Observational Astrophysics Flashcards
luminosity
energy radiated per unit time by a source
joules per second or watts
what is luminosity dependent on?
frequency (or wavelength)
objects of a particular colour radiate more power at frequencies corresponding to their colour
luminosity at a specific frequency (monochromatic luminosity)
L=L(v)
luminosity of a source in a frequency interval delta v centred on v0
L=L(v0)deltav
integral for luminosity
strictly speaking, luminosity should be the integral of L(v) dv between v0-1/2 delta v and v0+1/2 delta v
since delta v small, can approximate as L(v0)delta v
bolometric luminosity
total power by integrating across all frequencies
“energy per unit time radiated by all frequencies”
isotropic
uniform in all directions
what does assuming astrophysical sources are point sources that radiate isotropically allow?
relate luminosity to apparent brightness or flux
flux falls off with…
the square of the distance because of the surface area of a sphere increasing with the square of its radius
radiant flux, F
energy per unit time crossing a unit area perpendicular to the direction of light propagation
watts per square metre
flux for an isotropic point source
F=L/4piD^2
how is flux density denoted and units
denoted by:
F(v),S(v), Fv or Sv
unit is janksy (Jy)
why does flux density need to be defined
frequency dependent
if observe at frequencies v1 and v2, flux in this interval is…
F= integral Sv dv between v1 and v2
bandwidth of interval
delta v= v2-v1
mean frequency
v bar = 1/2(v2+v1)
if delta v is small or Sv is either flat or varies linearly with frequency, then
F=Sv bar delta v
integrated flux = flux density x bandwidth
solid angle
fraction of sky covered by an extended source
steradian (sr)
solid angle formula
omega = A/D^2
A = area
D=distance
solid angle of whole sky
4piD^2/D^2=4pi sr
solid angle for a spherical source
A=piR^2 so omega = pi(R/D)^2
show from trig that theta/2=R/D
so can be written as omega=pi(theta/2)^2
theta in radians
what is the need to introduce specific intensity?
an extended source may deliver the same flux density as a point source but is spread over a small area of the sky.
an extended source will not be equally bright across their entire projected area.
specific intensity
flux density of the source (through a plane perpendicular to the direction of the source) per unit solid angle
if rays arrive at an angle, the flux is
reduced by cos theta
specific intensity formula
Ivcos theta = dSv/d omega
small angle approx used so cos disappears
flux density in terms of specific intensity
integral of the specific intensity over the solid angle of the source
if Iv is constant over the source on the sky then…
Sv=Iv omegas
what does integral of specific intensity over frequency give you?
intensity
intensity has units of
energy per unit time per unit solid angle crossing a unit area perpendicular to the direction of propagation.
if Iv is constant over the frequency band then
I=Iv deltav
if we integrate intensity of all frequencies…
we obtain bolometric intensity
brightness temperature derivation
use taylor expansion to show exp(hv/kT)-1 is approx =hv/kT
then can plug into specific intensity equation for blackbody and rearrange for
Tb=c^2Iv/2v^2k
when will brightness temp correspond to the actual temp of the source
if its a blackbody and hv«kT
bolometric apparent magnitude
mbol=-2.5log10F+const (pogson)
vega defined to have mbol=0 so used as calibrator
Johnson system
set of standard filters from near ultraviolet to infrared
transmission function T
defines the fraction of light transmitted by the filters as a function of frequency (or wavelength)
for bolometric apparent magnitude , T(v)=
1 at all frequencies
difference between the Johnson magnitudes defines a
colour index which gives information on the temperature of the star
bolometric correction
difference between the bolometric magnitude and the Johnson V band
measure of what fraction of light is observable in the visible band
BC=mbol-V
extinction
effect where some light is absorbed on the way to us
colour excess or reddening
Eb-v=(B-V)-(B-V)0
why is the sky blue
light encounters atmosphere
scatters light
blue scattered more than red
refractor
objective lens forms a real image of source in focal plane
eyepiece forms a real image of source
magnification (ratio of angular size of virtual image to that of source)=fobj/feye=Dobj/Deye
can either view a virtual image by…
through the eyepiece or place a detector at the focal plane
reflectors
cheaper to make and shorter
types of reflectors
newtonian
cassegrain
gregorian
ritchy-chretein
cassegrain and gregorian
both have parabolic mirrors
gregorian has concave, elliptical secondary mirror
cassegrain has a convex, hyperbolic secondary mirror
ritchey-chretein
special form of the cassegrain with two hyperbolic mirrors
free from spherical abberation at a flat focal plane so good for wide field and photographic observations
reflector properties
virtual image viewed through eyepiece
collection area of obj = pi(Dobj/2)^2
light observed through eyepiece with area=pi(Deye/2)^2
flux density increased by (Dobj/Deye)^2
in reflector, secondary mirror is always
smaller, light squished into smaller area
reflector solid angle
increase by (Dobj/Deye)^2
illumination, J
energy per second per unit area in the focal plane
same units as flux
energy collected per second by the telescope aperture
flux density x bandwidth x area
focal ratio
F=f/D
rewriting illumination using focal ratio
J approx I/F^2
increasing illumination…
reduces the time it takes the detector to collect a given number of photons.
