +answers

Second Year Problem sheet 3
7th March 2002
Course ASTR2B15: Techniques in modern astronomy
(Answers to be handed in on Friday 14th March)
1. A telescope is equipped with an adaptive optics system that includes a deformable
mirror with effective actuator spacing, mapped back onto the primary mirror, of
20cm. If the seeing coherence length at 500nm is 0.20m and the fitting constant K
for the deformable mirror is 0.15, what is the maximum Strehl ratio that the
system can achieve at a wavelength of 1.6 microns?
[10]
2. Explain, including a sketch, how a stellar coronograph works. Include in your
explanation what the effect the focal plane mask has on the distribution in the
conjugate pupil plane of light from the central star.
What are the benefits that adaptive optics bring to coronography?
[5]
3. The Very Large Telescope Array is made up of many circular 26m diameter dish
radio telescopes. The maximum separation that can be achieved between
telescopes in the array is 27 kilometres. At a wavelength of 10cm, what is the
achievable angular resolution of a single telescope in the array and what is the
smallest achievable angular resolution of the whole array?
A radio source is emitting radiation at a frequency of 1 GHz. If heterodyne
detection of the signal is used with a local oscillator of 0.97 GHz, what will be the
frequencies of the various components of the signal coming out of the receiver?
(before filtering and rectifying).
[5]
4. A quarter wave plate is made of quartz that has a degree of birefringence J of
0.009. How thick must the quartz be to act as a quarter wave plate at a wavelength
of 550nm?
After an observation of light with a rotating polarizer no variation in intensity is
seen. On the addition of a quarter wave plate before the polarizer again no
variation is seen. What is the polarization state of the incident light? (Explain how
you arrived at your answer).
If linearily polarised light is incident on a half wave plate with the direction of
polarisation at angle of  to the optic axis of the plate, what is the state of
polarisation of the emergent light?
[5]
1. A telescope is equipped with an adaptive optics system that includes a
deformable mirror with effective actuator spacing, mapped back onto the
primary mirror, of 20cm. If the seeing coherence length at 500nm is 0.20m
and the fitting constant K for the deformable mirror is 0.15, what is the
maximum Strehl ratio that the system can achieve at a wavelength of 1.6
microns?
Coherence length at 500nm =0.2m
Coherence length at 1.6microns r1600
 1600 
 r500 

 500 
d
6
5
=0.808m
5
3
 2  K  
 r 
If d=0.2. k=0.15 and r=0.808m, then
5
3
 0.2
  0.0146
 0.808 
 2  0.15
Since
S.R.  exp  2
therefore maximum S.R.
S.R.  exp(0.0146)  0.985
[10]
2. Explain, including a sketch, how a stellar coronograph works. Include in
your explanation what the effect the focal plane mask has on the distribution
in the conjugate pupil plane of light from the central star.
What are the benefits that adaptive optics bring to coronography?
A stellar coronograph is a device that suppresses the light from a central bright
star preferentially to light from any object close to the star. This enables low
contrast objects close to the star to be studied.
Simple coronograph design
The coronograph consists of a small focal plane mask that sits over the image of
the central star.
This focal plane mask conditions the remaining light from the central star that
passes through to be at the edge of the pupil.
This light is then removed by a Lyot stop, which consists of an undersized
aperture at a pupil conjugate that masks off the light at the edge of the pupil.
The size of the Lyot stop is also important. If the stop is small then it will suppress
the central star well but the overall throughput of the system will be reduced.
Typically stop sizes with diameters of 80 percent of the full aperture are used.
The central secondary obstruction also needs to be masked out in the pupil plane
as light from the central star appears around its edge.
Advantages of adaptive optics and coronography
The image size is smaller and fixed and therefore a smaller focal plane mask can
be used and objects closer to the central star can be observed.
The faint objects will also be sharpened and hence brighter.
Therefore the combination of AO with coronagraphy will enable higher sensitivity
and spatial resolution imaging of regions close to the central star.
3. The Very Large Telescope Array is made up of many circular 26m diameter
dish radio telescopes. The maximum separation that can be achieved between
telescopes in the array is 27 kilometres. At a wavelength of 10cm, what is the
achievable angular resolution of a single telescope in the array and what is
the smallest achievable angular resolution of the whole array?
A radio source is emitting radiation at a frequency of 1 GHz. If heterodyne
detection of the signal is used with a local oscillator of 0.97 GHz, what will be
the frequencies of the various components of the signal coming out of the
receiver? (before filtering and rectifying).
Resolution of individual telescope is given by Rayleigh Criteria as
θ =1.22 λ /d
where λ is the wavelength and d the telescope diameter. Therefore at a
wavelength of 10cm and with a telescope diameter of 26m. The resolution is
θ =1.22 x0.1/26=4.69x10-3rad
= 968 arcsec.
Resolution of whole array is given by
θ =λ /d
where λ is the wavelength and d the greatest telescope separation diameter.
Therefore at a wavelength of 10cm and with a greatest separation of 27km. The
resolution is
θ =0.1/27000=3.7x10-6rad
= 0.76 arcsec.
Frequencies coming out of heterodyne receiver are
2nd harmonic of signal
2 GHz
2nd harmonic of oscillator 1.94GHz
Sum frequency
1.97GHz
Intermediate frequency
0.03GHz - 30MHz
DC signal
4. A quarter wave plate is made of quartz that has a degree of birefringence J of
0.009. How thick must the quartz be to act as a quarter wave plate at a
wavelength of 550nm?
After an observation of light with a rotating polarizer no variation in
intensity is seen. On the addition of a quarter wave plate before the polarizer
again no variation is seen. What is the polarization state of the incident light?
(Explain how you arrived at your answer).
If linearily polarised light is incident on a half wave plate with the direction
of polarisation at angle of  to the optic axis of the plate, what is the state of
polarisation of the emergent light?
For the 1/4 wave plate
Optical path difference o. p.  ( ne  no ) d
where d is the depth of the plate
Phase difference 
If (ne no )  0.009 ,
2

(ne  no )d
  550nm , phase difference = 
2
550x10 9
d

 1.53x10 5 m
4(ne  no ) 4 * 0.009

If no variation is seen with rotating polariser then light must either be circularly
polarised, unpolarised or a mixture of both.
If a quarter wave plate is added and no variation is seen again in the intensity then
the light must be unpolarised, sine if there was an circularly polarised component
this would be converted to linearly polarised light which would give an intensity
maximum.
If linearly polarised light is incident on a 1/2 wave plate at an angle of  to the optics
axis the light will stay linearly polarised but the plane of polarisation will be rotated
by 2.