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

Review: Electromagnetic Radiation Increasing energy 10 -15 m 10 -9 m 10 -6 m Increasing wavelength 10 -4 m 10 -2 m 10 3 m • 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.

Electromagnetic Fields Direction of “Travel”

Sinusoidal Fields • BOTH the electric field

E

field

B

and the magnetic have “sinusoidal” shape

Wavelength  of Sinusoidal Function  

Wavelength

 is the distance between any two identical points on a sinusoidal wave.

Frequency n of Sinusoidal Wave time 1 unit of time (e.g., 1 second) 

Frequency:

the number of wave cycles per unit of time that are registered at a given point in space. (referred to by Greek letter n [ nu])  n is inversely proportional to wavelength

“Units” of Frequency

c

     meters second meters cycle   n   cycles second   1   cycle second    1 "Hertz" (Hz)

Wavelength and Frequency Relation  Wavelength is proportional to the wave velocity v.

 Wavelength is inversely proportional to frequency.

 e.g., AM radio wave has long wavelength (~200 m), therefore it has “low” frequency (~1000 KHz range).

 If EM wave is not in vacuum, the equation becomes n  v  where v  c

n

Light as a Particle: Photons  Photons are little “packets” of energy.

 Each photon’s energy is proportional to its frequency.

 Specifically, energy of each photon energy is

E = h

n Energy = (Planck’s constant) × (frequency of photon) h  6.625 × 10 -34 Joule-seconds = 6.625 × 10 -27 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

http://scienceworld.wolfram.com/physics/PlanckLaw.html

Planck’s Radiation Law • Wavelength of MAXIMUM emission  max is characteristic of temperature T • Wavelength  max  as T   max http://scienceworld.wolfram.com/physics/PlanckLaw.html

Sidebar: The Actual Equation  2

hc

2  5

hc e



kT

1  1 • Complicated!!!!

–

h = Planck’s constant = 6.63 ×10 -34 Joule - seconds

–

k = Boltzmann’s constant = 1.38 ×10 -23 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

Sidebar: The Actual Equation  2

hc

2  5

hc e



kT

1  1 • Complicated!!!!

–

h = Planck’s constant = 6.63 ×10 -34 Joule - seconds

–

k = Boltzmann’s constant = 1.38 ×10 -23 Joules -K -1

–

c = velocity of light = 3 ×10 +8 meter - seconds -1

– –

T = temperature

[K]  =

wavelength

[meters]

Shape of Planck Curve http://csep10.phys.utk.edu/guidry/java/planck/planck.html

• “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 http://csep10.phys.utk.edu/guidry/java/planck/planck.html

• 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 http://csep10.phys.utk.edu/guidry/java/planck/planck.html

• 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

 3  max [ meters  

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:  3  max [ meters  

T

(

recall that human vision ranges from 400 to 700 nm, or 0.4 to 0.7 microns

)

 max for T = 5700-K • Wavelength of Maximum Emission is:  max   3

m

0.508



m

5700 (in the visible region of the spectrum)  508

nm

 max for T = 7000-K • Wavelength of Maximum Emission is:  max   3

m

0.414



m

 414

nm

7000 (very short blue wavelength, almost ultraviolet)

Wavelength of Maximum Emission for Low Temperatures • If T << 5000-K (say, 2000-K), the wavelength of the maximum of the spectrum is:  max  2.898 10  3

m

2000 1.45



m

 1450

nm

(in the “near infrared” region of the spectrum) • The visible light from this star appears “reddish”

Why are Cool Stars “Red”?

Less light in blue Star appears “reddish” 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

 (  m)  max Visible Region

Wavelength of Maximum Emission for High Temperatures • T >> 5000-K (say, 15,000-K), wavelength of maximum “brightness” is:  max  2.898 10  3

m

15000 0.193



m

 193

nm

“Ultraviolet” region of the spectrum Star emits more blue light than red  appears “bluish”

Why are Hotter Stars “Blue”?

More light in blue Star appears “bluish” 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

 max  (  m) Visible Region

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

   8 Joules 2 m -sec-K 4 , Stefan-Boltzmann constant – –

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 (  where R is star’s radius) R 2 , • 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 m

   10     •

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)

M

   10  

F

   10

pc

    

m

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

F

   10

pc

     10 1

pc

  2 2 1 distance  distance 10 pc   2

Simpler Equation for Absolute Magnitude

M

   10        distance 10 pc    2     

m

   10    distance 10 pc    

m

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” http://zebu.uoregon.edu/~soper/Stars/hrdiagram.html

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

“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”  90% of stars on Main Sequence  10% are White Dwarfs <1% are Giants http://www.anzwers.org/free/universe/hr.html

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:

L star L Sun

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 – B: “blue”,  max   350 nm 450 nm – V: “visible” (= “green”),  max  550 nm – R: “red”,  max  – I: “infrared,  max 650 nm  750 nm – sometimes “II”, farther infrared,  max  850 nm

90 80 70 60 50 50 40 30 20 10 0 0 Transmission (%) Filter Transmittances

U,B,V,R,I,II Filters

Visible Light R I II U B V 300 400 500 600 700 800 Wavelength (nm) Wavelength (nm) 900 1000 1100 U V B I R II

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 0.3 0.4 0.5 0.6 0.7 0.8

 (  m) Visible Region

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 0.3 0.4 0.5 0.6 0.7 0.8

 (  m) Visible Region

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)