Download Where is the Sun in the Milk Way?

Where is the Sun in the Milk Way? •  The distance of the Sun to the center of our Galaxy: 8 kpc •  Why do we see the objects in different colors? –  for instance, why does an object look red? Blackbody Radia8on •  Why do we see the objects in different colors? –  for instance, why does an object look red? –  because it absorbs most colors except red, which is being reflected! •  Blackbody is a physical enDty that absorbs all radiaDon that falls on it, at all wavelengths. •  RadiaDon emanaDng from a blackbody is due uniquely to its thermal energy. •  Planck func8on: –  Max Planck showed that a blackbody with temperature T emits a conDnuous spectrum of radiaDon characterized by a funcDon Bν(T), called “Planck funcDon”. (erg s-­‐1 cm-­‐2 Hz-­‐1 sr-­‐1 in CGS or J s-­‐1 m-­‐2 Hz-­‐1 sr-­‐1 in SI), (J=Joules) –  Planck funcDon depends only on T and ν and is given by the following expression 3
2h
ν
1
Bν (T) = 2 hν
c
e kT −1
k is the Boltzmann constant, k = 1.380658 x 10−16 erg/K c is the speed of light, c = 2.99793458 x 1010 cm/s €
ν is the frequency at which the spectral energy is emiced h is the Planck constant, h = 6.6260755 x 10−27 erg s •  Planck func8on is isotropic and thus independent of the direc8on •  Can also be wricen per unit wavelength (Bλ) and Bν dν = −Bλ dλ
dν c
2hc 2 1
Bλ = −Bν
= 2 Bν = 5
hc
dλ λ
λ
e λkT −1
(erg s-­‐1 cm-­‐1 cm-­‐2 sr-­‐1 or erg s-­‐1 Å-­‐1 cm-­‐2 sr-­‐1) €
Planck distribuDons (Bλ) as a funcDon of wavelength for T = 2000, 6000 and 12 000 K. λmax associated to each funcDon. Visible part of the spectrum is also shown. •  The monochromaDc flux (Fν) is defined as the quanDty of energy in the spectral range between ν and ν+dν emiced per unit surface, per unit Dme in units of erg s-­‐1 cm-­‐2. •  For a blackbody, Fν= πBν •  The energy distribuDon emiced by a blackbody leads to two laws: –  Stefan-­‐Boltzmann law: gives the total power output per unit area, F, of a blackbody €with temperature T ∞
∞
F = Fν dν = πBν dν = σT 4
0
0
∫
∫
σ is the Stefan-­‐Boltzmann constant, σ = 5.67051 x 10−5 erg cm-­‐2 K-­‐4 s-­‐1 €
–  Wien’s law: the wavelength λmax, at which the Bλ is at its maximum, varies inversely with temperature λmax
0.2898 K cm
=
T
–  It explains why hocer blackbodies (or stars) are blue and cooler ones are red €
–  Exercise: Calculate the €λmax for the Sun. •  The surface temperature of the Sun, T=5800 K λmax
0.2898 K cm 0.2898
=
=
≅ 5 ×10 −5 cm = ? Å
T
5800
Luminosity, Effec8ve Temperature, Flux and Magnitudes •  What is Luminosity (L) of a star? –  The luminosity of a star is defined as the radiaDve power output emanaDng from its surface, in other words, the total energy output from a star’s (or an object’s) surface and given in units of “erg s-­‐1” –  It is independent of distance –  and important to understand the energy producDon of a star •  To obtain the luminosity (L), the radiaDon field emiced over the enDre electromagneDc spectrum and over the enDre surface of the star must be integrated! Teff : effecDve temperature, is the temperature of a blackbody which has the same radius R* and luminosity L* of a star. F : Flux energy emiced from an object’s (star) surface per unit area at per unit Dme €
L∗ = 4πR∗2σTeff4
⎛ L ⎞1/ 4
Teff = ⎜ ∗ 2 ⎟
⎝ 4πR∗ σ ⎠
L∗
4
F=
=
σ
T
eff
2
4
π
R
∗
•  At a distance d larger than R✽ from the center of the star ⎛ R ⎞2
L∗
4
∗
F(d) = σTeff ⎜ ⎟ =
2
d
4
π
d
⎝
⎠
d €
L •  This equaDon shows the effect of the geometrical diluDon of the flux as a funcDon of distance from a star. •  It’s also called the “Inverse-­‐Square Law” •  Magnitude is a relaDve scale that measures the logarithmic value of the radiaDve flux. •  DefiniDon of magnitude is given by #F &
m2 − m1 = −2.5 log% 2 (
$ F1 '
m1 and m2 are the apparent magnitudes of two stars F1 and €
F2 are their observed fluxes •  Absolute magnitude, M: the magnitude of a star at a distance of 10 pc #L &
M 2 − M 1 = −2.5 log% 2 (
$ L1 '
•  The difference between the apparent and the absolute magnitude of a star is related to its distance d (in parsecs) to the observer via the equaDon ≡ !!m − M = −5 + 5log d
•  The value m – M is oqen called the “distance modulus”. €
–  Exercise: Knowing that the apparent visual magnitude of the Sun is m = -­‐26.73, calculate its absolute magnitude, M? •  The Sun is by definiDon at a distance of one astronomical unit (AU) from the Earth. •  Since 1 AU = 1.496 x 1013 cm = 4.848 x 10−6 pc, the distance modulus equaDon Mv = 4m.84