AST14 Observational Astronomy II PDF

AST14 - Observational Astronomy II Light, Definitions and Formulas THE NATURE OF LIGHT We study light because we can’t touch the celestial objects themselves We study light through flux, magnitude, and color filters...

AST14 - Observational Astronomy II Light, Definitions and Formulas THE NATURE OF LIGHT We study light because we can’t touch the celestial objects themselves We study light through flux, magnitude, and color filters BRIEF HISTORICAL DEVELOPMENT 1800s o William Herschel discovered Uranus and measured the rainbow’s temperature o He also made a thermometer that detects warmth outside the visible spectrum o Infrared light Mid-1800s o Infrared astronomy was used for quantitative science o A cow was the first detected object in infrared (‘di ko sure) 1940s o Radio astronomy Late 1900s o Ultraviolet, X-Ray, Millimeter and Submillimeter observations became possible o Maxwell’s Electromagnetic theory LIGHT In Classical Physics, light is a travelling wave with oscillating electric field with perpendicular oscillating magnetic field Energy of wave is inversely proportional to its wavelength c = vλ where o c = speed of light o v = frequency o λ = wavelength In Quantum Mechanics, light is a particle photon with energy given by E = hv = hλ/c where o h = Planck’s constant (6.626 x 10-34 J/Hz or Js) ! Light is both a particle and a wave simultaneously. !! The method of observation determines if it has features of a wave or particle. IDIOSYNCRASIES OF RESEARCH COMMUNITIES X-Ray astronomers characterize photons by energies o A similar approach is adopted in studying cosmic rays UV to IR observers refer to them by wavelength Millimeter and Radio regimes describe things in terms of frequency ELECTROMAGNETIC SPECTRUM Continuous but is divided into a number of bands Different bands were referred to depending on the research community Gamma Rays - X-Rays - UV - IR - Radio  QUANTUM NATURE OF LIGHT Photons - emitted when there is energy state exchange Late 1800s o Heinrich Hertz - studied photoelectric effect Early 1900s o Albert Einstein - light is quantized o Photons - discrete packets where energy is delivered QUANTUM THEORY Quantization of energy in a bound system applies to every energy change between two states The energy, spins, momentums, molecular vibrations, and rotations of electrons are all quantized o The Sun’s core is ~15 million K o The Sun’s core is made of plasma o Most emission is of the free-bound (electron capture) variety so the Sun should exhibit a continuous emission spectrum, but it doesn’t. The Sun has bright emission lines (Balmer and Lyman Series) Absorption of photons has a directional bias that causes absorption lines. Balmer o H𝛼 - 3 to 2 transition (λ = 656.3 nm) o Hβ - 4 to 2 transition (λ = 486.1 nm) Lyman o Ly 𝛼 - 2 to 1 transition (λ = 121.6 nm) o Ly β - 4 to 2 transition (λ = 102.6 nm) MEASURING LIGHT Luminosity - total amount of energy emitted across a specific wavelength range Brightness - amount of energy it emits through a specific wavelength range over a solid angle Absolute Magnitude - brightness of an object as it would be at a set distance o Solar system objects - 1 AU o Extrasolar - 10 pc Bolometric Luminosity - luminosity or brightness across the entire spectrum BLACK BODY Stellar spectra can be modeled as perfect absorbers and emitters and they reflect next to no light, this means that their output follows the pattern given by Planck’s Law. In general, astronomical bodies emit their light isotropically (equal in all directions) Flux:  THE MAGNITUDE SCALE Instruments typically return raw observing results in the form of counts per second (photon strikes on receiving area divided by the time taken) A logarithmic scale; British astronomer Norman Pogson introduced the equation Magnitude is dependent not only on physical characteristics of that object but also the distance from the observer, amount of material between the observer, detector used, and the transmission characteristics of the instrument determined by a filter. As an example, a star/galaxy may be magnitude 10 in R band but magnitude 12 in B band. The magnitude of extended and resolved objects is the integrated magnitude, the combined light over