#### Transcript Lecture 1 Coordinate Systems - Department of Physics & Astronomy

```Review Ch 1-6
• The Celestial Sphere
– History
– Positions
• Celestial Mechanics
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Elliptical Orbits
Newtonian Mechanics
Kepler’s Laws
Virial Theorem
• Continuous Spectrum of Light
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Stellar Parallax
The Magnitude Scale
The Wave Nature of light
Quantization of Energy
The Color Index
•Special Relativity
–Lorentz Transformations
–Time and Space in Relativity
–Relativistic Momentum
–Redshift
•Interaction of Light and Matter
–Spectral Lines
–Photons
–The Bohr Model of the Atom
–Quantum Mechanics and WaveParticle Duality
•Telescopes
–Basic Optics
–Optical Telescopes
–Infrared,UV,X-ray and GammaRay astronomy
Midterm Exam 1
• The exam
– Part I - in-class
• Conceptual questions
• “Easy” calculations (bring a calculator)
• Ten sheets (back and front) of notes (no xerographic reduction)
– Part II - Take-home
• Application of the material to Astrophysical Problems
• More of a pedagogical learning exercise
• Preparation
– Material from textbook ch 1-6
• Concepts
• Equations (no derivations in classroom portion of exam)
– Examples from textbook
– Homework problems!!! See solutions on web…
Possible Problem Topics
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Motions of heavenly bodies
Position on celestial sphere
Elliptical Orbits
Newtonian Mechanics and Kepler’s Laws
Escape Velocity
Stellar Parallax
Magnitude Scale, Flux, Luminosity,…
Basic Properties of Light
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Kirchoff’s Laws
Spectrographs
Photons
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Wien’s and Stefan/Boltzmann
Planck Function (usage…not derivation)
Monochromatic Flux/Luminosity
Color Index, Bolometric Correction
Redshift
Spectral Lines
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Photoelectric, Compton effects
Energy in terms of eV-nm
•
Model of the Atom
– Wavelengths of emitted photons
Quantum Mechanics
– Wave-Particle Duality
– Uncertainty Principle
– Quantum numbers of Atomic States
– Pauli Exclusion Principle
Telescopes
– Basic (Snell’s Law/Reflection)
– Diffraction limit/Resolution/Seeing
– Brightness/Focal Ratio/Magnification
– Mounts
– Diameter/Resolving Power
Special Relativity
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Time Dilation
Length contraction
Redshift
Relativistic Momentum
Positions on the Celestial Sphere
The Altitude-Azimuth Coordinate System
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Coordinate system based on observers
local horizon
Zenith - point directly above the
observer
North - direction to north celestial pole
NCP projected onto the plane tangent
to the earth at the observer’s location
h: altitude - angle measured from the
horizon to the object along a great
circle that passes the object and the
zenith
z: zenith distance - is the angle
measured from the zenith to the object
z+h=90
A: azimuth - is the angle measured
along the horizon eastward from north
to the great circle used for the measure
of the altitude
Equatorial Coordinate System
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Coordinate system that results in
nearly constant values for the
positions of distant celestial
objects.
Based on latitude-longitude
coordinate system for the Earth.
Declination - coordinate on
celestial sphere analogous to
latitude and is measured in
degrees north or south of the
celestial equator
Right Ascension - coordinate
on celestial sphere analogous to
longitude and is measured
eastward along the celestial
equator from the vernal equinox
 to its intersection with the
objects hour circle
Hour circle
Positions on the Celestial Sphere
The Equatorial Coordinate System
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Hour Angle - The angle between
a celestial object ’ s hour circle
and the observer ’ s meridian,
measured in the direction of the
object ’ s motion around the
celestial sphere.
Local Sidereal Time(LST) - the
amount of time that has elapsed
since the vernal equinox has last
traversed the meridian.
Right Ascension is typically
measured in units of hours,
minutes and seconds. 24 hours of
RA would be equivalent to 360.
Can tell your LST by using the
known RA of an object on
observer’s meridian
Hour circle
What is a day?
The period (sidereal) of earth’s revolution
about the sun is 365.26 solar days. The
earth moves about 1 around its orbit in
24 hours.
•
Solar day
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Is defined as an average interval
of 24 hours between meridian
crossings of the Sun.
