Astrophysics for Icarus

Transcription

1 Astrophysics for Icarus Kelly Lepo 1. Black Body Radiation Like lots of things in astronomy, black body radiation is a terrible name. When you see the phrase, think hot, glowing thing. For example, an electric element on a stove glows red because it is hot. The element behaves like a black body. Lots of things in astronomy, like stars, disks of gas, white dwarfs and neutron stars can be modeled as hot, glowing black bodies. This will get you an answer that is more or less right, depending on the exact details of the object. Planck s law describes how energy is emitted by a black body at a given temperature. The black body flux (the energy per second from a star, received on a one square meter detector) given off by a spherical star at a given temperature at a given wavelength is: B λ (λ, T ) = 2πhc2 (λ) 5 1 e hc (λ)k bt 1 (1) in units of: erg m 2 nm 1 s 1. Where λ is wavelength (units: nanometers), T is temperature (units: Kelvin), k B is the the Boltzmann constant, h is the Planck constant, and c the speed of light (units: meters/second). If you plot B λ vs. λ for different temperatures, you get something that looks like Fig: 1. For a simulation of how a flux vs. wavelength curve changes with temperature, see 2. Observed flux The amount of flux we receive at a detector on Earth at a particular wavelength depends on the inverse square law of light: ( ) 2 F λ (λ, T ) = 2πhc2 1 R (2) (λ) 5 e hc (λ)k bt 1 r Where R is the radius of your star and r is the distance from the Earth to the star. Usually R is given in solar radii (how many times bigger or smaller the object is than the

2 Fig. 1.

3 sun) and r is given in parsecs (a unit of distance used in astronomy). Note that you will have to convert these measurements into a common unit (like m or cm) to get the right units in the end. This means that the farther the star is from us, the less flux we will receive. It also means that the larger the radius of the star, the more flux we receive. 3. Astronomical Magnitudes Mostly for historical reasons, astronomers often use the idea of magnitudes to say how bright an object appears at a particular wavelength. They originate from the Greek astronomer Hipparchus who ranked stars from brightest (0th magnitude) to faintest (6th magnitude). Since our eyes are logarithmic photon detectors, this makes magnitude a logarithmic measure of flux. Fainter objects have a larger magnitude (bigger number). Brighter objects have a smaller magnitude (smaller number). The magnitude system is defined relative to a reference point like the star Vega. This is a weird and counter-intuitive system, but we are stuck with it, so we will have to deal. Magnitude is defined as: m = m log 10 ( F F 0 ) (3) where m is the magnitude of your object, m 0 is the magnitude of your reference object, F is the flux of your object and F 0 is the flux of your reference object. When we get data from a telescope, it is often in the form of the object s magnitude as seen through a certain filter. This means that we have a logarithmic measure of the integrated flux our detector measurers over the wavelengths that the filter covers. For example: ( 595nm m V = m v,0 2.5 log F (λ)dλ ) 507nm 10 λf v,0 (4) However, this depends on you knowing the response function of your filter. One way to approximate this is to simply use the flux at the center of the filter: ( ) F (551nm) m V = m v,0 2.5 log 10 (5) F v,0

4 By measuring the magnitude of our object at different filters (like U, B, V and R), we can get an idea of the black body flux of our object, and hence how hot our object is. See figs. 2 and 3. One system used to measure the brightness of stars (which we are going to use for this exercise) is the Johnson system, which is set so the star Vega has m = 0 at all filters. We can look up the value for f 0, the center wavelength and the wavelength range of the filter in a table, like

5 Fig. 2. A hotter star. Notice there is more flux in the U filter than the R filter, so the object has a smaller m U magnitude and a larger m R

6 Fig. 3. A cooler star. Notice there is more flux in the R filter than the U filter, so the object has a smaller m R magnitude and a larger m U. Also note that both magnitudes are larger than the hotter star, meaning it is a fainter object.