We have already seen that astronomers use *filters* to isolate parts of the spectrum, and so measure monochromatic flux. The amount of the spectrum that a filter allows through is known as the *bandpass*. Filters are usually categorized into narrow-band filters, which have a bandpass of order 10 nm and typically isolate a spectral line, and broad-band filters, which have a bandpass of order 100 nm. The central wavelength of the filter bandpass is known as the *effective wavelength*. Most modern filters are constructed of different coloured glasses, often in conjunction with thin-film coatings to help define the bandpasses and minimise reflection at the surfaces.

Photometry of a source in a set of filters provides crude spectral information about the source. Well-defined sets of filters are known as *photometric systems*. A photometric system with too many filters, each with a very narrow bandpass, would make it difficult to detect sufficient photons from a source, and strong absorption/emission features in the spectrum might adversely affect some of the bandpasses. Conversely, a photometric system with too few filters, each with a very wide bandpass, would provide insufficient spectral information.

The most widely used photometric system today is the *UBVRI* system, also known as the Johnson-Morgan-Cousins system (see Prof Vik Dhillon's notes for an excellent discussion of the history of this system). The filters cover the whole range of optical wavelengths - *UBV* covers the "Ultraviolet", "Blue" and "Visual" ranges, whilst *RI* cover the "Red" and "Infrared" range. The UBVRI filter set is shown in figure 14 below.

As discussed in PHY104, it is common practice to refer to apparent magnitudes in a particular filter by the name of the filter. If the apparent magnitude of a star in the *V-*band is \(m_V = 15.5\), this is often simply written as \(V=15.5\). The difference in magnitude between two filters is called a *colour index*. If a star's magnitudes in two filters are \(B=16.0\) and \(V=15.5\), the star has a colour index of \(B-V = 0.5\).

Although the *UBVRI* system is very widely used, it is not the only photometric system. Another system you may well come across is that used by the Sloan Digital Sky Survey (SDSS). This uses five filters, named *u'g'r'i'z'*. The SDSS filter set has filters with higher transmissions than *UBVRI*, making them good for faint sources. The SDSS filters also have minimal overlap between filters. It is likely that this filter set will become more common, and eventual dominant over time. The SDSS filter set is shown in figure 16.

Recall that the definition of apparent magnitude is

\[m= -2.5 \log_{10} F + c.\]

It can be seen that the magnitude scale depends upon our choice of the constant \(c\). The most logical way of choosing this scale is to choose a *zero-point*, i.e to choose a star that has a magnitude of 0. We can then measure the magnitudes of all other stars with respect to this one.

The A0V star Vega was chosen as this so-called *primary standard* because it indeed does have a magnitude close to zero as determined by Hipparchos' original magnitude scale, it is easily observable in the northern hemisphere for more than 6 months of the year, it is non-variable, relatively nearby (and hence unreddened by interstellar dust), and it has a reasonably flat and smooth optical spectrum. However, because Vega is too bright to observe with modern telescopes and instruments without saturating their detectors, and because it is not always observable, an all-sky network of fainter *secondary standards* has also been defined, where the magnitude of each star relative to Vega has been carefully calibrated. Over the years, refinements in the definition, number and measurement accuracy of the primary and secondary standards has resulted in the apparent magnitude of Vega now being 0.03 in the V-band, and it is also thought that Vega may be slightly variable, but for the purposes of this course we can ignore this few per cent offset and assume it is 0 in all bands.