the whole surface of the object is surface brightness Apparent magnitude - brightness as seen from earth Absolute magnitude - important characteristic of a star Interstellar reddening - major problem in galactic astronomy; gas and dust scatters blue light over red light and the more the scatter, the redder it appears PHOTOMETRIC CALIBRATIONS For objects with flat spectral energy distributions, the Vega and AB systems are the same For extended objects (galaxies/nebulae), the usual magnitude is the integrated magnitude CONCEPT OF ZERO POINT  PHOTOMETRIC SYSTEM Well-defined passbands with a known sensitivity to incident radiation For each photometric system, a set of primary standard stars is provided Passbands - range of frequencies or wavelengths that can pass through a filter Primary standard stars - series of stars that have had their light output in various passbands Johnson-Morgan or UBV photometric system o Most commonly adopted standardized photometric system, developed in 1953 o First standardized photometric system o Employed for classifying stars according to their colors Johnson-Cousins UBVRI photometric system o Common extensions of the original system that provides redder passbands PHOTOMETRIC LETTERS Each letter designates a section of light of the EM spectrum NUV - major groups near-ultraviolet Center on V band - visible light NIR - near infrared MIR - part of mid-infrared Use if U, B, V, R, I dates from 1950s J to N bands were labelled following on from near-infrared’s closest-to-red band, I Later the H band was inserted, then Z in the 1990s, and finally Y without changing earlier definitions 1 Nanometer = 10 Angstroms Broadband - passbands wider than 30 nm Intermediate band - passbands between 10 and 30 nm wide Narrow band - passbands less than 10 nm wide COLOR INDICES Defined by taking the difference in magnitudes at two different wavelengths Negative values indicate that a star is hot (most radiation coming from shorter wavelengths) Basic Knowledge of the Sky WAYS TO SPECIFY A LOCATION Degrees - Latitude xx.xxxx degrees N/S; Longitude yyy:yyyy E/W of Greenwich Degrees, Min, Sec - Latitude xx:xx:xx N/S; Longitude yyy:yy:yy E/W Timezones for Longitude - Latitude xx:xx:xx N/S; Longitude yy h yy m yy s E/W RA & DEC Global coordinates o Any star has the same RA and Dec for all observations on Earth o Position remains the same o Fixed to the sky Ecliptic - path of zodiac signs RA Celestial longitude The point at which the sun appears to cross the equator from south to north Vernal equinox DEC Celestial latitude Measured away from celestial equator Projected sphere SEXIDECIMAL NOTATION Dec (degrees:arcmin:arcsec) & RA H:M:S o RA 06:45:09, Dec 16:42:58 Arc o Geometric angle o Circle - 360 degrees o 1 degree = 60 arcmin = 3600 arcsec o 1 arcmin = 60 arcsec Caveat o On celestial equator, 1 h = 15 degrees o Far from the celestial equator, distance of object moves in 1 h can be smaller o When near the poles, star trails are much shorter  ALT & AZ Local coordinates - observer dependent Change over just a few minutes as stars rise, move, and set ALTITUDE Angular distance above local horizon 0 to 90 degrees (zenith) AZIMUTH Angular distance from local north along horizon Due North = az - 0 degrees Due East = az - 90 degrees Due South = az - 180 degrees Due West = az - 270 degrees  SPHERICAL EXCESS (E) Non-negative quantity defined to be the deviation of the sum of the interior angles E = -α + β + γ - 180 degrees Area (S) = S + R2E ELLIPTIC COORDINATES Ecliptic with the origin being the first point of Aries Ecliptic latitude (β) and ecliptic longitude (δ) SUPER GALACTIC SYSTEM Extragalactic Flat 2d structure formed by distribution of galaxies (super galactic plane) SGl - super galactic latitude SGb - super galactic longitude Uncertainties and Error Analysis Measuring Errors - repeated measurement allows the results to gradually and asymptotically approach what we may have missed (?) Error - difference between observed/calculated value and true value Illegitimate Errors - originate from mistakes or blunders in measurements or computations; usually apparent Uncertainty - random fluctuations in measurements Systematic Errors o Limit the precision and accuracy of results in defined