the magnitude scale - physics and astronomy...

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

Classify brightness of stars by using magnitudes

Orion Constellation


Betelgeuse (α Ori) m=0.45

Rigel (β Ori) m=0.15

ζ Ori m=1.85

δ Ori m=2.40

κ Ori m=2.05

η Ori m=3.35

ο Ori m=4.70

Orion Constellation

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 / meter2.

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


Mathematically, consider a star of instrinsic Luminosity, L,

surrounded by a spherical shell of radius, R.

Area of Sphere = 4πR2.

The Flux = F = L / (4πR2).

L

R


Example : Luminosity of the Sun is L⊙ = 3.839 x 1026 W. What is the flux of the Sun at a distance of 1 AU = 1.496 x 1011 m ?

F = L / (4πR2) = 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 = 2.063 x 106 AU ?

F1 / F2 = (R2 / R1)2 = (2.063 x 106 AU / 1 AU )2

= 4.3 billion times lower than solar irradiance !


•  Present-day magnitude scale is defined such that the 1 magnitude corresponds to a change in flux by a factor of 2.512.

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).


•  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).


•  Present-day magnitude scale is defined such that one magnitude corresponds to a change in flux by a factor of 2.512.

mA - mB = -2.5 x log10( FA / FB )

•  m⊙ = -26.83 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 = 10-0.4 x (m⊙

- mg) = 10-0.4 x (-26.83 - 30) = 6 x 1022


Absolute Magnitude, M, is the magnitude of a an object if it were placed at a distance of 10 parsecs (definition).

10+0.4(m - M) = (F10 / F) = (d / 10 pc)2

Solving for d: (d / 10 pc) = 100.2(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.


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.


Example: What is the absolute magnitude of the Sun ?

Msun = msun - 5 log10(d / 10 pc)

msun = -26.83 and d=1 AU = 4.85 x 10-6 pc.

Msun = +4.74

The distance modulus is: msun - Msun = -31.57


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 = +0.55

The distance modulus is: mVega - MVega = -0.55.


Compare the flux and luminosity of the Sun and Vega.

msun - mvega = -2.5 log10( Fsun / FVega )

Fsun / FVega = 10-0.4(msun-mvega) = 10-0.4(-26.83 - 0) = 54 billion !

Lsun / LVega = Fsun (dsun)2 / FVega (dVega)2 = 0.021

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 !


Orion Constellation




ζ Ori m=1.85

δ Ori m=2.40

κ Ori m=2.05

η Ori m=3.35

ο Ori m=4.70

m=6.20

Orion Constellation

Scientific Process

Make Observations (take data)

Ask Questions

Suggest Hypothesis

Make Predictions

Make new Experiments

to Test Predictions

Results of new Experiments does not support hypothesis. Revise hypothesis

or choose new one.

Test supports hypothesis, make additional predictions and test them

too. Repeat ad nausem.

From Malcolm Gladwell’s, Outliers

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.

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

U B V

R I

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

Wavelength (nm)

% T

rans

mitt

ance

http://hubblesite.org/gallery/behind_the_pictures/meaning_of_color/toolbox.php

For more examples of images from many-colored filters, see:

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.

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) = 10-0.4(U-B)

As U - B gets smaller, 10-0.4(U-B) gets bigger, and the flux at 365nm gets larger than the flux at 440 nm.

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:

T = 30,000 K

T = 10,000 K

6,000 K

3,000 K 1000 K

100 1000 10,000

U filter B filter

V filter

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 !)

Wavelength [nm]

Log

Flux

per

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.

T = 30,000 K

T = 10,000 K

6,000 K

3,000 K 1000 K

100 1000 10,000

Log

Flux

per

nm U filter

B filter V filter

Wavelength [nm]

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)

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 [ (ehc/(550nm)kT - 1) / (ehc/(450nm)kT - 1) ]

hc/k =(6.626 x 10-34 J s) x (2.998 x 108 m / s) / (1.38x10-23 J / K) = 0.0144 [m * K]

∴ B440(42000K) / B550(42000K) = (3.05)x[ 0.865 / 1.180 ] = 2.236

B440(10000K) / B550(10000K) = (3.05) x [ 12.71 / 25.38 ] = 1.527

42,000K => B - V = -2.5 x log10(2.236 x 98nm/89nm) + CB-V = -0.33

10,000K => 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)


T = 3600 K Appears Redder


T = 13,000 K Appears Bluer

Orion Constellation

The Color Index�Color is related to temperature�

Recall relation of Flux per unit wavelength for blackbody

radiation.

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 !

Combine Luminosity and Color information for Stars

Recall that the luminosity has a strong temperature (T) dependence

L = 4πR2 σT4

Now we know that for objects that emit like blackbodies, their color has a temperature dependence. This is similar to Wein’s Law:

λmax = C / T (C is a constant)

This means that stars (which emit like blackbodies) can be classified on a Luminosity - Temperature plot.

Combine Luminosity and Color information for Stars

This is the Hertzprung-Russell (HR) diagram, which is a stellar classification system developed by Ejnar Hertzprung and Henry Norris Russel in Denmark around 1910.

The HR diagram relates the magnitudes and colors of stars as a function of their temperature. We will return to this later this semester.

Ejnar Hertzsprung Henry Norris Russell

Abs

olut

e M

agni

tude

(M

) (L

umin

osity

)

Temperature (Color, B-V) Hotter

Brighter

Theoretical HR diagram

Abs

olut

e M

agni

tude

(M

) (L

og L

umin

osity

)

Log Temperature (Color, B-V) Hotter

Brighter

Text HR diagram where data points show measurements from 22,000

real stars from the Hipparcos satellite.

30,000 K 7500 K 5000 K 4000 K 3000 K

(Lines are Theoretical, expected luminosities and

temperatures of stars)

the magnitude scale - physics and astronomy...

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