Filter | \(\lambda_{\rm eff}\) (nm) | \(\Delta \lambda\) (nm) | \(m_{\rm vega}\) | \(F_{\nu} {\rm (W\,m}^{-2}{\rm Hz}^{-1}{\rm )}\) | \(F_{\lambda} {\rm (W\,m}^{-2}{\rm nm}^{-1}{\rm )}\) | \(N_{\lambda} {\rm (photons\,s}^{-1}{\rm cm}^{-2}{\rm \unicode{xC5}}^{-1}{\rm )}\) |
---|---|---|---|---|---|---|

U | 360 | 50 | 0 | \(1.81 \times 10^{-23}\) | \(4.19 \times 10^{-11}\) | 759 |

B | 430 | 72 | 0 | \(4.26 \times 10^{-23}\) | \(6.91 \times 10^{-11}\) | 1496 |

V | 550 | 86 | 0 | \(3.64 \times 10^{-23}\) | \(3.61 \times 10^{-11}\) | 1000 |

R | 650 | 133 | 0 | \(3.08 \times 10^{-23}\) | \(2.19 \times 10^{-11}\) | 717 |

I | 820 | 140 | 0 | \(2.55 \times 10^{-23}\) | \(1.14 \times 10^{-11}\) | 471 |

**Table 1: **Characteristics of the *UBVRI* photometric system. \(m_{\rm vega}\) refers to the magnitude of the star Vega in the filter. By definition, the magnitude of Vega is actually \(V=0.03\) and all the colours (e.g *B-V*) are zero. Magnitudes defined this way are referred to as the *Vega magnitude system*. The final three columns give the flux expected at the top of the Earth's atmosphere from a zero magnitude source, in various units.

Once we know that the magnitude of Vega is defined as zero, this allows us to calculate the value of *c*, and also some physical characteristics of the photometric system. Table 1 uses the definition of Vega as having \(m=0\) to list the characteristics of the *UBVRI* system. All values are taken from Chris Benn's ING Signal program. From left-to-right are tabulated the filter name, the effective wavelength (\(\lambda_{\rm eff}\)), the bandpass (\(\Delta \lambda\); FWHM), the approximate apparent magnitude of Vega (\(m_{\rm vega}\)), the band-averaged monochromatic fluxes in both frequency (\(F_{\nu}\)) and wavelength (\(F_{\lambda}\)) units of a *V=0* A0V star, and the photon flux (\(N_{\lambda}\)) in units of photons s^{-1} cm^{-2} Å^{-1} (the latter units are used for convenience as they result in more easily remembered values).

Previously, we saw how to extract the sky-subtracted signal from an object, measured in *counts*, from an image. Once this has been done, it is useful to convert to a magnitude in a photometric system. To see why this is important imagine two astronomers observing the same star, through the same filter, but with very different telescopes. Clearly an astronomer with a large telescope is going to measure more counts than the unfortunate astronomer who has a small telescope. It is therefore not very meaningful to share our results with others in units of counts.

Converting a measurement in counts into a calibrated magnitude involves five steps:

- Divide the number of counts by the exposure time, to get a measure of flux in
*counts per second*. - Calculate the
*instrumental magnitude*, from the counts per second - Determine the
*extinction coefficient*, and correct the instrumental magnitude to the above-atmosphere value. - Repeat the above steps for a standard star and use the resulting above-atmosphere instrumental magnitude of the standard star to calculate the zero point.
- Use the zero point to transform the above-atmosphere instrumental magnitude of the target star to the required photometric system.

**Instrumental Magnitudes**Let us call the sky-subtracted signal from our target object, in counts, \(N_t\). We convert this to an

\[ m_{\rm inst} = -2.5 \log_{10} \left( N_t / t_{\exp} \right), \]

where \(t_{\rm exp}\) is the exposure time of the image in seconds. The instrumental magnitude depends on the characteristics of the telescope, instrument, filter and detector used to obtain the data. It is therefore meaningless to compare two instrumental magnitudes taken under different situations, without first putting them on a calibrated scale.

