The Magnitude Scale. The Color Index.

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1 The Magnitude Scale The Color Index.

2 The Magnitude Scale Measuring the brightness of astronomical objects While cataloging stars in the sky, the Greek Astronomer Hipparchus developed the magnitude system, which is still used by astronomers today. Hipparchus gave the brightest stars an apparent magnitude, m = 1 and the faintest stars, m = 6. Note that fainter stars have higher magnitude values

3 Classify brightness of stars by using magnitudes Orion Constellation

4 Classify brightness of Betelgeuse (α Ori) m=0.45 stars by using magnitudes δ Ori m=2.40 ο Ori m=4.70 ζ Ori m=1.85 η Ori m=3.35 Orion Constellation κ Ori m=2.05 m=6.20 Rigel (β Ori) m=0.15

The Magnitude Scale Measuring the brightness of astronomical objects Brightness of an object is measured in terms of its radiant flux, F, received from the object.

5 The Magnitude Scale Measuring the brightness of astronomical objects Brightness of an object is measured in terms of its radiant flux, F, received from the object. F is the total amount of light energy of all wavelengths that crosses a unit area. Flux is the number of Joules of light energy per second per one square meter. Flux has units of Watts / meter 2. The measured flux depends on the intrinsic Luminosity, L and its distance from the observer. Luminosity has units of Watts (Energy per second). Same object located farther from the Earth, would appear less bright - it would have lower flux

6 The Magnitude Scale Measuring the brightness of astronomical objects R L Mathematically, consider a star of instrinsic Luminosity, L, surrounded by a spherical shell of radius, R. Area of Sphere = 4πR 2. The Flux = F = L / (4πR 2 ).

7 The Magnitude Scale Measuring the brightness of astronomical objects Example : Luminosity of the Sun is L = x W. What is the flux of the Sun at a distance of 1 AU = x m? F = L / (4πR 2 ) = 1365 W m -2 This value is defined as the solar irradiance (also called solar constant, S on the inside cover of your book). What is the flux of the Sun at a distance of 10 parcsecs = x 10 6 AU? F1 / F2 = (R2 / R1) 2 = (2.063 x 10 6 AU / 1 AU ) 2 = 4.3 billion times lower than solar irradiance!

8 The Magnitude Scale Measuring the brightness of astronomical objects Present-day magnitude scale is defined such that the 1 magnitude corresponds to a change in flux by a factor of ma - mb = -2.5 x log10( FA / FB ) If FA = FB x 2.512, then -2.5 x log10 (FA / FB ) = -2.5 x log10 (2.512) =-2.5 x (0.4) = -1 mag. Thus, ma - mb = -1 mag. Star A is 1 magnitude brighter than Star B. (Star B is 1 mag fainter than Star A).

9 The Magnitude Scale Measuring the brightness of astronomical objects In addition, present-day magnitude scale is defined such that the star Vega has a magnitude of 0 (by definition). ma - mb = -2.5 x log10( FA / FB ) If FA = FB x 2.512, then -2.5 x log10 (FA / FB ) = -2.5 x log10 (2.512) =-2.5 x (0.4) = -1 mag. Thus, ma - mb = -1 mag. Star A is 1 magnitude brighter than Star B. (Star B is 1 mag fainter than Star A).

10 The Magnitude Scale Measuring the brightness of astronomical objects Present-day magnitude scale is defined such that one magnitude corresponds to a change in flux by a factor of ma - mb = -2.5 x log10( FA / FB ) m = mag for the Sun (denoted by Greek symbol ). The faintest galaxies yet observed have mg = 30 mag (observed by the Hubble Space Telescope). This corresponds to a flux ratio of m - mg = -2.5 x log10( F /Fg ) F / Fg = x (m - m g ) = x ( ) = 6 x 10 22

11 The Magnitude Scale Measuring the brightness of astronomical objects Absolute Magnitude, M, is the magnitude of a an object if it were placed at a distance of 10 parsecs (definition) (m - M) = (F10 / F) = (d / 10 pc) 2 Solving for d: (d / 10 pc) = (m-M) m - M = 5 log10(d / 10 pc) or m - M = 5 log10(d) - 5, for d in units of parsec m-m is defined as the Distance Modulus.

