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.

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:

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, *λ*_{max}, which depends on the temperature of the blackbody. If *T* increases, *λ*_{max} decreases

**Wien’s displacement law**: the product of *T* and *λ*_{max} for a blackbody is constant.

**Stefan-Boltzmann equation**: *T* is also related to *L* (**luminosity**) through the surface area of the star, 4*πR*^{2}. *T*_{e} 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*.

And related to *F*_{surf} (**surface flux**) simply by:

For the 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 |

*M _{U}*,

*M*and

_{B}*M*(

_{V}**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** is independent of *distance*.

For given regions of the spectrum (that is, chunks of the spectrum spanning from *λ* to *λ +* *dλ*), we can define **monochromatic luminosity** (*L _{λ}*) and

**monochromatic flux**(

*F*), based on Planck’s function.

_{λ}How to go from **color index** to **flux** (or viceversa)?

Example: From *F _{λ}* we can obtain the

**color index**(

*U-B*):

- Integrate
*F*over a range of wavelengths, for_{λ}*U*and*B*.*S*and_{U}*S*are the sensitivity function (function of_{B}*λ*) of the detector. It takes into account how well the detector detects light of a certain wavelength.

Note: For rough results, you can approximate the integral as*Bλ**λ*(The value of Planck’s function at the wavelength*λ*times the bandwidth). The same applies for*B*:

Look at the table above. For*U*, the peak has a wavelength of 3650 Å, and the bandwidth is 680 Å. - Apply:

The “*C*“s are just some rather arbitrary constants. - The analogous development in 1 and 2 applies to (
*B*–*V*).

**Color-color diagram**:

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: