Color Index

Previous: Magnitude

Any body at a temperature above zero emits light. A body which adsorbs all of the light incident upon it is an ideal emitter. Such a body radiates light in a characteristic spectrum and is called a blackbody.

This characteristic spectrum is called blackbody radiation.

Stars and planets, roughly speaking, can be considered blackbodies. Although they are not ideal blackbodies.

Spectra of of a blackbody at different temperatures

Spectra of of a blackbody at different temperatures. Source: wikipedia

The blackbody spectrum is given by Planck’s function. It is a function of wavelength λ (or frequency ν) of the light emitted, and temperature T of the blackbody:

plancks-function

The Planck’s function can be used to make a connection between observed properties of a star (radiant flux and apparent magnitude) and its intrinsic properties (radius and temperature).

The blackbody spectrum has a peak at a certain wavelength, λmax, which depends on the temperature of the blackbody. If T increases, λmax decreases

Wien’s displacement law: the product of T and λmax for a blackbody is constant.

wiens-law

Stefan-Boltzmann equationT is also related to L (luminosity) through the surface area of the star, 4πR2. Te instead of just T means the effective temperature of a star. It is the temperature that the star would have based on the emitted spectrum, if it was a blackbody.

stefan-boltzmann-eq

stefan-boltzmann-ct

And related to Fsurf (surface flux) simply by:

flux-stefan-boltzmann

For the Sun:

t-sun

lambda-sun

On measuring the spectrum of light of a star, distinguish between:

  • Measuring magnitudes over the entire spectrum. These are bolometric magnitudes.
  • Measuring magnitudes over a certain wavelength region defined by the sensitivity of the detector (for instance, the human eye is only sensible to the region of visible light). This way we can define the color of a star, as the light emitted within one region of the spectrum.

UBV system:

Spectrum regions U (ultraviolet magnitude) B (blue magnitude) V (visual magnitude)
Filter/[Å] 3650 4400 5500
Bandwidth/[Å] 680 980 890

MUMB and MV (absolute color magnitudes) can be determined if the distance d to the star is known.

Color Index: Differences (U-B) and (B-V). Magnitudes decrease with increasing brightness. Thus smaller values of (B-V) denote a bluer star.

color-index

Color index is independent of distance.

Bolometric correctionbc

For given regions of the spectrum (that is, chunks of the spectrum spanning from λ to λ + ), we can define monochromatic luminosity (Lλ) and monochromatic flux (Fλ), based on Planck’s function.

L-b

F-b

How to go from color index to flux (or viceversa)?

Example: From Fλ we can obtain the color index (U-B):

  1. Integrate Fλ over a range of wavelengths, for U and B . SU and SB are the sensitivity function (function of λ) of the detector. It takes into account how well the detector detects light of a certain wavelength.
    int-F-l
    Note: For rough results, you can approximate the integral as  Δλ (The value of Planck’s function at the wavelength λ times the bandwidth). The same applies for B:
    int-approx
    Look at the table above. For U, the peak has a wavelength of 3650 Å, and the bandwidth is 680 Å.
  2. Apply:
    u-b-approx
    The “C“s are just some rather arbitrary constants.
  3. The analogous development in 1 and 2 applies to (BV).

Color-color diagram:

Effective_temperature_and_color_index

Color-color diagram showing differing plots for real stars and for blackbodies. Source: Wikipedia

Stars don’t follow the same pattern in the diagram as blackbodies, simply because stars are not true blackbodies. Some light is adsorbed as it travels through a star’s atmosphere. So temperature is not the only factor.

Summarizing pic:

color-index-summary

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