The relationship between instrumental magnitudes and calibrated magnitudes can be understood as follows. The counts per second \(N_t / t_{\exp}\) is proportional to the flux, \(F_\lambda\). Hence

\[ m_{\rm inst} = -2.5 \log_{10} \left( \kappa F_\lambda \right) = -2.5 \log_{10} F_\lambda + c',\]

therefore, instrumental magnitudes are offset from calibrated magnitudes by a constant:

\[ m_{\rm calib}= m_{\rm inst} + m_{\rm zp},\]

where the constant, \(m_{\rm zp}\), is known as the

We can derive a simple equation for the extinction correction by assuming the atmosphere is a series of thin plane-parallel layers. Figure 17 shows such a layer, of thickness \(dx\) at an altitude \(x\). The path length through the layer for light from a star at a zenith distance \(z\) is equal to \(dx \,/ \cos z = dx \sec z\). The term \(\sec z\) is known as the

If the monochromatic flux from an object incident on the layer is \(F_\lambda\) then the flux absorbed by the layer \(dF_\lambda\) will be proportional to both \(F_\lambda\) and the path length through the layer. Therefore:

\[ dF_\lambda = - \alpha_\lambda F_\lambda \sec z \, dx,\]

where the constant of proportionality, \(\alpha_\lambda\) is known as the *absorption coefficient, * with units of m^{-1}. The absorption coefficient is a function of the composition and density of the atmosphere, and hence the altitude of the layer, \(x\). We can re-arrange this equation (and drop the \(\lambda\) subscripts for clarity) to give

\[ \frac{dF}{F} = - \sec z \, \alpha \, dx.\]

Integrating the equation above for \(x\) values from the top of the atmosphere, \(t\), to the bottom \(b\), we obtain

\[ \int_t^b \frac{dF}{F} = - \sec z \int_t^b \alpha dx.\]

Hence

\[ \frac{F_b}{F_t} = \frac{F}{F_0} = \exp \left( -\sec z \int_t^b \alpha dx \right),\]

where for clarity we have renamed the above-atmosphere flux \(F_t = F_0\) and the flux measured at the ground by the observer \(F_b = F\). Remembering that the difference between two magnitudes is \(m_1 - m_2 = -2.5 \log_{10} (F_1/F_2) \), we can write (via some magic with the change of base formula),

\[m-m_0 = -2.5\log_{10} (F/F_0) = 2.5 \sec z \log_{10}(e) \int_t^b \alpha dx .\]

We define the *extinction coefficient*, \(k\), as:

\[k = 2.5 \log_{10}(e) \int_t^b \alpha dx,\]

to finally obtain:

\[ m = m_0 + k \sec z = m_0 + kX,\]

where \(m_0\) is the magnitude of a star observed with no extinction (i.e above the atmosphere) and \(m\) is the magnitude of a star observed at the Earth's surface at zenith distance \(z\). As an example, if the extinction coefficient from a site in the *V*-band is \(k=0.15\) magnitudes/airmass then a star would appear 0.15 magnitudes fainter at the zenith than it would appear above the atmosphere, and 0.3 magnitudes fainter than above the atmosphere when at a zenith distance of 60^{o}.

The dominant source of extinction in the atmosphere is Rayleigh scattering by air molecules. This mechanism is proportional to \(\lambda^{-4}\), which means that extinction is much higher in the blue than in the red. The extinction can also vary from night to night depending on the conditions in the atmosphere, e.g. dust blown over from the Sahara can increase the extinction on La Palma during the summer by up to 1 magnitude. Table 2 lists the extinction coefficients on a typical (undusty) night on La Palma in *UBVRI*. For reference, the night sky brightness on La Palma when the Moon illumination is 0% (Dark), 50% (Grey) and 100% (Bright) illuminated is also listed. All values in table 2 have been taken from Chris Benn's ING signal program.

