Deducing Temperatures and Luminosities of Stars (and other objects…)

Review: Electromagnetic Radiation EM radiation is the combination of time- and space- varying electric + magnetic fields that convey energy. Physicists often speak of the “particle-wave duality” of EM radiation. –Light can be considered as either particles (photons) or as waves, depending on how it is measured Includes all of the above varieties -- the only distinction between (for example) X-rays and radio waves is the wavelength. Gamma Rays Ultraviolet (UV) X Rays Visible Light Infrared (IR) Microwaves Radio waves m m 10 3 m m10 -9 m10 -4 m Increasing wavelength Increasing energy

Electromagnetic Fields Direction of “Travel”

Sinusoidal Fields BOTH the electric field E and the magnetic field B have “sinusoidal” shape

Wavelength  of Sinusoidal Function Wavelength is the distance between any two identical points on a sinusoidal wave.

Frequency of Sinusoidal Wave Frequency: the number of wave cycles per unit of time that are registered at a given point in space. (referred to by Greek letter  nu])  is inversely proportional to wavelength time 1 unit of time (e.g., 1 second)

“Units” of Frequency

lWavelength is proportional to the wave velocity v. lWavelength is inversely proportional to frequency. le.g., AM radio wave has long wavelength (~200 m), therefore it has “low” frequency (~1000 KHz range). lIf EM wave is not in vacuum, the equation becomes Wavelength and Frequency Relation

Light as a Particle: Photons lPhotons are little “packets” of energy. lEach photon’s energy is proportional to its frequency. lSpecifically, energy of each photon energy is E = h Energy = (Planck’s constant) × (frequency of photon) h  × Joule-seconds = × Erg-seconds

Planck’s Radiation Law Every opaque object at temperature T > 0 K (a human, a planet, a star) radiates a characteristic spectrum of EM radiation –spectrum = intensity of radiation as a function of wavelength –spectrum depends only on temperature of the object This type of spectrum is called blackbody radiation

Planck’s Radiation Law Wavelength of MAXIMUM emission max is characteristic of temperature T Wavelength max  as T  max

Sidebar: The Actual Equation Two “competing” terms that only depend on wavelength and temperature Constants: –h = Planck’s constant = 6.63 × Joule - seconds –k = Boltzmann’s constant = 1.38 × Joules -K -1 –c = velocity of light = 3 ×10 +8 meter - seconds -1

Temperature dependence of blackbody radiation As temperature T of an object increases: –Peak of blackbody spectrum (Planck function) moves to shorter wavelengths (higher energies) –Each unit area of object emits more energy (more photons) at all wavelengths

Shape of Planck Curve “Normalized” Planck curve for T = 5700 K –Maximum value set to 1 Note that maximum intensity occurs in visible region of spectrum

Planck Curve for T = 7000 K This graph also “normalized” to 1 at maximum Maximum intensity occurs at shorter wavelength –boundary of ultraviolet (UV) and visible

Planck Functions Displayed on Logarithmic Scale Graphs for T = 5700 K and 7000 K displayed on same logarithmic scale without normalizing –Note that curve for T = 7000 K is “higher” and peaks “to the left”

Features of Graph of Planck Law T 1 < T 2 (e.g., T 1 = 5700 K, T 2 = 7000 K) Maximum of curve for higher temperature occurs at SHORTER wavelength : – max (T = T 1 ) > max (T = T 2 ) if T 1 < T 2 Curve for higher temperature is higher at ALL WAVELENGTHS  More light emitted at all if T is larger –Not apparent from normalized curves, must examine “unnormalized” curves, usually on logarithmic scale

Wavelength of Maximum Emission Wien’s Displacement Law Obtained by evaluating derivative of Planck Law over T (recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns)

Wien’s Displacement Law Can calculate where the peak of the blackbody spectrum will lie for a given temperature from Wien’s Law: (recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns)

Wavelength of Maximum Emission is: (in the visible region of the spectrum) max for T = 5700 K

Wavelength of Maximum Emission is: (very short blue wavelength, almost ultraviolet) max for T = 7000 K

Wavelength of Maximum Emission for Low Temperatures If T << 5000 K (say, 2000 K), the wavelength of the maximum of the spectrum is: (in the “near infrared” region of the spectrum) The visible light from this star appears “reddish”

Why are Cool Stars “Red”? (  m) max Visible Region Less light in blue Star appears “reddish”

T >> 5000 K (say, 15,000 K), wavelength of maximum “brightness” is: “Ultraviolet” region of the spectrum Star emits more blue light than red  appears “bluish” Wavelength of Maximum Emission for High Temperatures

Why are Hotter Stars “Blue”? (  m) max Visible Region More light in blue Star appears “bluish”

Betelguese and Rigel in Orion Betelgeuse: 3,000 K (a red supergiant) Rigel: 30,000 K (a blue supergiant)