The earth actually rotates about its
axis by nearly 361 in one solar
day.
Sidereal day
–
Time between consecutive
meridian crossings of a given star.
The earth rotates exactly 360
w.r.t the background stars in one
sidereal day = 23h 56m 4s
Precession of the Equinoxes
• Precession is a slow wobble
of the Earth’s rotation axis
due to our planet’s
nonspherical shape and its
gravitational interaction with
the Sun, Moon, etc…
• Precession period is 25,770
years, currently NCP is
within 1 of Polaris. In
13,000 years it will be about
47 away from Polaris near
Vega!!!
• A westward motion of the
per year.
Elliptical Orbits 3
Kepler’s Laws of Planetary Motion
Kepler’s First Law: A planet orbits the Sun in an ellipse,
with the Sun at one focus of the ellipse.
Kepler’s Second Law: A line connecting a planet to the
Sun sweeps out equal areas in equal time intervals
Kepler’s Third Law: The Harmonic Law
P2=a3
Where P is the orbital period of the planet measured in
years, and a is the average distance of the planet
from the Sun, in astronomical units (1AU = average
distance from Earth to Sun)
Kepler’s First Law
Kepler’s First Law: A
planet orbits the Sun in
an ellipse, with the Sun at
one focus of the ellipse.
• a=semi-major axis
• e=eccentricity
• r+r’=2a - points on
ellipse satisfy this
relation between sum
of distance from foci
and semimajor axis
b 2 = a 2 (1- e 2 )
a(1- e 2 )
r=
1+ e cosq
A = pab
Kepler’s Second Law
Kepler’s Second Law:
A line connecting a
planet to the Sun
sweeps out equal
areas in equal time
intervals
Kepler’s Third Law
Kepler’s Third Law:
The Harmonic Law
P2=a3
• Semimajor axis vs
Orbital Period on a loglog plot shows harmonic
law relationship
Newton’s Laws of Motion
• Newton’s First Law: The Law of Inertia. An object at rest will
remain at rest and an object in motion will remain in motion in a
straight line at a constant speed unless acted upon by an
external force.
• Newton’s First Law: The net force (thesum of all forces) acting
on an object is proportional to the object’s mass and its
resultant acceleration.
n
Fnet = å Fi = ma
i=1
• Newton’s Third Law: For every action there is an equal and
opposite reaction
Newton’s Law of Universal Gravitation
• Using his three laws of motion along
with Kepler’s third law, Newton
obtained an expression describing the
force that holds planets in their
orbits…(derivation in book, done on
blackboard)
Mm
F =G 2
r
Work and Energy
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Energetics of systems
Potential Energy
Kinetic Energy K = 1 mv 2
2
Total Mechanical Energy
Conservation of Energy
Gravitational Potential
Mm
energy
U = -G
r
• Escape velocity
v esc = 2GM /r
Derivation of Kepler’s Third Law
• Integration of the
expression of the 2nd
law over one full period
dA 1 L
=
dt 2 m
•
• Results in
A=
1L
P
2m
2
4
p
P2 =
a3
G(m1 + m2 )
• Derived on pp 48-49
Stellar Parallax
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•
Trigonometric Parallax:
Determine distance from
“triangulation”
tan q = B /d
d = B /tanq
Parallax Angle: One-half the
maximum angular displacement
due to the motion of Earth about
the Sun (excluding proper
motion)
d=
1AU 1
» AU
tan p p
PARSEC/Light Year
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•
1 radian = 57.2957795 = 206264.806”
Using p” in units of arcsec we have:
206,265
d»
AU
p"
Astronomical Unit of distance:
PARSEC = Parallax Second = pc
1pc = 2.06264806 x 105 AU
•
The distance to a star whose parallax
angle p=1” is 1pc. 1pc is the distance at
which 1 AU subtends an angle of 1”
d»
•
•
1
pc
p"
Light year : 1 ly = 9.460730472 x 1015 m
1 pc = 3.2615638 ly
•Nearest star proxima centauri has a
parallax angle of 0.77”
•Not measured until 1838 by Friedrich
Wilhelm Bessel
•Hipparcos satellite measurement
accuracy approaches 0.001” for over
118,000 stars. This corresponds to a a
distance of only 1000 pc (only 1/8 of
way to centerof our galaxy)
•The planned Space Interferometry
Mission will be able to determine
parallax angles as small as 4
distance measurements of objects up
to 250 kpc.
The Magnitude Scale
• Apparent Magnitude: How bright an object appears.
Hipparchus invented a scale to describe how bright
a star appeared in the sky. He gave the dimmest
stars a magnitude 6 and the brightest magnitude 1.
Wonderful … smaller number means “bigger”
brightness!!!
• The human eye responds to brightness
logarithmically. Turns out that a difference of 5
magnitudes on Hipparchus’ scale corresponds to a
factor of 100 in brightness. Therefore a 1
magnitude difference corresponds to a brightness
ratio of 1001/5=2.512.
• Nowadays can measure apparent brightness to an
accuracy of 0.01 magnitudes and differences to
0.002 magnitudes
• Hipparchus’ scale extended to m=-26.83 for the
Sun to approximately m=30 for the faintest object
detectable
Flux, Luminosity and the Inverse Square Law
•
Radiant flux F is the total amount of
light energy of all wavelengths that
crosses a unit area oriented
perpendicular to the direction of the
light’s travel per unit
time…Joules/s=Watt
•
Depends on the Intrinsic Luminosity
(energy emitted per second) as well
as the distance to the object
•
Inverse Square Law:
F=
L
4pr 2
Absolute Magnitude and Distance Modulus
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•
Absolute Magnitude, M: Defined to
be the apparent magnitude a star
would have if it were located at a
distance of 10pc.
Ratio of fluxes for objects of
apparent magnitudes m1 and m2 .
F2
= 100(m1 -m 2 )/ 5
F1
•
Taking logarithm of each side
æ F1 ö
m1 - m2 = -2.5log10ç ÷
è F2 ø
•Distance Modulus: The connection
between a star’s apparent magnitude, m ,
and absolute magnitude, M, and its
distance, d, may be found by using the
inverse square law and the equation that
relates two magnitudes.
F10 æ d ö
(m-M )/ 5
100
=
=ç
÷
F è10 pc ø
2
Where F10 is the flux that would be received
if the star were at a distance of 10 pc and d
is the star’s distance measured in pc.
Solving for d gives:
d =10(m-M +5)/ 5 pc
The quantity m-M is a measure of the
distance to a star and is called the star’s
distance modulus
æ d ö
m - M = 5log10 (d) - 5 = 5log10 ç
÷
è10 pc ø
Einstein’s Postulates of Special Relativity
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•
The Principle of Relativity. The
laws of physics are the same in
all inertial reference frames.
The Constancy of the Speed of
Light. Light moves through
vacuum at a constant speed c
that is independent of the motion
of the light source.
A reference frame in which a mass point thrown
from the same point in three different (non coplanar) directions follows rectilinear paths each
time it is thrown, is called an inertial frame.
– L. Lange (1885) as quoted by Max von Laue in
his book (1921) Die Relativitätstheorie, p. 34, and
translated by Iro).
Proper Time And Time Dilation
t2 - t1 =
(t2¢ - t1¢ ) + (x 2¢ - x1¢ )u /c 2
1- u 2 /c 2
(x 2¢ - x1¢ ) = 0
Proper Length and Length Contraction
•
Measure positions at endpoints
at same time in frame S’ and in
frame S, L’=x2’-x1’
(x 2 - x1 ) - u(t2 - t1 )
¢
¢
x 2 - x1 =
1- u 2 /c 2
L
¢
L=
1- u2 /c 2
Lmoving = Lrest 1- u2 /c 2 = Lrest / g
Redshift
zº
lobs - lrest
lrest
n obs = n rest
lobs = lrest
1- v r /c
1+ v r /c
1+ v r /c
1- v r /c
velocity of object by
measuring shift in spectral
lines….
• See example 4.3.2 p99
• For v<<c we have
z=
1+ v r /c
-1
1- v r /c
z +1=
1+ v r /c
1- v r /c
v r (z + 1) 2 -1
=
c (z + 1) 2 + 1
vr
l - lrest
= z = obs
c
lrest
Relativistic Momentum and Energy
Momentum
p=
mv
1- u /c
2
2
= gmv
Kinetic Energy
K = mc 2 (g -1)
Total Energy
E = gmc 2
Rest Energy
E = mc 2
Momentum Energy Relation
E 2 = p2c 2 + m 2c 4
Speed of Light
•
Ole Roemer(1644-1710) measured the
speed of light by observing that the
observed time of the eclipses of Jupiter’s
moons depended on how distant the Earth
was from Jupiter. He estimated that the
speed of light was 2.2 x 108 m/s from
these observations. The defined value is
now c=2.99792458 x 108 m/s (in vacuum).
