Photometry by MikeJenny

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```									Photometry
Measuring Energy
• Photometry measures the             • Spectroscopy measures energy
energy from a source using a          over a wide range of
narrow range of wavelengths.          wavelengths.
– Visual wavelengths from             – Visual spectrum
400-700 nm                         – UV, IR spectra
– Narrower slice of                   – Full EM spectra
wavelengths
• Spectroscopy requires
• Photometry uses filters to select     instruments to get at each
wavelengths.                          wavelength separately.
– Interferometer
Luminosity of Stars
• Luminosity measures how much energy is produced.
– Absolute brightness L

• Relative luminosity is usually based on the Sun.
• Astronomers measure luminosity relative to the Sun.
– LSun = 1 L
– LSirius = 23 L

• Stars range from 0.0001 L to 1,000,000 L .
Magnitude
• The observed brightness is

m  n  2.5 log( En / Em )   • The magnitude scale was
originally 6 classes.
– Effectively logarithmic
For 1 unit of magnitude:
• The magnitude (m) was made
En
 101 2.5  2.512          formal in 1856.
Em                             – Lower numbers brighter
– 6m at the limit of human
vision
Brightness Magnified
• Images from a telescope must
fe                    fo
fit within the pupil.             P D                 M
– Brightness proportional to       fo                    fe
the aperture squared
– Ratio of observed to natural
R
Ltelescope

D M 2  1
• No increase for extended                 Leye           P2
objects from magnification.
– Eg. M31(> moon)
– Light on more rods
– Exclusion of other light
Point Source Magnified
• Point sources are smaller than
one pixel (or rod).
– No increase in image size         D2
G 2
from magnification               P
• The ratio of brightness increase
is the light grasp G.              G  2 10 4 (m 2 ) D 2
– Pupil size 7 mm

• The limiting magnitude comes       mm in  16 .8  5 log10 D
from the aperture.
– CCD 5 to 10 magnitudes                                in meters
better
8” aperture is 13.3m
Apparent Magnitude
• The observed magnitude             • Some bright stars (app. mag.):
depends on the distance to the        – Sun                  -26.7
source.                               – Sirius               -1.4
– Measured as apparent               – Alpha Centauri       -0.3
magnitude.
– Capella              0.1
– Rigel                0.1
• The scale is calibrated by stars
within 2° of the north celestial      – Betelgeuse           0.5
pole.                                 – Aldebaran            0.9

• These are all brighter than first
magnitude (m = 1.0)
Distance Correction
• Brightness falls off as the
d     2

 100 
M  m  2.5 log      
square of the distance d.
     
• Absolute magnitude M
M  m  5  5 log d                   recalculates the brightness as if
the object was 10 pc away.
– 1 pc = 3 x 1016 m = 3.26 ly

M  m  5  5 log d  AD
• The absolute magnitude can be
AD = 0.002 m/pc in galactic plane     corrected for interstellar
Absolute Magnitude
• Distance is important to           • Some bright stars (abs. mag.):
determine actual brightness.          – Sun                  4.8
– Sirius               1.4
– Alpha Centauri       4.1
• Example: 2 identical stars            – Capella              0.4
A is 7 pc, B is 70 pc from Earth      – Rigel                -7.1
The apparent brightness of B is       – Betelgeuse           -5.6
1/100 that of A                       – Aldebaran            -0.3
The magnitude of B is 5 larger.
• These are quite different than
their apparent magnitudes.
Imaging
• Photographic images used the     • CCDs can directly integrate the
width of an image to determine     photoelectrons to get the
intensity.                         intensity.
– Calibrate with known stars       – Sum pixels covered by
– Fit to curve                        image
– Subtract intensity of nearby
D  A  B log 10 I                 dark sky

• Data is corrected for reddening
due to magnitude and zenith
angle.
Solar Facts
– R = 7  105 km = 109 RE        • Composition:
– Mostly H and He
• Mass :
– M = 2  1030 kg                • Temperature:
– M = 333,000 ME                    – Surface is 5,770 K
– Core is 15,600,000 K
• Density:
– r = 1.4 g/cm3                  • Power:
– (water is 1.0 g/cm3, Earth is      – 4  1026 W
5.6 g/cm3)
Hydrogen Ionization

ep = p2/2m                                 • Particle equilibrium in a star is
dominated by ionized hydrogen.

n=3                                      • Equilibrium is a balance of
n=2                                        chemical potentials.

 g H n nQp 
n=1                                         H n   mH n c  kT ln 
2                   
 nH 
       n   
 g p nQp 
  p   m p c  kT ln 
2
 n 

 H n    e     p                                p 
 g e nQe 
 e  me c  kT ln 
2
 n      
 e 
Saha Equation