12 The Magnitude Scale Measuring the brightness of astronomical objects Absolute Magnitude, M, is the magnitude of a an object if it were placed at a distance of 10 parsecs (definition). It is intrinsic to an object and never changes. (Like an object s Luminosity.) Apparent Magnitude, m, is the magnitude of an object as it appears to be. It depends on how far away the object is from the observer. (Like an object s Flux.) They are related by the Distance Modulus, DM = m - M.

13 The Magnitude Scale Measuring the brightness of astronomical objects Example: What is the absolute magnitude of the Sun? Msun = msun - 5 log10(d / 10 pc) msun = and d=1 AU = 4.85 x 10-6 pc. Msun = The distance modulus is: msun - Msun =

14 The Magnitude Scale Measuring the brightness of astronomical objects Example: What are the absolute magnitude and distance modulus of the Vega? MVega = mvega - 5 log10( d / 10 pc ) mvega = 0. and dvega = 7.75 pc. MVega = The distance modulus is: mvega - MVega =

15 The Magnitude Scale Measuring the brightness of astronomical objects Compare the flux and luminosity of the Sun and Vega. msun - mvega = -2.5 log10( Fsun / FVega ) Fsun / FVega = (m sun -m vega ) = ( ) = 54 billion! Lsun / LVega = Fsun (dsun) 2 / FVega (dvega) 2 = The Sun appears to be more than 50 billion times brighter than Vega, but the Sun has only 2.1% of the Luminosity of Vega. Distance matters!

16 Classify brightness of stars by using magnitudes Orion Constellation

17 Classify brightness of Betelgeuse (α Ori) m=0.45 stars by using magnitudes δ Ori m=2.40 ο Ori m=4.70 ζ Ori m=1.85 η Ori m=3.35 Orion Constellation κ Ori m=2.05 m=6.20 Rigel (β Ori) m=0.15

18 The Color Index The apparent and absolute magnitudes covered so far are bolometric magnitudes (bolometric comes from the word bolometer which is an instrument that measures the increase in temperature in the flux it receives at all wavelengths). In practice, detectors measure an object s flux within a certain wavelength region defined by the sensitivity of the detector. Astronomers use measurements of an object s flux within two (or more) different filters to measure an object s Color Index.

19 Blue = 329 nm Green = 656 nm Red = 673 nm

20 % Transmittance U B V R I Wavelength (nm) The color of an object can be measured precisely by using filters that measure the relative flux of the object within narrow wavelength ranges. Some astronomical filters are: U : ultraviolet, filter centered at 365 nm B : blue, filter centered at 440 nm V : visual, filter centered at 550 nm R : red, filter centered at 630 nm I : infrared, filter centered at 900 nm

21 For more examples of images from many-colored filters, see: meaning_of_color/toolbox.php

22 The Color Index The Color index is defined as the difference between the magnitude of an object measured in two different colors: Definitions: U, B, V (other capital letters) refer to the apparent magnitude measured in that filter. MU, MB, MV refer to the absolute magnitude measured in that filter.

23 The Color Index The Color index is defined as the difference between the magnitude of an object measured in two different colors: U - B is the color index between ultraviolet and blue light. B - V is the color index between blue and visual light. Note that U - B = MU - MB and B - V = MB - MV Because magnitudes decrease with increasing flux, an object with smaller color index said to be bluer than an object with higher color index. Example: U - B = -2.5 log10 [ F(365nm) / F(440nm) ] F(365) / F(440) = (U-B) As U - B gets smaller, (U-B) gets bigger, and the flux at 365nm gets larger than the flux at 440 nm.