The meter is derived from this value.
•
Measurement of speed of light is the same
for all inertial reference frames!!!
Special Relativity
(will come back to this topic..soon)
16.5 minutes for light
to travel 2AU
The Wave Nature of Light
• Light impinging on
double slit
• Exhibits Inerference
pattern
Interference condition
ì nl
ï
d sinq = í
1
ïî(n - )l
2
(n=0,1,2,…for bright fringes)
(n=1,2,…for dark fringes)
INTERFERENCE
http://vsg.quasihome.com/interfer.htm
WAVE
Electromagnetic Waves
Electromagnetic Wave speed
Light is indeed an
Electromagnetic Wave
Waves are Transverse
Electromagnetic Spectrum
Region
Gamma Ray
X-Ray
Wavelength
nm
1 nm<10 nm
Ultraviolet
10 nm<400 nm
Visible
400 nm<700 nm
Infrared
700 nm<1 mm
Microwave
1mm<10 cm
10 cm<
Particle-like nature of light
Photons
• Photon = “Particle of
Electromagnetic “stuff””
Light is absorbed and emitted in
tiny discrete bursts
Failure of Classical Theory
• Photo-electric effect (applet)
•
•
•
Any object with temperature above
absolute zero 0K emits light of all
wavelengths with varying degrees of
efficiency.
An Ideal Emitter is an object that
absorbs all of the light energy incident
upon it and re-radiates this energy with
a characteristic spectrum.Because an
Ideal Emitter reflects no light it is known
as a blackbody.
Wien’s Law: Relationship between
wavelength of Peak Emission max and
temperature T.
Blackbody
lmax T = 0.002897755mK
•
Stefan-Boltzmann equation: (Sun example)
4
L = AsT
L:Luminosity A:area T:Temperature
•Planck used a mathematical “sleight of
hand” to solve the ultraviolet catastrophe.
•The energy of a charged oscillator of
frequency f is limited to discrete values
of Energy nhf.
•During emission or absorption of light
the change in energy of an oscillator is
hf.
•The mean energy at high frequencies
tends to zero because the first allowed
oscillator energy is so large compared
to the average thermal energy available
kBT that there is almost zero probability
that this state is occupied.
u( l,T) =
•Planck seemed to be an “Unwilling
revolutionary”.He viewed this “quantization”
merely as a calculational trick…Einstein
viewed it differently…light itself was
quantized.
8phc
l5 (e hc / lk T -1)
B
Stefan-Boltzman derived
etotal
c
=
4
òl
¥
=0
u(l,T)dl = ò
0
2phc 2
dl
l5 (e hc / lkB T -1)
2p 5 kB 4
=
T = sT 4
2 3
15c h
4
etotal
¥
Monochromatic Luminosity and Flux
Monochromatic Luminosity
Ll dl = 4p 2 R2 Bl dl
8p 2 R 2 hc 2 / l5
Ll dl = hc / lkB T
dl
e
-1
distance r from the model star is:
2
Ll
2phc 2 / l5 æ R ö
Fl dl =
dl = hc / lkB T ç ÷ dl
4p × r 2
e
-1 è r ø
mbol = -2.5log10
(ò
¥
0
)
Fl dl + Cbol
S
Fd is the number of Joules of starlight energy
with wavelengths between  and dthat
arrive per second per one square meter of
detector aimed at the model star, assuming that
no light has been absorbed or scattered during
its journey from the star to the detector. Earth’s
atmosphere absorbs some starlight, but this can
be corrected. The values of these quantities
usually quoted for stars have been corrected and
would correspond to what would be measured
above Earth’s atmosphere.
Why do we keep the
wavelength dependence?
Filters!!!
Sf(
The Color Index
UVB Wavelength Filters
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•
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Bolometric Magnitude: measured over all
wavelengths.
UBV wavelength filters: The color of a star may
be precisely determined by using filters that
transmit light only through certain narrow
wavelength bands:
– U, the star’s ultraviolet magnitude.