ways o Errors that will make results different from the true values o Faulty calibration of equipment Accuracy - how close Precision - how well the result has been determined Random Errors - fluctuations in observations that yield different results each time the experiment is repeated Discrepancy - difference between 2 results/best estimates Parent/Sample Distribution - always expect a discrepancy between 2 measurements o Parent - probability of getting particular observation in a single measurement o Sample - measurements we have made are from parent distribution Infinite measurements - describe exactly the distribution of data point If the distribution results from random errors in measurement, then it is very likely that it can be described in terms of Gaussian or normal error distribution Parameters of parent distribution - Greek letters Experimental estimates of the parameters of the parent distribution - Latin letters p(x) - limit of a very large number of observations, the fraction dN of observation of variance x that yield values between x and xdx (ano raw) STANDARD DEVIATION Measure of the dispersion of the expected observation about the mean Parameter easier to use analytically and justified fairly Root mean square of the deviation and is associated with the second moment of data about the mean VARIANCE Limit of average of squares of deviation from the mean  SCIENTIFIC NOTATION Used to avoid ambiguity, round only final results Left most number that is not 0 - most significant Right most number that is not 0 - least significant ERROR ANALYSIS Fluctuations in readings (systematic errors) Instrument Uncertainties - lack of perfect precision in the measuring instrument DISCREPANCY AND TOLERANCE Best estimate (xbest) - value of the quantity without uncertainty Absolute uncertainty (delta x) - sdev of the problem distribution for obtaining a particular value Discrepancy - absolute value of difference between xbest Tolerance - maximum discrepancies that we’re willing to tolerate DISCREPANCY TEST Agreement Discrepancy < tolerance = they agree Fails of discrepancy test o Mistake o Systematic errors o Statistics/anomaly RANDOM PROCESS Statistical uncertainties Not from lack of precision but from overall statistical fluctuations Constellations, Asterisms, Bright Stars Constellations - imaginary figures based on stories and mythology of different cultures Asterisms - collection of stars that often have no association except their location in a particular region of the celestial sphere o The Plough - Ursa Major o Summer Triangle - Deneb (Cygnus), Vega (Lyra), and Altair (Aquila) Stars belonging to constellations are and asterisms were named using Greek letters o Alpha being the brightest, then Beta, Gamma, up until Zeta BAYER CATALOGUE Johann Bayer in 1603 Consists approximately of 2500 bright stars Used by most astronomers Used Greek letters and constellation names  FLAMSTEED CATALOGUE Astronomer Royal John Flamsteed in 1712 Improved version of Bayer Catalogue Geographically limited, not covered entirety of the sky Used numbers instead of Greek letters Ordered the naming by RA instead of magnitude Betelgeuse in Bayer’s is alpha Ori, while it is 68 Ori in Flamsteed’s BONNER DURCHMUSTERUNG CATALOGUE Bonn Observatory between 1799 and 1875 and extended in 1886 by Cordoba Observatory Contains 325000 stars with magnitude 9.5 Divided into a series of 1 degree wide strips Stars are number within the strip in order of their RA Betelgeuse = BD+07 1055 MESSIER CATALOGUE Charles Messier in the 18th century Passionate for observing comets Created as non-comet objects to avoid confusion for later observation Many objects are not visible in the southern hemisphere 110 of them are favorites of many amateur astronomers i.e. M31 (Andromeda Galaxy) and M45 (Pleiades) Involves deep sky objects that are too faint to be observed via naked eye CALDWELL CATALOGUE Patrick Caldwell-Moore in 1990s List of objects omitted by the Messier Catalogue 109 objects including star clusters, planetary nebulae, and galaxies UNITED STATES NAVY OBSERVATORY CATALOGUE USNO A2.0 - 500 million objects with R and B magnitudes down to magnitude 20 USNO-B1.0 - excess of one billion objects down to magnitude 21 Accessible via