• The masses in H are related.
mHn c2  mpc2  mec2  e n
– Small amount en for
degeneracy

g (H n )  gn ge g p  4n2   • Protons and electrons each have
half spin, gs = 2.
– H has multiple states.

n ( H n ) g n e n           • The concentration relation is the
     e     kT
Saha equation.
ne n p    nQe
– Absorption lines
Spectral Types
• The types of spectra were originally                      •
classified only by hydrogen              • Type   Temperature
absorption, labeled A, B, C, …, P.           O      35,000 K
B      20,000 K
• Understanding other elements’                A      10,000 K
lines allowed the spectra to be
ordered by temperature.                      F       7,000 K
G       6,000 K
• O, B, A, F, G, K, M                          K       4,000 K
• Oh, Be A Fine Guy/Girl, Kiss Me              M       3,000 K
• Our Brother Andy Found Green
Killer Martians.
Spectral Classes
• Some bright stars (class):
– Sun                  G2
– Sirius               A1       • Detailed measurements of
– Alpha Centauri       G2         spectra permit detailed classes.
– Capella              G8
– Rigel                B8
– Betelgeuse           M1       • Each type is split into 10 classes
– Aldebaran            K5         from 0 (hot) to 9 (cool).

• Temperature and luminosity are
not the same thing.
Filters
• Filters are used to select a restricted bandwidth.
– Wide: Dl ~ 100 nm
– Intermediate: Dl ~ 10 nm
– Narrow: Dl < 1 nm

• A standard set of optical filters dates to the 1950’s
– U (ultraviolet – violet): lp = 365 nm, Dl = 70 nm
– B (photographic): lp = 440 nm, Dl = 100 nm
– V (visual): lp = 550 nm, Dl = 90 nm
Filter Sets
• Other filter sets are based on a   • CCDs have are good in IR, so
specific telescope.                  filter sets have moved into IR as
– HST: 336, 439, 450, 555,          well.
675, 814 nm                       – U, B, V, R, I, Z, J, H, K, L,
– SDSS: 358, 490, 626, 767,              M.
907 nm                            – Example M : lp = 4750 nm,
Dl = 460 nm
• The standard intermediate filter
set is by Strömgren.
– u, b, v, y, b
– bw: lp =486 nm, Dl=15 nm
Color Index
• The Planck formula at relates the
intensity to the temperature.                        2c 2 h
– Approximate for T < 104 K
Wl (l , T )               e  hc / lkT
l5
• Two magnitude measurements at                  hc   hc
different temperatures can          TB V              0.65 10 4 K
lB k lV k
determine the temperature.
– Standard with B and V filters                             TB V 
B  V  2.5 log 10 exp        
– Good from 4,000 to 10,000 K                                T 
7090 K
T
( B  V )  0.71
Stellar Relations
• The luminosity of a star should   • Some bright stars:
be related to the temperature.       – Sun        G2     4.8
– Blackbody formula                 – Sirius     A1     1.4
– Depends on radius                 – Alpha Centauri G2 4.1
– Capella G8        0.4
L  4R 2T 4                  – Rigel      B8     -7.1
– Betelgeuse M1     -5.6
– Aldebaran K5      -0.3
Luminosity vs. Temperature

-20
-15
• Most stars show a relationship
-10                      between temperature and
Abs. Magnitude

-5                      luminosity.
0            Sun         – Absolute magnitude can
5                          replace luminosity.
10                        – Spectral type/class can
15                          replace temperature.
20
O B A F G K M
Spectral Type
Hertzsprung-Russell Diagram
• The chart of the stars’
luminosity vs. temperature is
called the Hertzsprung-Russell
diagram.

• This is the H-R diagram for
hundreds of nearby stars.
– Temperature decreases to
the right
Main Sequence

-20
• Most stars are on a line called
-15                                the main sequence.
-10
Abs. Magnitude

-5
Sirius                 • The size is related to
0                                 temperature and luminosity:
5                     1 solar      – hot = large radius
15                                  – cool = small radius
20
O B A F G K M
Spectral Type
Balmer Jump
• The color indexes can be
measured for other pairs of
filters.

• The U-B measurement brackets
the Balmer line at 364 nm.
– Opaque at shorter
wavelength

• This creates a discontinuity in
energy measurement.
– Greatest at type A
– Drop off for B and G
Michael Richmond, RIT
Photometric Comparison
• Stellar classification is aided by different response curves.
Bolometric Magnitude
• Bolometric magnitude measures
BC  mbol  V                          the total energy emitted at all
BC  M bol  M V                       wavelengths.
– Modeled from blackbody
– Standard filter V
– Zero for main sequence
            
L  3 10 28 W 10 0.4 M bol
stars at 6500 K

                  
  2.5 10 8 W m 2 10 0.4 mbol
• Luminosity is directly related to
absolute bolometric magnitude.
– Flux to apparent bolometric
magnitude

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