Measured through filter centered at 365nm
and effective bandwidth of 68nm.
– B,the star’s blue magnitude. Measured
through filter centered at 440nm and
effective bandwidth of 98nm.
– V,the star’s visual magnitude. Measured
through filter centered at 550nm and
effective bandwidth of 89nm
U,B,and V are apparent magnitudes
Sensitivity Function S()
Filter Response
U = -2.5log10
(ò
¥
0
)
Fl SU dl + CU
æ
U - B = -2.5log10 ç
ç
è
B = -2.5log10
ò
ò
Fl SU dl ö
0
÷+C
U -B
¥
Fl SB dl ÷ø
¥
0
(ò
¥
0
)
Fl SB dl + CB
æ
U - B = -2.5log10 ç
ç
è
V = -2.5log10
ò
ò
¥
0
¥
0
Fl SU dl ö
÷+C
U -B
Fl SB dl ÷ø
http://astro.unl.edu/naap/blackbody/animations/blackbody.html
(ò
¥
0
)
Fl SV dl + CV
Spectral Type, Color and Effective Temperature
for Main-Sequence Stars
Spectral Type
B-V
Te(K)
O5
-0.45
35,000
B0
-0.31
21,000
B5
-0.17
13,500
A0
0.00
9,700
A5
0.16
8,100
F0
0.30
7,200
F5
0.45
6,500
G0
0.57
6,000
G5
0.70
5,400
K0
0.84
4,700
K5
1.11
4,000
M0
1.24
3,300
M5
1.61
2,600
40,000
35,000
30,000
Temperature
25,000
20,000
15,000
10,000
5,000
0
-1
-0.5
0
0.5
1
1.5
B-V
From Frank Shu, An Introduction to Astronomy(1982), Adapted from C.W. Allen, Astrophysical Quantities
Note that this table does not quite agree with our text!!!!
2
Spectral Type, Color and Effective Temperature
for Main-Sequence Stars
Spectral Type, Color and Effective Temperature
for Main-Sequence Stars (continued)
Interstellar Reddening
One also needs to correct color indices for
interstellar
reddening. As the light
propagates through interstellar dust, the
blue light is scattered preferentially making
objects appear to be redder than they
actually are…
Fraunhofer Spectral Lines
• Josef von Fraunhofer(17871826) had catalogued 475
dark spectral lines in the solar
spectrum.
• Fraunhofer showed that we
can learn the chemical
composition of the stars.
Identified a spectral line of
sodium in the spectrum of the
Sun
Spectral Lines
Kirchoff’s Laws of Spectra
• Robert Bunsen (1811-1899)
created a burner that
produced a “colorless”
flame ideally suited for
studying the spectra of
heated substances.
• Gustav Kirchoff (1824-1887)
and Bunsen designed a
spectroscope that could
analyze the emitted light.
• Kirchoff determined that 70
of the dark lines in the solar
spectrum corresponded to
the 70 bright lines emitted
by iron vapor.
• Chemical Analysis by
Spectral Observations
“Spectral Fingerprint”.
•A hot dense gas or hot solid object produces
a continuous spectrum with no dark spectal
lines
•A hot, diffuse gas produces bright spectral
lines (emission lines)
•A cool, diffuse gas in fron of a sources of a
continuous spectrum produces dark spectral
lines (absorption lines) in the continuous
spectrum.
Spectral Lines
Application of Spectral Measurements
• Stellar Doppler Shift
• Galactic Doppler Shifts
• Quasar Doppler Shifts
Spectral Lines
Spectrographs
• Spectroscopy
• Diffraction grating
equation
dsinq = nl
(n=0,1,2,…)
• Resolving Power
Photons
Photoelectric Effect
• Photoelectric Effect
• Kinetic energy of ejected
electrons does not depend
on intensity of light!
• Increasing intensity will
produce more ejected
electrons.
• Maximum kinetic energy of
ejected electrons depends
on frequency of light.
• Frequency must exceed
cutoff frequency before any
electrons are ejected
Einstein took Planck ’ s assumption of
quantized energy of EM waves seriously.
