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Midterm Review
AST443Stanimir Metchev
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Administrative
• Homework 2:– problems 4.4, 5.1, 5.3, 5.4 of W&J– due date extended until Friday, Oct 23
• drop off in my office before 6pm
• Midterm: Monday, Oct 26– 8:20–9:20pm in ESS 450– closed book– no cheat sheet– no need to memorize obscure constants or formulae
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Midterm Review• Coordinates and time• Detection of light:
– telescopes, detectors• Radiation:
– specific intensity, flux density, etc– optical depth, extinction, reddening– differential refraction
• Magnitudes and photometry:– apparent, absolute; distance modulus– CMDs and CCDs– photometry, PSF fitting
• Statistics– testing correlations and hypotheses– non-parametric vs. parametric tests– one-tailed vs. two-tailed tests– one-sample, two-sample tests
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Celestial Coordinates
• horizoncoordinates– altitude/elevation (a),
azimuth (A)– zenith, zenith angle
(z)• observer’s latitude
– angle PZ
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Celestial Coordinates
• equatorialcoordinates– right ascension
• R.A., α– declination
• DEC, δ
• cf., Earth’s longitude,latitude
• meridian, hour angle(HA)
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Celestial Coordinates• ecliptic coordinates
– ecliptic longitude (λ)– ecliptic latitude (β)
• vernal equinox (ϒ)– Earth passes through ecliptic
on Mar 20/21– origin of equatorial and
ecliptic longitude• Earth’s axial tilt:
– ε = 23.439281º– a.k.a., “obliquity of the
ecliptic”
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Celestial Coordinates• galactic coordinates
– galactic longitude (l; letter“ell”)
• l = 0 approx. toward galacticcenter (GC)
• definitionlNCP = 123º (B1950.0)
– galactic latitude (b)• NGP definition (B1950.0)α = 12h 49mδ = 27.4º
– Sagittarius Al = 359º 56′ 39.5″b = –0º 2′ 46.3″
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Coordinate Transformations• equatorial ↔ ecliptic
• equatorial ↔ horizontal!
cos"cos# = cos$ cos%
cos" sin# = cos$ sin%cos& ' sin$ sin&
sin" = cos$ sin% sin& + sin$ cos&
cos$ sin% = cos" sin# cos& + sin" sin&
sin$ = sin"cos& ' cos" sin# sin&
!
cosasinA = "cos# sinHA
cosacosA = sin#cos$ " cos#cosHAsin$
sina = sin# sin$ + cos#cosHAcos$
cos# sinHA = "cosasinA
sin# = sinasin$ + cosacosAcos$
φ ≡ observer’s latitude
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Equatorial CoordinateSystems
• FK4– precise positions and motions of 3522 stars– adopted in 1976– B1950.0
• FK5– more accurate positions– fainter stars– J2000.0
• ICRS (International Celestial Reference System)– extremely accurate (± 0.5 milli-arcsec)– 250 extragalactic radio sources
• negligible proper motions– J2000.0
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Astronomical Time• sidereal time
– determined w.r.t. stars– local sidereal time (LST)
• R.A. of meridian• HA of vernal equinox
– sidereal day: 23h 56m 4.1s• object’s hour angle
HA = LST – α
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Astronomical Time• sidereal time
– determined w.r.t. stars– local sidereal time (LST)
• R.A. of meridian• HA of vernal equinox
– sidereal day: 23h 56m 4.1s• object’s hour angle
HA = LST – α• solar time
– solar day is 3 min 56 seclonger than sidereal day
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Astronomical Time• universal time
– UT0: determined from celestial objects• corrected to duration of mean solar day• HA of the mean Sun at Greenwich (a.k.a., GMT)
– UT1: corrected from UT0 for Earth’s polar motion• 1 day = 86400 s, but duration of 1 s is variable
– UTC: atomic timescale that approximates UT1• kept within 0.9 sec of UT1 with leap seconds• international standard for civil time• set to agree with UT1 in 1958.0
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Astronomical Time• tropical year
– measured between successive passages of the Sun through the vernalequinox
– 1 yr = 365.2422 mean solar days• mean sidereal year
– Earth: 50.3″/yr precession in direction opposite of solar motion– 365.2564 days
• Julian calendar– leap days every 4th year; 1 yr = 365.25 days– t0 = noon on Jan 1st, 4713 BC
• Gregorian calendar– no leap day in century years not divisible by 400 (e.g., 1900)– 1 yr = 365.2425 days
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Coordinate Epochs• Coordinates are given at B1950.0 or J2000.0 epochs
– Besselian years (on Gregorian calendar; tropical years)– Julian years (Julian calendar)
• Gregorian calendar is irregular– complex for precise measurements over long time periods
• Julian epoch:– Julian date: JD = 0 at noon on Jan 1, 4713 BC– J = 2000.0 + (JD – 2451545.0) / 365.25– J2000.0 defined at
• JD 2451545.0• January 1, 2000, noon
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Focusing
• focal length (fL), focal plane
• object size (α, s) in the focal planes = fL tan α ≈ fLα
• plate/pixel scaleP = α/s = 1/fL
– Lick observatory 3m• fL = 15.2m, P = 14″/mm
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Energy and Focal Ratio• Specific intensity:
– Planck law– [erg s–1 cm–2 Hz–1 sterad–1] or [Jy sterad–1]
• Integrated apparent brightness
Ep ∝ (d / fL)2 : energy per unit detector area
• focal ratio: ℜ ≡ fL / d– “fast” (< f/3) vs. “slow” optics (>f/10)– fast data collection vs. larger magnification
magnification = fL / fcamera
!
