Oct. 9, Discussion of Measurement uncertainties (cont.) Measurements always have uncertainties, which can be estimated in our labs (and in your everyday “observations” of the world, economy, etc.) If make N measurements, m i, first derive their average or mean value A m = [m 1 + m 2 + … m N ]/N and then the deviation, D i, of each (i = 1…N)) measurement about the average, D i = m i – A m. Make a Table of m i and D i and then calculate the average of all the D i ‘s Report this average scatter, A D = [D 1 + D 2 + … D N ]/N as your 1 st estimate of overall uncertainty, “mean error” (M.E.) “Root Mean Square” (rms) error is an even better estimate (and often available as a key on a calculator; or in EXCEL…). But don’t work it out by hand; Mean Error is ok. rms = sqrt[ sum((D 1 ) 2 + ((D 2 ) 2 + …(D N ) 2 )/N] Your fractional error F.E. = M.E./A m (or = rms/A m ) is overall summary of your uncertainty. If you have 2 measured values (e.g. d and r in DL1 solar pinhole measurements), just report the F.E. for the larger of the two as overall F.E.

Oct. 9, Introduction to light: direct link to the stars… Light is our direct measure of physical properties of stars (and planets, galaxies, etc.) “Light” is an electromagnetic wave (see Text, fig. 5-6 = Tf5-6) of alternating electric vs. magnetic fields which propagate at speed of light, c = 3 x 10 5 km/sec and which differ only in wavelength (distance between successive “crests” of the wave) from radio (λ = 100km) to visible (λ = 500nm = 0.5micron) to highest energy cosmic gamma-rays (λ = nm = cm!); see Tf5-7. EM waves described by their wavelength, λ, or frequency, ν, where ν = c/ λ. Frequency ν measured in Hz = cycles/sec; FM radio has ν ~100MHz = 10 8 Hz; optical light has ν ~10 14 Hz

Oct. 9, Quantity and Quality of light Light is both a wave and a particle, a photon, with energy E ν proportional to its frequency: E = h ν, where h = Planck’s const Quantity of light we detect (with our eye; or a telescope…) is measured by a quantity called photon flux, or number of photons per unit area over some range of ν : F phot = photons/(cm 2 – sec) Energy flux of light we detect is F = E ν F phot = erg/ cm 2 – sec, where the erg is a unit of energy (= Joules, where the Joule is a larger unit of energy) Quality of light we detect is measured by the spectrum of light, or distribution of its wavelength λ or frequency ν Spectrum of light is (for now…) a continuum, or smoothly varying distribution

Oct. 9, Continuum spectrum of a Black Body The most important continuum spectrum in the Universe is a Black Body spectrum, which describes the Sun and even the very origin of the Universe (Big Bang)! What is a Black Body? A uniformly glowing object (usually solid, or in any case very “thick” or opaque to its own radiation) that emits with “perfect” emissivity. Its spectrum is remarkable, Tf5-10: Key properties: Intensity (or flux) and color of peak both vary with source temperature, T

Oct. 9, Two powerful laws of BB radiation measure T & F The shift of the peak λ, to have a maximum flux (brightness) at wavelength λ max, allows the BB spectrum temperature to be directly measured since Wien’s Law relates the two: λ max = /T (for λ max in meters) or λ max = 2.95 x 10 6 /T (for λ max in nm, more usual unit) So Sun, with surface temp. T = 5500K, has λ max = 540nm The energy flux of the BB source (as measured per unit area on its surface) is proportional to the temperature T to 4 th power, by the Stefan-Boltzmann Law: F = σ T 4 where σ = 5.7 x W/(m 2 - K 4 ) where K is the temperature in degrees Kelvin (note: σ = 5.7 x erg/(cm 2 - sec - K 4 ) if you use cm units instead or m for R and erg/sec instead of W) The total energy flux emitted from a BB source of radius R is its luminosity: L = 4πR 2 F = 4πR 2 σ T 4 The total energy flux detected from a BB source with luminosity L at distance d is F = L / 4πd 2 (this is the inverse square law for light)