SAO DS9 UCAC2 and UCAC3 - high precision astrometry down to magnitude 16 in F band Naval Observatory Merged Astrometric Dataset (NOMAD) - merged catalogue of Hipparcos, Tycho-2, UCAC2, USNO-B1.0, and 2MASS NOMAD covers BVRJKH bands down to mag 20 HIPPARCOS AND TYCHO CATALOGUE 118218 source catalogues Objects were numbered and prefixed with HIP Betelgeuse = HIP 27989 Tycho-1 and Tycho-2 contains 1 million and 2.5 million entries Tycho is named in 3 numbers separated by a dash and prefixed with TYC Betelgeuse = TYC 129-1873-1  HUBBLE GUIDE STAR CATALOGUE Pointing catalogue for Hubble Space Telescope 1 billion objects with J and F magnitudes to mag 21 Multiple star systems excluded Betelgeuse = GSC 00129-01873 SLOAN DIGITAL SKY SURVEY Very large and very deep sky survey by Sloan Foundation 2.5 m optical telescope at the Apache Point Observation Sloan - f/5 telescope with 3 degree field of view. It has 120 megapixels ccd and a 640 fibre integrated fibre spectrograph 500 million stars with mag 22 1 million spectra of galaxies 2MASS 2 Microns All Sky Survey that observes NIR with two 1.3 m telescopes in Arizona and Chile 300 milliob stars and 1 million galaxies Betelgeuse = 2MASS J05551028C724255 NGC AND IC CATALOGUE New General Catalogue of Nebulae and Cluster of Stars - non-stellar catalogue sources John Louis Emil in 1888 7840 objects including Messier Catalogue M31 = NGC 224 Index Catalogues (IC) with 5386 more objects UPPSALA GENERAL CATALOGUE Northern hemisphere catalogue of bright galaxies to mag 14.5 2921 galaxies prefixed with UGC THE BURNHAM, AITKEN, AND WASHINGTON DOUBLE STAR CATALOGUE Generational series of double star catalogues initiated by Sherburne Wesley Burnham who published 13665 pairs of double stars Included the Aitken double stars catalogue (17180) Washington Double star Catalogue (115769 stars with position, mag, and color type) o Noted by their J2000 position and prefixed with WDS HENTY DRAPER CATALOGUE 359083 spectrally classified stars (1886 and 1949) with mag 9 or 11 Spectral classification with eh standard OBAFGKM spectrographic classification scheme Objects are numbered HDm HDC or HDEC Betelgeuse = HD 39801  GENERAL CATALOGUE OF VARIABLE STARS Variable and suspected variable stars American Association of Variable Star Observers (AAVSO) keeps the variable star catalogue with the International Variable Star Index 20 million variable stars SIMBAD Astronomical database operated by Centre de Donnees Astronomique de Strasbourg (CDS) in France Provides basic data, cross identification, bibliography, and measurements for objects outside solar system MINOR PLANET CENTRE Data and orbital elements for minor planets, comets, and natural satellites Asteroids and comets (minor celestial bodies), assessing potential impact hazards INFRARED SCIENCE ARCHIVE Infrared Processing and Analysis Center (IPAC) at Caltech Storing and distributing IR astronomical data VIZIER CDS Offers access to a vast collection of astronomical catalogs and data tables Users can perform queries based on various criteria Measuring Light STARS TO DETECTOR Flux/Flux Density (F) o Brightness of a celestial object (irradiance) o Total energy from a source that crosses a unit/area/time o F = Energy/time x area o F = L/4πr2 Powerhouse of the source - luminosity/radial flux Detectors - brightness of a star; power delivered to a sensor by EM radiation Retina - photon counter P = AF Bolometric Flux (Fbol) o Integrated over all wavelengths Monochromatic Flux o Flux per wavelength o Spectral irradiance Attenuation o Effective amount of light lost in aggregate for a number of sight lines o Scatters light to the line of sight Sets of agreed-upon response functions = standard photometric system Quantum Efficiency e(λ) - fraction of incoming photons that are actually detected  MAGNITUDE SYSTEM 2nd century BC - Hipparcus 1856 - Norman Pogson Zero-point - Vega o Average of many stars per calibrating scale in absolute flux units o Vega is slightly variable (0.03 mag) Absolute Magnitude (10 pc) o M = -2.5 log L + G’ o Specifying abs mag = specifying luminosity Distance Modulus o Difference between an object’s apparent and absolute magnitude o Depends on d o m - M = 5 log r - 5 (r in pc) Magnitude (capital letter depending