I(",T) =2h" 3
c2
1
eh" kT
#1
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Optical Telescope Architectures
Also:• Schmidt-Cassegrain
• spherical primary (sph. aberration), corrector plate; cheap for large FOV• no coma or astigmatism; severe field distortion
• Ritchey-Chrétien• modified Cassegrain with hyperbolic primary and hyperbolic convex secondary• no coma; but astigmatism, some field distortion
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Imaging through a TurbulentAtmosphere: Seeing
• FWHM of seeing disk– θseeing <1.0″ at a good site
• r0: Fried parameter– θseeing = 1.2 λ/r0
– r0 ∝ λ6/5 (cos z)3/5
– θseeing ∝ λ–1/5
• t0: coherence time– t0 = r0 / vwind
– vwind ~ several m/s– t0 is tens of milli-sec
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Basic Concept of aSemi-Conductor Detector
• electron-hole pair generation• doping:
– n-type (electrons)– p-type (holes)– creates additional energy levels
within band gap– increases conductivity
• silicon (Si)– band gap: Eg = 1.12 eV
• cut-off wavelength λc = 1.13µm
– free-electron energy: 4 eV (3000Å)– 1 photon -> 1 electron
!
"c =hc
Eg
=1.24µm
Eg (eV)
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Basic Concept of a CCD Pixel:A P-N Photo Diode
• depleted region– low conductivity– can support an E field
• net positive charge (higher charge density near top)• additional E-field applied• subsequently generated electrons get trapped in potential well near top
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Charge Transfer
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Front- vs. Back-Illumination
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Read Noise
• electrons / pix / read• sources
– A/D conversion not perfectly repeatable– spurious electrons from electronics (e.g.,
from amplifier heating)• alleviated through cooling• nowadays: <3–10 electrons
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Dark Current
• electrons / pixel / second• source
– thermal noise at non-zero detector temperature
• higher at room temperature (~10,000)• at cryogenic temperatures
– LN2, –100 C– 0.1–20 e–/pix/s
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Detector Calibration• bias frames
– non-zero bias voltage– 0s integrations
• dark frames– equal to science integrations
• flat field frames– QE of detector pixels is non-uniform in 2-D– QE is dependent on observing wavelength
• bad pixels
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Sky and Telescope Calibration• sky background images• photometric calibration:
– airmass curve, filter transmission• sky transmission
– spectrum of a star of a known spectral energy distribution• astrometric calibration
– binary with a known orbit– star-rich field with precisely known positions: HST observations of
globular clusters• point-spread function calibration
– nearby bright star (for on-axis calibration)– star-rich field (2-d information on PSF)
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Aperture Photometry• object flux = total counts – sky counts• estimation of background
– Npix, bkg > 3 Npix, src– use rbkg >> FWHM, whenever possible
• enclosed energy P(r)– “curve of growth”
• optimum aperture radius r– SNR(r) first increases, then decreases with r
• Fig. 5.7 of Howell– dependent on PSF FWHM and source brightness
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28source: Kitt Peak National Observatory
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Radiation• specific intensity Iν
– dE = Iν dt dA dν dΩ [erg s–1 cm–2 Hz–1 sterad–1] or [Jy sterad–1]– 1 Jy = 10–23 erg s–1 cm–2 Hz–1 = 10–26 W m–2 Hz–1
– surface brightness of extended sources (independent of distance)• spectral flux density Sν
– Sν = ∫ Iν dΩ [erg s–1 cm–2 Hz–1] or [Jy] or [W m–2 Hz–1]– point sources, integrated light from extended sources
• flux density F– F = ∫ Sν dν [erg s–1 cm–2] or [W m–2]
• power P– P = ∫ F dA = dE / dt [erg s–1] or [W]– received power: integrated over telescope area– luminosity: integrated over area of star
• conversion to photon counts– energy of N photons: Nhν
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Blackbody Radiation (Lecture 4)• Planck law
– specific intensity
• Wien displacement law T λmax= 0.29 K cm
• Stefan-Boltzmann law F = σ T 4
– energy flux density– [erg s–1 cm–2]
• Stellar luminosity– power– [erg s–1]
• Inverse-square law L(r) = L* / r2
!