24 The Color Index The relation between apparent magnitude and flux are related by: U = -2.5 x log10( Fλ x SU(λ) dλ ) + CU Where the integral is over all wavelengths. The Sensitivity Function, SU, is the fraction of the objects flux that is detected as a function of wavelength in the U filter (each filter has a sensitivity function), like those shown here:

25 Log Flux per nm T = 30,000 K T = 10,000 K 6,000 K 3,000 K U filter B filter V filter 1000 K ,000 Wavelength [nm] Objects with different blackbody temperatures have different amounts of light measured in the UBV filters. (Measuring the relative amount of light at even shorter or longer wavelengths would give us even more information!)

26 Log Flux per nm T = 30,000 K T = 10,000 K 6,000 K 3,000 K U filter B filter V filter 1000 K ,000 Wavelength [nm] We calculate the amount of light in a filter by integrating the filter and the object s spectrum, e.g., for the U-filter: U = -2.5 x log10( Fλ x SU(λ) dλ ) + CU Where Fλ is the Flux per nm of the object, SU is the filter s Sensitivity function and CU is a constant.

27 The Color Index A very hot star has a surface temperature of 42,000 K and a less hot star has a surface temperature of 10,000 K. Estimate their B- V colors (given that CB - CV = CB-V = 0.65): B = -2.5 x log10( Fλ x SB(λ) dλ ) + CB V = -2.5 x log10( Fλ x SV(λ) dλ ) + CV Approximate that (where Bλ(T) is the Planck function at wavelength λ and temperature T) Fλ x SB(λ) dλ = B440(T) ΔλB and Fλ x SV(λ) dλ = B550(T) ΔλV where ΔλB = 98 nm and ΔλV = 89 nm (approximate as square filters)

28 The Color Index B - V = -2.5 x log10( B440(T) ΔλB / B550(T) ΔλV ) + CB-V B440(T) / B550(T) = (550/440) 5 x [ (e hc/(550nm)kt - 1) / (e hc/(450nm)kt - 1) ] hc/k =(6.626 x J s) x (2.998 x 10 8 m / s) / (1.38x10-23 J / K) = [m * K] B440(42000K) / B550(42000K) = (3.05)x[ / ] = B440(10000K) / B550(10000K) = (3.05) x [ / ] = ,000K => B - V = -2.5 x log10(2.236 x 98nm/89nm) + CB-V = ,000K => B - V = -2.5 x log10(1.527 x 98nm/89nm) + CB-V = 0.09 Stars with higher temperatures have lower color index (bluer colors) Stars with lower temperatures have higher color indexes (redder colors)

29 The Color Index Color is related to temperature Betelgeuse (α Ori) m=0.45 T = 3600 K Appears Redder Recall relation of Flux per unit wavelength for blackbody radiation. Rigel (β Ori) m=0.15 T = 13,000 K Appears Bluer Orion Constellation

30 The Color Index The Bolometric Correction is defined as the difference between an object s bolometric magnitude (the magnitude corresponding to the flux over all wavelengths) and its visual (V) magnitude. BC = mbol - V = Mbol - MV where mbol = -2.5 x log10( Fλ dλ ) + Cbol Note that there is no sensitivity function (like for the magnitude measured in each color filter). For the bolometric magnitude, the integral is over all wavelengths!

31 What have we learned? The magnitude scale is a logarithmic scale that quantifies the relative brightness of an object compared to another object. The apparent magnitude is the brightness of the object (related to flux) (often relative to the Sun). The absolute magnitude is the brightness of the object at a fixed distance (10 pc), related to Luminosity. The Color Index is the relative brightness of an object in two different bandpasses (wavelengths). It is related to the logarithm of the ratio of an object s flux at two different wavelengths (blue light relative to red light, for example.

The Magnitude Scale Measuring the brightness of astronomical objects

The Magnitude Scale Measuring the brightness of astronomical objects While cataloging stars in the sky, the Greek Astronomer Hipparchus developed the magnitude system, which is still used by astronomers