increasing aperture diameter
+ increased illumination and better angular resolution
-more expensive to build and house
increase focal length
+ increase image size in focal length so it covers more detector pixels
-decreases illumination, requiring longer exposure times
photo multiplier tubes - incident photon strikes a…
photocathode, held at a potential of 1kV relative to the anode on the other side of a vacuum tube
photomultiplier tubes - what are the anode and cathode separated by?
a series of dynodes at successively more positive potentials
(amplification stage)
photomultiplier tubes - the electron emitted from the cathode is…
accelerated towards the first dynode where they have enough energy to release several more electrons
(repeated and a cascade of electrons reach the anode)
con of photomultiplier tubes
little directional information
(think of as single pixel) so poor imaging
quantum efficiency approx 10%
describe image intensifiers
photoelectrons emitted by cathode are accelerated down the evacuated tube by a voltage difference of around 15 kV and strike a phosphor screen, producing an image
photons emitted strike a second cathode and process repeats
intensity of image increases with each stage
what does image intensifiers give you
maintain poisson info so get a 2D image rather than a single number
think of as night vision goggles
quantum efficiency of image intensifiers
20 to 30%
describe a CCD
semiconductor array of pixels
electron released when a photon strikes the semiconductor
bias voltage draws electrons into a potential well where they are stored
stored charge read out by manipulating the bias voltage
charge moves across the CCD line by line before being read out pixel by pixel (show)
high quantum efficiency
until saturation, no of electrons stored prop to no of incident photons
saturation
point where electrons spill over
pros of CCDs
readout noise is very low
readout rate very high
quantum efficiency around 90% nowadays
sloan CCD array
6 columns of 5 CCDs
Different filter on each CCD (wavelengths from UV to IR, create false colour images)
operates in two modes
two modes of sloan CCD array
stare mode - telescope is tracking, single exposure
dirft scan mode - target takes about 1 minute to drift across on the CCD columns
pixel capacity
number of electrons that a pixel (potential well) can store
for long exposure, the number of electrons produced…
may exceed the pixel capacity
dynamic range
range over which detector response is linear
brightest object before saturation/ faintest object detectable is approx…
60dB
what does adding many short exposures together in software do?
prevents bleeding effect
systematic errors which need to be eliminated from CCD observations
Bias
Dark (thermal) current
response factor
Bias
signal from the CCD due to residual charge in the CCD and readout noise rather than photons
think of as electron already in the bucket so offset
how to measure bias
taking an exposure of zero seconds followed by read out the CCD chip
dark (thermal) current
signal from electrons produced in the absence of light due to thermal emission in the CCD
will vary pixel to pixel but grow over time
response factor
signal readout from each pixel differs from the true signal due to variations in the sensitivity across the CCD
non uniform response in the CCD
for pixel at (x,y) can model this distortion by response factor r(x,y)
what can r(x,y) be estimated from
a flat field observation: an exposure for a uniform light source for which the ‘true’ signal should be uniform
*new flat field required each observing session since irregularities (eg dust) can very
relation between systematic effects
signal measured from pixel = true signal x response factor + bias at pixel + dark current from pixel
what is r(x,y) prop to?