" =2# 5
k4
15c 2h
3= 5.67 $10%5erg cm–2 s–1 K–4
!
L*
= 4"R*
2#Teff
4
!
I(",T) =2h" 3
c2
1
eh" kT
#1
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Blackbody Radiation (Lecture 4)
Teff, Sun = 5777 KT λmax= 0.29 K cm
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Magnitudes (Lecture 4)• apparent magnitude: m = –2.5 lg F/F0
– m increases for fainter objects!– m = 0 for Vega; m ~ 6 mag for faintest naked-eye stars– faintest galaxies seen with Hubble: m ≈ 30 mag
• 109.5 times fainter than faintest naked-eye stars– dependent on observing wavelength
• mV, mB, mJ, or simply V (550 nm), B (445 nm), J (1220 nm), etc
• bolometric magnitude (or luminosity): mbol (or Lbol)– normalized over all wavelengths
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Magnitudes and Colors(Lecture 4)
• magnitude differences:– relative brightness
V1 – V2 = –2.5 lg FV1/FV2• ∆m = 5 mag approx. equivalent to F1/F2 = 100
– color
B – V = –2.5 (lg FB/FV – lg FB,Vega/FV,Vega)
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Extinction and Optical Depth(Lecture 4)
• Light passing through a medium can be:– transmitted, absorbed, scattered
• extinction at frequency ν over distance sdLν(s) = –κν ρ Lν ds = –L dτνLν = Lν,0e–τ = Lν,0e–κρs =Lν,0e–s/l
Aν = 2.5 lg (Fν,0/Fν) = 2.5 lg(e)τν = 0.43τν mag– medium opacity κν [cm2 g–1], density ρ [g cm–3]– optical depth τν = κν ρs [unitless]– photon mean free path: lν = (κν ρ)–1 = s/τν [cm]AV = mV – mV,0
• reddening between two frequencies (ν1, ν2)Eν1,ν2 = mν1 – mν2 – (mν1 – mν2)0 [mag]
– (mν1 – mν2)0 is the intrinsic color of the star
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Interstellar Extinction Law
extinction is highest at ~100 nm = 0.1 µmunimportant for >10 µm
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Interstellar Extinction Law
• AV / E(B–V) = 3.1– AV / E(J–K) = 5.8– AV / E(V–K) = 1.13– Aλ / E(J–K) = 2.4 λ–1.75 (0.9 < λ < 6µm)
• AV ≈ 0.6 r / (1000 ly) mag– b < 2º (galactic latitude)
• AV ≈ 0.18 / sin b mag– b > 10º
• NH / AV ≈ 1.8 x 1021 atoms cm–2 mag–1
– atoms of neutral hydrogen (H I)
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Atmospheric Extinction
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Photometric Bands: Visible
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Photometric Bands: Near-Infrared
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Extinction and Reddening: CMD• Legend:
– arrow: AV = 5 magextinction
– solid line: mainsequence
– dotted line: substellarmodels
– crosses: knownbrown dwarfs
– solid points: browndwarf candidates
AV = 5 mag
Metchev et al. (2003)
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Extinction and Reddening: CCD• Legend:
– arrow: AV = 5 magextinction
– solid line: mainsequence + giants
– dotted line: substellarmodels
– crosses: knownbrown dwarfs
– solid points: browndwarf candidates
AV = 5 mag
Metchev et al. (2003)
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OBAFGKM + LT
higherionizationpotentialspecies
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Statistics: Basic Concepts• Binomial, Poisson, Gaussian distributions
– calculating means and variances
• Central Limit Theorem:“Let X1, X2, X3, …, Xn be a sequence of n independent andidentically distributed random variables each having finiteexpectation µ > 0 and variance σ2 > 0. As n increases, thedistribution of the sample average approaches the normaldistribution with a mean µ and variance σ2 / n irrespective of theshape of the original distribution.”
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A bizarre p.d.f. p(x) with µ = 0, σ2 = 1
p.d.f. of sum of 2 randomvariables sampled from p(x)(i.e., autoconvolution of p(x))
p.d.f. of sum of 3 randomvariables sampled from p(x)
p.d.f. of sum of 4 randomvariables sampled from p(x)
source: wikipedia
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Statistics: Basic Concepts
• probability density function (p.d.f.)– density of probability at each point– probability of a random variable falling within a given interval
is the integral over the interval
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46Hubble (1929)
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Student’s t Distribution
k = d.o.f.
source: wikipedia
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F Distribution
source: wikipedia
d1, d2 = d.o.f.
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χ2 Distribution
source: wikipedia
k = d.o.f.
χ2
f (χ2,k)
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χ2 Distribution
• probability that measured χ2 or higheroccurs by chance under H0
source: wikipedia