Filter | \(\lambda_{\rm eff}\) (nm) | \(k\) (mags/airmass) | \(m_{\rm sky}\) (mags/arcsec^{2}) |
||
---|---|---|---|---|---|

Dark | Grey | Bright | |||

U | 360 | 0.55 | 22.0 | 20.0 | 17.7 |

B | 430 | 0.25 | 22.7 | 20.7 | 18.4 |

V | 550 | 0.15 | 21.9 | 19.9 | 17.6 |

R | 650 | 0.09 | 21.0 | 19.7 | 17.5 |

I | 820 | 0.06 | 20.0 | 18.9 | 16.7 |

**Correcting for extinction**

To measure the extinction on a particular night, it is necessary to measure the signal from a non-variable star at a number of different zenith distances. Inspecting the equation \(m = m_0 + k \sec z\), it can be seen that the extinction would be given by the gradient of a plot of the instrumental magnitude of the star versus \(\sec z\) and the \(y\)-intercept would give the above-atmosphere instrumental magnitude. Although such a plot would give the most accurate answer, it is also possible to obtain an estimate of \(k\) from just two measurements of the instrumental magnitude of a star at two different zenith distances: subtracting \(m_{z_1} = m_0 + k \sec z_1\) from \(m_{z_2} = m_0 + k \sec z_2\) eliminates \(m_0\), allowing \(k\) to be derived.

Note that no explicit extinction correction is required when performing relative photometry. This is because the target and comparison stars are always observed at the* same airmass* and hence suffer the *same extinction*. Hence, when the target signal is divided by the comparison star signal to correct for transparency variations, the variation due to extinction present in the comparison star is removed from the target star.**Aside: Secondary Extinction**For very accurate photometry, the wide bandpass of broad-band filters has to be taken into account when correcting for extinction.

\[ m = m_0 + k \sec z + k_2 C \sec z,\]

where \(C\) is the

\[m_{\rm zp} = m_{\rm std} - m_{{\rm std},0,i}.\]

The calibrated magnitude of our standard star \(m_{\rm std}\) can be looked up in a catalog. The calibrated magnitude of our target star, \(m\), can then be found using:

\[m = m_{\rm zp} + m_{0,i},\]

where \(m_{0,i}\) is the above-atmosphere instrumental magnitude of our target star.

Each filter in a photometric system will have a different zero point. Once the zero point has been measured for a particular telescope, instrument, filter and detector combination, it should remain unchanged, although dirt and the degradation of the coatings on the optics will cause minor changes to the zero point on long timescales. To determine the zero points for the UBVRI system, the photometric standards measured by Landolt can be used.

**Figure 18: **A plot by Michael Richmond showing two different V filters and the spectra of hot and cold stars, demonstrating why correcting for colour terms is necessary when performing high-accuracy photometry (see text for details).

Fortunately, it is straightforward to correct for this systematic error by observing a field with many standard stars possessing a range of colours. The advantage of observing a single field is that all of the stars will then be at the same airmass and hence extinction effects are cancelled out. If this is not possible, you can use many observations of standard stars in different fields. We have already seen that:

\[m_{\rm zp} = m_{\rm std} - m_{{\rm std},0,i}.\]

Hence, if the telescope, instrument, filter and detector combination being used matches that of the photometric system perfectly, we can write:

\[m_{\rm std} - m_{{\rm std},0,i} - m_{\rm zp} = 0.\]

In practice, however, the observer's equipment is never identical to that used to define the photometric system, resulting in the above equation being modified to:

\[m_{\rm std} - m_{{\rm std},0,i} - m_{\rm zp} = c C,\]

where \(c\) is the *colour term*, and \(C\) is the colour index (e.g \(B-V\)). Hence, the colour term is equal to the gradient in a plot of \(m_{\rm std} - m_{{\rm std},0,i} - m_{\rm zp}\) against \(C\), i.e a plot of the difference between the catalogue magnitudes of the standard stars and their calibrated magnitudes as a function of colour; the \(y\) intercept is set to zero by using a zero point calculated from a star of \(B - V = 0\). An example of such a plot is shown in figure 19, which has a gradient of 0.072. You can see from this plot that use of colour terms are necessary to obtain accuracies of order 0.01 magnitudes (i.e 1% in flux). If your requirements are less accurate, you can ignore colour terms, as we will do for the remainder of this course.

**Figure 19: **A plot of the difference between the catalogue magnitudes of a set of standard stars and their calibrated magnitudes (y axis) as a function of colour (x axis). The gradient of the line is equal to the colour term.