flat field signal - bias at pixel - dark current
sensitivity of an instrument or detector
the smallest signal that it can measure which is clearly not noise
noise
random signal from another undesired source
how to measure reliability of an observation
SNR
expected signal level / expected noise level
minimum SNR
not trusted unless at least 3
(preferably much higher)
signal can be easily detected after background subtracted because…
SNR is large
S is large compared to N
low SNR
signal, background and noise have comparable sizes
difficult to estimate the amount of background to subtract
what is fluctuation in no of photons expected in given time interval
uncertainty
axioms for counting photons
- photons arrive independently in time
- average photon arrival rate is constant
if observed photons satisfy axioms, then
number of photons observed will follow a poisson distribution
expectation value of the number photons
E(N)-<N>=Rtau</N>
N=no of photons
mean photon arrival rate is R per second
tau = exposure time
series of observations - wouldn’t expect exactly the same no of photons for each. Instead…
average number of photon counts
<N>=R_tau
</N>
as Rtau increase, shape of poisson distribution
becomes more symmetric
tends towards a normal/gaussian distribution
variance of N
measure of the spread of the distribution
= mean squared standard deviation of N from <N></N>
for a poisson distribution, var(N)=
R_tau
estimate of the photon rate
R hat = No of obs/tau
quote our experimental estimate of the mean no of photons in tau as
Nobs +/- sqrt (Nobs)
since standard deviation is sqrt(Nobs)
total noise variance
found by adding together all the individual sources
sigma total ^2 = sigma poisson ^2 + sigma other ^2 +…
sources of poisson noise
fluctuations in the photon count from the sky
dark current: thermal fluctuations in a CCD
no of photons collected is…
total energy collected by detector / hv0
total energy = FAT = SvA deltav tau
correcting for combined quantum efficiency of telescope and detector
Ntot= n SvA delta v tau/h vo
n is fraction of photons that produce response in the detector
maximise SNR
bigger telescope
larger observing time
large bandwidth
when the noise of dominated by counting statistics of photons, the sensitivity of a telescope…
only increases with the square root of the aperture’s area
in radio astronomy, noise is often dominated by
noise in the receiver electronics
when noise of dominated by detector electronics, the sensitivity of the telescope…
increases in proportion to its aperture
when is it not true that increasing bandwidth increases SNR
if source only emits over a narrow frequency
will just increase noise without increasing any signal
line profile
specific intensity or flux density as a function of frequency
shape of line profile characterised by a measure of its width
why is spectroscopy important
allows us to deduce many physical characteristics of planets, stars and galaxies
learn about chemical elements from
frequency
learn about chemical abundances from
relative intensities
learn about bulk velocity from
frequency
learn about temp, pressure, gravity from
line width
learn about spread of velocities from
line width
learn about magnetic and electric fields from
‘fine structure’ in lines
eg Zeeman splitting
spectral resolving power
characterises effectiveness of filter or any spectroscopy instrumentation
R=vo/delta v = lambda0/delta lambda
useful spectral resolving power
approx 10^5 or higher
ie sensitive to Doppler shifts of a few km/s
much higher spectral resolving powers can be achieved by…
using a prism or diffraction grating
difference in prism and diffraction grating dispersing light
prism disperses blue light more strongly than red light
diffraction grating disperses red light more strongly than blue light
for diffraction grating with light incident at right angles, diffracted light has an intensity maximum when path difference satisfies
asin theta = n lambda
a = spacing between grating lines (not number of grating lines)
n = order of the maximum
if light is incident at an oblique (not perpendicular) angle…
maxima occurs at pd which satisy
a(sin theta + sin alpha)=n lambda
how to get expression for angular dispersion
differentiating maxima expressions
how can higher angular resolution be achieved
large n
small a and/or large theta
linear dispersion
when focal length taken into account
dx=fdtheta
relationship between linear and angular dispersion
linear dispersion = f x angular dispersion
intensity profile from a monochromatic light source diffracted by an infinite diffraction grating
a series of infinitely thin peaks equally spaced in sin theta
grating with N rules
for the mth opening, phase difference with 0th opening is
m (2pi/lambda) a sin theta
what limits the resolving power of a diffraction grating
the finite width of the peak
we can only resolve two lines if they are…
separated by at least their width
R=lambda/delta lambda= Nn
key features of a spectrometer design
aperture - slit - collimating lens - grating - focussing lens - detector
(in that order)
collimating lens
diverging rays become parallel again
after grating, individual colours…
parallel
eg all red rays are parallel
slit cuts out
unwanted light
angular size at collimating lens defines
range of angles entering the grating
diameter of the collimating lens
approx the width of the grating so little light is lost
focussing lens will produce a diffraction pattern that
spreads out the light over a width around lambda/Dfocus
we want lambda/Dfocus < lambda/Dgrating so
Dfocus > D grating
light from a distance point source arrives at a telescope as
plane waves which are diffracted as they pass through circular aperture
can work out airy disc pattern depending on
size of aperture and wavelength of incident light
resolving power - simplified 1D analysis
integrate diffraction pattern across a slit of width D