Light consisted of massless photons
whose energy was:
E photon = hn =
hc
l
K max = E photon - F = hn - F =
hc
l
-F
Einstein was awarded the Nobel Prize in
1921 for his work on the photo-electric
effect
Photons
Compton Scattering
•
•
•
•
•
Compton Scattering
Wikipedia entry
Arthur Holly Compton (18921962) provided convincing
evidence that light manifests
particle-like properties in its
interaction with matter by
considering how (x-ray) photons
can “collide” with a free electron
h
at rest.
lC =
= 0.00243nm
me c
Compton
Wavelength:
Conservation of energy and
momentum leads to the following: is the characteristic change in
wavelength in the scattered photon.
Showed that photons are
massless yet carry momentum!!!
E photon = hn =
hc
l
= pc
Rutherford-Bohr Model of the Atom
• Ernest Rutherford (1871-1937)
• Rutherford Scattering
Observation consistent with scattering
from a very small (10,000 times smaller
radius than the atom) dense object… The
nucleus
Wavelengths of Hydrogen
Bohr’s Semi-classical Model of the Atom
• Bohr assumed for the
electron proton system
– to be subject to
Coulomb’s Law for
electric charges
– Quantization of angular
momentum for the
electron orbit. L=nh/2
• Quantization condition
prevents electron from
energy
• Discrete energy levels for
electron orbit
Bohr’s Semi-classical Model of the Atom
Bohr’s Semi-classical Model of the Atom
Bohr’s Semi-classical Model of the Atom
Bohr’s Semi-classical Model of the Atom
Bohr Hydrogen atom and spectral lines
DeBroglie Matter Wave
Quantum Mechanics and Wave-Particle Duality
• Fourier
• Wave Packets
Uncertainty Principle
Additional Quantum Numbers and splitting of spectral lines
• Normal Zeeman Effect
Can measure magnetic fields by
examining spectra!!!!
Spin and the Pauli Exclusion Principle
No two electrons can share the same set of four
quantum numbers
Chemistry….The periodic table….
Electron Degeneracy Pressure and White Dwarfs
• Exclusion Principle for
Fermions (spin-1/2
particles) and
uncertainty principle
provide electron
degeneracy pressure
that is the mechanism
that prevents the further
collapse of white
dwarves….
Reflection and Refraction
Reflection
q1 = q2
Refraction
n1 sinq1 = n2 sinq2
Reflection and Refraction
Lenses
• Focal length
curvature , R1and R2,
of the lens surfaces
and the index of
refraction of the lens
material
• Convention for sign of
concave --> negative
convex --> positive
• Wavelength
dependent
1
1 1
= (nl -1)( - )
f
R1 R2
Lensmaker formula for
thin lenses
Mirrors
• For spherical mirrors
• f=R/2
• Wavelength independent
Focal Plane
•
Image size proportional to focal
length
y = f tanq
y » fq
Plate Scale
dq 1
=
dy f
Resolution and Rayleigh Criteria
Diffraction
• Single Slit diffraction
• Consider a point in the slit
D/2 away from another point
in slit. The pair of rays from
these two points will exhibit
destructive interference if
they arrive at a point on the
screen being 1/2 wavelength
out of phase
D
l
sinq =
2
2
sin q =
l
D
• Now divide the slit into
fourths
D
l
sinq =
4
2
sin q = 2
l
D
sinq = m
l
D
Condition for minimum in
light intensity on screen
Resolution
Single Slit Diffraction
Diffraction from a Circular Aperture
Airy Disk
Intensity as a function of angle from
center of the image is given by a Bessel
function
Diffraction limited spatial
resolution for a telescope
with an aperture of diameter
D is given by:
Rayleigh Criterion for Resolution
• Somewhat arbitrary
definition of barely
resolvable is that the
maximum of one of the Airy
disks lies at the first
minimum of the other Airy
disk
http://astronomy.swin.edu.au/cosmos/R/Resolution
Atmospheric Seeing
• Why do stars twinkle?
• Stars are so distant as to
effectively be point sources.
• Planets are resolvable.
• Plane wavefronts of light
from a distant point source
become distorted as the
wavefronts passes through
a turbulent layer in the
atmosphere.
• The index of refraction in
the turbulent layer is not
uniform
Brightness of Image
Refracting Telescope
• Angular
Magnification
Reflecting Telescopes
Telescope Mounts
• Altitude Azimuth
Equatorial