Observationelle Værktøjer

Magnitudes and their measurements

Measuring a magnitude means measuring the amount of light from an object on the sky with a given instrument at a given time. The instrument and observing conditions defines a response function, which takes into account the transmission of the optics, the filter selected and the quantum efficiency of the detector, which we can include in one function $S_i(\lambda)$. The index i is indication of the choice of filter (i = V is one choice) If the flux at the distance of the earth from the object is $f(\lambda)$, we will get a measurement, recorded as n_ph detected photons directly by a photoelectric photometer (or similar if integrated over an area of a CCD image), where

$$n_{ph} = \int_{-\infty}^{\infty} S_i(\lambda)f(\lambda)d\lambda$$ (1)

For historical reasons we now define the imagnitude by

$$m_i = -2.5\log(n_{ph}) + K_i$$ (2)

where the constant K_i depends on $S_i(\lambda)$. Magnitudes are used to compare the brightness of stars in different spectral regions and mostly occurs as differences:

$$m_{V,2} - m_{V,1} = -2.5\log(n_{ph,2}/n_{ph,1})$$ (3)

where the magnitude difference is related to the ratio of photon counts. As we are using the same filter system V, the constant disappears.

Extinction

The atmosphere causes absorption of the light and reduces the flux from the star. The flux received at the telescope is related to the flux above the atmosphere by the relation

$$f_{\lambda}(z) = f_{\lambda}(0)\exp(-\tau_{\lambda}\sec(z))$$ (4)

where z is the zenith distance and $\tau_{\lambda}$ is the optical depth of the atmosphere. Converting to magnitudes this gives a relation between the magnitude measured at the ground and above the atmosphere

$$m_{\lambda}(z) = m_{\lambda} + p_{\lambda}sec(z)$$ (5)

where $p_{\lambda}$ is called the extinction coefficient. By measuring a star at different zenith distances z and plotting $m_{\lambda}(z)$ versus $\sec(z)$, one can derive the extinction coefficient $p_{\lambda}$. This requires a stable photometric night. Cirrus clouds will spoil the measurements. Once the extinction is removed, we can discuss magnitudes as they would have been measured outside the Earth's atmosphere.

Colours

If we use different filters it is possible to define a colour by

$$m_j - m_i = -2.5\log(n_j/n_i) + K_{ji}$$ (6)

where we again have an arbitrary constant K_ji. Examples of colours are m_B-m_V or just B-V. In order to compare magnitudes and colours measured at different telescopes with different filteres and detectors, a system of standard stars are setup, which defines the standard system. If you measure magnitudes and colours for these stars you will get magnitudes and colours, which we will call system measurements. For the standard stars you want to transform your measurements to the standard system, which you can by measuring enough standard stars to enable you to determine the coefficients of linear transformations (for the B and V magnitudes)

B_i,std = a₀ + a₁B_i,instr + a₂(B_i,instr - V_i,instr) (7)

and

V_i,std = b₀ + b₁V_i,instr + b₂(B_i,instr - V_i,instr) (8)

By observing enough standard stars covering a range in magnitudes and colours, the six coefficients can be determined from a least square solution to the set of equations for all standard stars. The magnitudes should be corrected for extinction, before solving for the transformation. Once the transformation is known, it can be used for all stars measured. Sometimes it can be difficult to observe standard stars, because they are too bright for a large telescope, but the grid of stars defined as standards is expanding.

The distance

The magnitudes defined above is the apparent magnitudes as measured from our position in the Universe. To compare stars at different distances the absolute magnitude is defined, written with a capital M, which is the magnitude one would observe, if the object was 10 parsecs away. If we also include the effect of interstellar absorption due to dust and gas in the Galaxy, we get the relation

$$m_V = M_V - 5 + 5\log(d) + A_V$$ (9)

where d is the distance in parsecs and A_V the absorption in the V system. This latter absorption is related to the colour excess

E(B-V) = (B-V)_obs - (B-V)₀ (10)

which can be measured by the effect observed in Colour-Magnitude diagrams for open clusters. It can also be determined from multicolour measurements. The relation between absoprtion and colour excess is sort of universal due to the similarity of dust clouds. Thus one can use a relation

A_V = 3.4E(B-V) (11)

to fix the interstellar absorption once the reddening E(B-V) is measured. Absolute magnitudes of stars can be directly comapred, as the distance has now been eliminated. Still the measurement depends on the filter system. In order to compare the energy output in terms of the total luminosity of the star L, one must be able to calculate or measure the Bolometric Correction, which relates the absolute bolometric magnitude to the V magnitude

M_Bol = M_V + B.C. (12)

Once the B.C. is known for a given star, and it depends on spectral type and gravity of the star, one can compare the luminosities of stars. Often one will use the Sun as the reference and express the luminosity in solar units

$$M_{Bol} - M_{Bol,\odot} = -2.5\log(L/\hbox{$\rm\thinspace L_{\odot}$ })$$ (13)

where the absolute bolometric magnitude for the Sun is wellknown. Remember that the luminosity is related to the effective temparature T_effand the stellar radius by

$$L = 4\pi R^2\sigma T_{eff}^4$$ (14)

As you will realize, the transformation of observed magnitudes into the basic properties of stars in terms of L, R and T_eff is not straightforward and depends on a number of uncertain calibrations.