consider light emerging the aperture at theta to axis
the diffracted signal at an angle theta is obtained by
integrating e^i2pi theta x/lambda
from x=-D/2 to x=D/2
sinc x
sin x /x
maximum at x=0 and zeros occur at x=+/-mpi
when light from two point sources observed, telescope will produce
image that is combination of the diffraction pattern of each of the stars
airy discs would overlap if they are too close
we regard two stars as resolvable if
central max of diffraction pattern of one star coincides with first min of other
angular sep satisfies theta min = 1.22lambda/D
if we observe a point source which is not monochromatic, observed intensity is
integral of intensity pattern at each observed wavelength
combined intensity pattern washes out
the side lobes and the secondary maxima are not visible
diffraction blurs point sources over an angle
theta is approx lambda/D
for an extended source, each point of the source is
blurred by the diffraction pattern which we call the point spread function (PSF)
convolved
we say the source function is convolved with the PSF
rayleigh criterion defines
theoretical angular resolving power of a telescope
diffraction limited
if we resolve features down to the rayleigh criterion
for small telescopes of aperture a few cms, the diffraction limit is
larger than the typical size of the seeing disc so these telescopes are not limited by seeing
for ground based optical telescopes, D>1m, diffraction limit
is much smaller than the seeing disc
so theoretical angular resolving limit is not achieved and we say that its seeing limited
resolving power - more exact 2D analysis
integrate the diffraction pattern over a circular aperture of diameter D
earth’s atmosphere is opaque to EM radiation apart from
two windows, one in optical band and another in radio band
model of the atmosphere as series of parallel slabs
slab thickness dl
intensity I incident perpendicular to slab
dI=-Ikdl where k=absorption coefficient
optical depth
Iobs=I0e^-t
if t=0
atmosphere transparent Iobs=I0
if t«1
atmosphere optically thin I obs approx=I0
if t>1
atmosphere optically thick
Iobs«I0
observing stat at zenith angle theta
path length through slab of thickness dl=
dl/cos theta = dlsec theta
refraction - approach of modelling atmosphere as plane-parallel slabs and light incident at theta valid for
angles less than 60 degrees
for larger angles, need to account for Earth’s curvature
air molecules, dust and water vapour all
scatter light
effect depends on the size of the scattering particle
particle size a, where a»λopt
particles scatter all wavelengths equally
atmospheric water droplets fall into this category, why clouds appear white
scattering for particle sizes a approx= λopt
scattering strength prop. to 1/λ
why cigarette smoke has bluish tinge
scattering for particle size a«λopt
scattering strength prop. to 1/λ^4 so blue light strongly scattered
rayleigh scattering
why daytime sky is blue and red at sunset when blue scattered out of line of sight
scattering is anisotropic and
light from the sky is polarised
loss of intensity due to scattering
dI=-Isigmadl
what is scintillation
‘twinkling’of starlight caused by turbulence in the atmosphere
air cells of varying density and refractive index are continually passing through the line of sight to the star changing the illumination
cells deflect the light from a point source over a
seeing disk (angular radius approx 3arcsec)
if telescope aperture is such that D is approx r0(typical scale of the cells)
see rapid variations in positions and brightness of the image as individual cells move across the line of sight
if telescope aperture is such that D»r0
image formed from many cells added together
average brightness of image approx constant but rapid variations in position, size and shape of seeing disk
radio astronomy affected by scintillation, not from atmosphere but from
turbulence in the interstellar medium and the interplanetary medium
typical size of cells much larger
extremely high energy gamma ray detectors
energy >10^12 eV
do not detect these photons directly but see effect they have on atmosphere
they collide with atoms to create shower of subatomic particles
produces cherenkov radiation
high energy gamma rays
E=100mEv
need to get above most of atmosphere to detect
photons produce positron-electron pairs
track path through spark chambers
low energy gamma rays and x-rays
1keV<E<20keV
use propoetional counter: tube filled with gas
xray ionises gas atom and liberates an electron
electron accelerates towards anode, ionsing more atoms
leads to cascade of electrons
low energy gamma rays and x-rays
E>20 keV
x-rays would pass straight through gas
instead used solid state detector or scintillation counter
x-ray enters and ionises atom, electron excites other electrons, impurity atoms capture electrons
captures lead to bright flahses detected by photomultiplier
we can image x-rays using
grazing incidence optics
series of nested surfaces of high conducting material
x-rays reflected for large incidence angles
UV and optical wavelenghts
1eV<E<100eV
CCDs cooled to reduce thermal noise
near infrared wavelengths
0.04eV<E<1eV
CCD technology using semiconductors with appropriate band gap so that weak NIR photons can still eject electrons to produce current
far infrared wavelengths
10^-3eV<E<4x10^-3eV
too weak for CCDs but too high frequency for radio techniques
bolometer measures temp increase in crystlas when struck by photons
radio wavelengths
E<10^-4eV
treat as wave
radio waves detected by antenna where changing EM field induces a current
pre-amplifier produces voltage prop. to the current
radio antennas
chooses direction of observation
collects radiation
converts radiation into AC signal
radio receiver
amplifies the signal by gain factor
selects frequency and bandwidth
processes and records signals
radio - availability of phase information means
radio observations can be combined from different telescopes