on band ata) o U = -2.5 log Fu + Cu o Cu = U-band ) point Color Index = quantitative measure of color Large B-V - redder and vice versa B-V = MB - MV (mag in the bands) Color indexes ~ Mbol BC - Bolometric Correction o BC = Mbol - V MEASURING LIGHT DETECTORS 1. Photon counting 2. Power Measuring Compare a star to a standard star with a known mag ms o m = ms - 2.5 log F/Fs o m = ms - 2.5 log P/A / Ps/A = ms - 2.5 log P/Ps COUNTING PHOTONS Photon count rate = n dot Rp (λ) - photon response function n dot λ - monochromatic photon flux - number of photons of λ Ephoton = hc/ λ n dot λ = F/hc/ λ Uncertainty of counts o δn = square root of n - standard error Signal-to-Noise Ratio o Inverse of fractional uncertainty o SNR = n/δn Detector noise + photon counting noise If they are uncorrelated, add 2 sources of noise in quadrature σdet = detected noise expressed in equivalent photon counts t = exposure time Uncertainty in mag o δm = (2.5/ln 10)( δn/n) = δn/n  ESTIMATING EXPOSURE TIME Probability Probability - number of events occurring divided by sample space Events - all possibly mutual outcomes of a sample space Example. Coin toss o Single toss - one event o Possible outcomes - heads or tails o Borel-Kolmogorov Paradox (nabanggit lang ni sir, idk if kasama,, feeling ko nde naman) o Set of possible outcomes - {H, T} ▪ curly braces mean all inside the bracket (sample space) o Multiple coin tosses (3 tosses) ▪ HHH ▪ HHT ▪ HTH ▪ THH ▪ HTT ▪ THT ▪ TTH ▪ TTT Cardinality - number of elements of a set Realization - performed events Example. Single die o Single toss - {1, 2, 3, 4, 5, 6} Random Variable (RV) - outcome is not deterministic Example. Coin toss o H, RV X = 1 o T, RV X = 0 Example. 2 coin tosses o RV: x1 = number of heads in toss 1 o RV: x2 = number of heads in toss 2 P(xi) = number of outcomes of event xi divided by the total number of all events  Frequentist Interpretation - only works in an ideal situation where we know everything o P(xi) - event o xi - event that at least 1 H show up PROBABILITY AXIOMS Axioms - simplest case possible that is the truth Postulate - something that you can incorporate 0≤p≤1 Sum of probabilities = 1 (all possible outcomes) Probabilities for mutually exclusive events add If A & B can’t occur at the same time, they are mutually exclusive o Example of a mutually exclusive event is coin toss because head and tail cannot occur at the same time What happens when A and B are not mutually exclusive? P (A U B) = P (A) + P (B) - P (A ∩ B) Example: probability of liking basketball (BB) or volleyball (VB) is P (BB U VB) = P (BB) + P (VB) - P (BB ∩ VB) Where: o P (BB U VB) = Probability of liking BB or VB o P (BB) = Probability of liking BB o P (VB) = Probability of liking VB o P (BB ∩ VB) = Probability of liking BB intersection VB ▪ kumbaga sa venn diagram, ito ‘yung overlap noong dalawang circles : ) ONE EVENT IS DEPENDENT (?) P (A,B) = P (A) P (B|A) Where: o P (A,B) = Probability that both A&B happens o P (A) = Probability of A happening o P (B|A) = Probability of B happening given A A & B ARE INDEPENDENT P (A,B) = P (A) P (B) In this case, the events are not mutually exclusive EXAMPLES 1. A card is drawn from a shuffled deck o What is the probability that the card is black? ▪ 𝑃 = # 𝑜𝑓 𝑏𝑙𝑎𝑐𝑘 𝑐𝑎𝑟𝑑𝑠 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 26 ▪ 𝑃= 52 1 ▪ 𝑃= 2  o A red nine? ▪ 𝑃 = # 𝑜𝑓 𝑟𝑒𝑑 𝑛𝑖𝑛𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 2 ▪ 𝑃= 52 1 ▪ 𝑃= 26 o A queen of spades? ▪ 𝑃 = # 𝑜𝑓 𝑞𝑢𝑒𝑒𝑛 𝑜𝑓 𝑠𝑝𝑎𝑑𝑒𝑠 𝑡𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑐𝑎𝑟𝑑𝑠 1 ▪ 𝑃= 52 2. In a room of 36 people, what is the probability that: a. 1 person was not born on Christmas ▪ There are 365 days in a year, since that person is not born on Christmas, we can subtract 1 from 365. ▪ 𝑃 = 364 365 b. No one in the room was born on Christmas 364 36 ▪ 𝑃 = ( ) 365 ▪ 𝑃 = 0. 91 c. At least one person in the room was born on Christmas 364 36 ▪ 𝑃 =1 − ( ) 365 ▪ 𝑃 = 0. 09 3. 2 coins were tossed after one another given that: coin 1 is a fair coin coin 2 is a biased coin that yields to H 2/3 of the time and to T 1/3 of the time a. Probability that at least 1H shows up  b. Probability that 2H show up ▪ P (HH) = 1/3 c. Probability that exactly 1H show up ▪ P (HT) + P (TH) = 1/6 + 1/3 = ½ PERMUTATIONS Order doesn’t matter Number of configurations Ex. N people (distinguishable) o How many ways can we arrange these people in rows with n chairs? Ex. How many ways are there of picking a committee with k members from N people [ N ≥ k ] ? o Approach 1: count permutations o Approach 2: multiply by a correction factor Number of permutations: 𝑁! o (𝑁−𝑘)! Number of combinations: 𝑁! o 𝑘!(𝑁−𝑘)! 2 𝑁! o 𝑁𝑘= (𝑁−𝑘)! Caveat: o If there are categories of memberships within a committee, you should use another correction factor PARTICLE STATISTICS Number of distinct configurations of N particles that can be assigned to k states  MAXWELL-BOLTZMANN N distinguishable particles k possible states Given N & k, how many configurations if N = 2 particles and k = 3 particles N particles, k choices of states General formula: o k k k … ___ = kN configurations BOSE - EINSTEIN N indistinguishable particles k possible states o H0 restriction on the number of particles per state Given N & k, how many configurations if N = 2 particles and k = 3 particles General formula: o No restriction on the # of particles/state FERMI-DIRAC N distinguishable particles k possible states o max of 1 particle per state Given N & k, how many configurations if N = 2 particles and k = 3 particles  General formula (N particles, k states) o N > k → not possible → 0 configurations o N < k → k possible states ← N particles 3! o kCN= 𝑘! 𝑁!(𝑘−𝑁)! 3! 2!(3−2)! = 2!1! =3 Random Variables (RV) Variable that is not fully deterministic Several possible outcomes (loosely) with each outcome having an associated probability Can be either discrete or continuous o Continuous - o Discrete - Ex. Single die (Discrete) o Tossing the die will result to multiple trials o Trial #i where i = 1, 2, 3, … o Outcome of trial i → Xi ▪ Xi Є {1, 2, 3, 4, 5, 6} - 1/6 ▪ RV: { , , , , , } Xi - random variable Toss two dice → consider X = X1 + X2 Outcome (x) Probability 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 P (X = X1 + X2 = x) P (X = x) - probability mass function If RV is continuous, e.g. Vx = randomly chosen molecule (velocity component along the x-axis)  o P (v < Vx < v + dv) = f(Vx) dVx o f(Vx) = probability density function AVERAGE/MEAN/EXPECTATION VALUE Perform an experiment/observation o Measure RV X o Take N measurements of X results: ▪ X1, X2, X3, …, XN 𝑁 1 ▪ x̄ = < x > = E(X) = 𝑁 ∑ 𝑋𝑖 𝑖=1 Alternative: o Possible that values of outcomes are repeated Moments (Deviation from the center): L = 0, 1, 2, 3, … !! Discrete: Summation !! Continuous: Integral VARIANCE, STDDEV  BINOMIAL DISTRIBUTION What’s the probability of getting 3 ones in 5 tosses of a die given that all trials are independent? Among the 5 trials, 3 trials must have X = 1 Examples: o Normalization o Calculation of mean o 2nd moment o Standard Deviation o Probability Calculation Ex. Biased Coin - probability P of turning out heads and q = 1 - P of turning out tails o N times ▪ Xi → RV of the number of heads on the ith toss ▪ Note that Xi = { 1 w/prob P 0 w/prob q = 1 - P} a. Calculate < Xi > , < Xi2 >, & σXi let XN = Xi + X2 + X2 + X3 + … + XN b. What is P (XN = m) when m = 0, 1, 2, 3, 4, …, N c. Calculate < XN >, < XN2 >, & σXN  PROBABILITY DENSITY FUNCTION Suppose the probability density function associated with finding a particle around the position x is f(x) = A exp [-λ (x-a)2] such that A, λ, and a are positive constants a. Sketch the graph b. Determine A (Normalization constant) ▪ Key Idea: Sum of all probabilities of all possibilities = 1 ▪ Options: Gaussian Integral Express in terms of Gamma Functions Example: The probability density of finding a particle in region x is a. Sketch the graph of P(x) b. Determine A c. Find < X > , < X2 >, & σX d. Determine that the particle is in the region a/3 < x < a/1 Sample Calculations of pdf of Functions of a random variable  Functions of a random variable let Prob( x < X < x + dx ) = fXi(x) dx → known and let: Google Colab Modules AST14 - Astronomical Coordinates: Catalog Cross-matching AST14 - Linear and Non-linear Fitting AST14 - Astronomical Coordinates in Python: Coordinates and Velocities AST14 - The Balmer Jump AST14 - Physics of Stellar Atmospheres AST14 - Basic Theory of Bayesian Modeling.ipynb