Blackbody curves for stars at temperatures of Betelgeuse and Rigel

Stellar Luminosity Sum of all light emitted over all wavelengths is the luminosity –brightness per unit surface area –luminosity is proportional to T 4 : L =  T 4 –L can be measured in watts often expressed in units of Sun’s luminosity L Sun –L measures star’s “intrinsic” brightness, rather than “apparent” brightness seen from Earth

Stellar Luminosity – Hotter Stars Hotter stars emit more light per unit area of its surface at all wavelengths –T 4 -law means that small increase in temperature T produces BIG increase in luminosity L –Slightly hotter stars are much brighter (per unit surface area)

Two stars with Same Diameter but Different T Hotter Star emits MUCH more light per unit area  much brighter

Stars with Same Temperature and Different Diameters Area of star increases with radius (  R 2, where R is star’s radius) Measured brightness increases with surface area If two stars have same T but different luminosities (per unit surface area), then the MORE luminous star must be LARGER.

How do we know that Betelgeuse is much, much bigger than Rigel? Rigel is about 10 times hotter than Betelgeuse –Measured from its color –Rigel gives off 10 4 (=10,000) times more energy per unit surface area than Betelgeuse But the two stars have equal total luminosities  Betelguese must be about 10 2 (=100) times larger in radius than Rigel –to ensure that emits same amount of light over entire surface

So far we haven’t considered stellar distances... Two otherwise identical stars (same radius, same temperature  same luminosity) will still appear vastly different in brightness if their distances from Earth are different Reason: intensity of light inversely proportional to the square of the distance the light has to travel –Light waves from point sources are surfaces of expanding spheres

Sidebar: “Absolute Magnitude” Recall definition of stellar brightness as “magnitude” m F, F 0 are the photon numbers received per second from object and reference, respectively.

Sidebar: “Absolute Magnitude” “Absolute Magnitude” M is the magnitude measured at a “Standard Distance” –Standard Distance is 10 pc  33 light years Allows luminosities to be directly compared –Absolute magnitude of sun  +5 (pretty faint)

Sidebar: “Absolute Magnitude” Apply “Inverse Square Law” Measured brightness decreases as square of distance

Simpler Equation for Absolute Magnitude

Stellar Brightness Differences are “Tools”, not “Problems” If we can determine that 2 stars are identical, then their relative brightness translates to relative distances Example: Sun vs.  Cen –spectra are very similar  temperatures, radii almost identical (T follows from Planck function, radius R can be deduced by other means) –  luminosities about equal –difference in apparent magnitudes translates to relative distances –Can check using the parallax distance to  Cen

Plot Brightness and Temperature on “Hertzsprung-Russell Diagram”

H-R Diagram 1911: E. Hertzsprung (Denmark) compared star luminosity with color for several clusters 1913: Henry Norris Russell (U.S.) did same for stars in solar neighborhood

Hertzsprung-Russell Diagram

 90% of stars on Main Sequence  10% are White Dwarfs <1% are Giants “Clusters” on H-R Diagram n.b., NOT like “open clusters” or “globular clusters” Rather are “groupings” of stars with similar properties Similar to a “histogram”

H-R Diagram Vertical Axis  luminosity of star –could be measured as power, e.g., watts –or in “absolute magnitude” –or in units of Sun's luminosity:

Hertzsprung-Russell Diagram

H-R Diagram Horizontal Axis  surface temperature –Sometimes measured in Kelvins. –T traditionally increases to the LEFT –Normally T given as a ``ratio scale'‘ –Sometimes use “Spectral Class” OBAFGKM –“Oh, Be A Fine Girl, Kiss Me” –Could also use luminosities measured through color filters

“Standard” Astronomical Filter Set 5 “Bessel” Filters with approximately equal “passbands”:  100 nm –U: “ultraviolet”, max  350 nm –B: “blue”, max  450 nm –V: “visible” (= “green”), max  550 nm –R: “red”, max  650 nm –I: “infrared, max  750 nm –sometimes “II”, farther infrared, max  850 nm

Filter Transmittances Visible Light U B V R I II Wavelength (nm) Transmittance (%)

Measure of Color If image of a star is: –Bright when viewed through blue filter –“Fainter” through “visible” –“Fainter” yet in red Star is BLUISH and hotter (  m) Visible Region L(star) / L(Sun)

Measure of Color If image of a star is: –Faintest when viewed through blue filter –Somewhat brighter through “visible” –Brightest in red Star is REDDISH and cooler (  m) Visible Region L(star) / L(Sun)

How to Measure Color of Star Measure brightness of stellar images taken through colored filters –used to be measured from photographic plates –now done “photoelectrically” or from CCD images Compute “Color Indices” –Blue – Visible (B – V) –Ultraviolet – Blue (U – B) –Plot (U – V) vs. (B – V)