20 Stellar Photometry

Naseer Iqbal

 

Learning Outcomes

 

After studying this module, you shall be able to

  • Learn about basics of Photometry, why photometry is important and what information it provides about the celestial source that is under observation.
  • Know about the basics of terminology used in the Photometry
  • To learn methods of Photometry and Point spread function.

 

1.  Introduction to photometry:

 

The most important information that we receive about celestial object is the amount of energy called flux, in the form of electromagnetic radiation. The science of measuring flux from celestial objects is called Photometry. This is an efficient way of carrying out brightness measurements of light from Astronomical object’s and therefore, plays key role for the characterization of an astrophysical target. Our eye does not respond equally to all wavelengths of visible light. Photometry accounts this, by weighing the power measured at each wave-length with the sensitivity of eye at that wavelength. Power at each wavelength is weighted by using luminosity function. It differs from radiometry, in  which  radiant energy (including light) is measured in terms of un-weighted absolute power. For example, the eye is more sensitive to green light than to red, this means that green source will be more luminous than a red source and both having same radiant flux. Radiant energy as a light source puts out very few lumens as it carries most of energy in the infra red with dim glow in the visible and thus outside the visible spectrum it does not contribute to photometric quantities at all. Photometry measures many observable parameters and helps us to obtain information and conclusions as following

  • The detection/presence of a source,
  • The apparent brightness of a source can be used for calculating its luminosity,
  • The morphology of the sources characterizes the type of object (extended / point like.)
  • The spectral energy distribution (SED) of the source allows a characterization of all important radiation components in the system
  • The colour characterizes temperature of a star, stellar population in a galaxy etc
  • The temporal variability of the source can be investigated with repeated observations. CCD photometry is used to obtain light curves of faint objects

 

Due to the interaction of light with the atoms and molecules in the object, the flux is often quite irregular on small wavelength scales. These irregularities in the flux as a function of wavelength can tell us lots about the object- what it is made of, how the object is moving and rotating, the pressure and ionization of the material in the object etc. The observation of these irregularities is called spectroscopy. A combination of good wavelength resolution i.e Spectroscopy and good flux calibration i.e photometry is called spectro-photometry. The spectro-photometry is used to obtain the SED of celestial objects. Spectro-photometry of an object contains more information than photometric scan of the same object in same wavelength range. Given this fact still one prefers to do low wavelength resolution photometry rather than higher resolution spectro-photometry because it is much easier to make photometric observations of faint objects than to make spectroscopic observations of the same object. Also the equipment required for CCD imaging photometry is cheaper and simpler than that needed for spectroscopy.

 

2.  Magnitudes:

 

Photometry is concerned with brightness measurement of electromagnetic radiation from astronomical sources. Measurements of a star’s brightness are usually obtained within a limited frequency range (the photometric band) and are therefore more directly linked to the flux density. Since at infrared the measured flux densities of stars are often weak, especially at infrared and radio wavelengths where the photons are not very energetic, the flux density is sometimes expressed in Jansky. However, The basic unit of astronomical photometry is the magnitude and Astronomers measure the visible brightness of stars in terms of magnitude from the very ancient times, originating from the earliest known astronomical catalog. This catalog divided the stars into six classes with the brightest one being the first class and the faintest one the sixth class. Norman Pogson in 1856 proposed a mathematical law such that logarithmic scale approximately agrees with these measurements i.e magnitude

 

related to logarithm of the diameter of the minimum size of the telescope to see the star called the limiting magnitude. 

 

?? = 16.8 + 5???10?                                           (1)

 

Where ?? limiting magnitude and d is diameter of the objective in meters.  So it was then generally decided to express magnitude directly in terms of flux coming from the star as

 

? = −2.5 ??? ?/?0                                                        (2)

 

F0 is fixed as reference to standard stars. These standard stars are chosen such that they are non variable and there magnitude is set to zero. The larger magnitude means fainter source.

 

3.  SOME BASIC TERMINOLGY:

 

Some frequently terms encountered while studying the photometry are as:

 

Absolute Magnitude:

 

Absolute magnitude has a distance dependence of brightness. It gives the intrinsic amount of brightness put out by the source. It is equal to apparent magnitude that the source situated at a distance of 10 parsec. Absolute magnitude satisfies the relation given by

 

m – M = 5 log ρ -5                                     (3)

 

Where r is the distance from the source in parsecs. If the interstellar magnitude adds a magnitude to the apparent magnitude then same amount should also be added to the right of the above equation.

 

Apparent Magnitude:

 

Apparent magnitude is the brightness of the star and it is defined by the formula (eq.2)

 

? = −2.5 ??? ?/?0 

 

Balmer decrement measures D, c1:

 

The discontinuity in the near ultraviolet stellar spectrum at the end of hydrogen Balmer series absorption is a special feature. This is sensitive to temperature as well as gravity. Therefore different photometric include monitoring on either side of the discontinuity to determine the jump in energy terms or the decrement value D. In uvby systems the (u-v)-(v-b) correlates with Balmer decrement and this is denoted by the special symbol c1.

 

Bolometric magnitude:

 

The bolometric magnitude is the total power of the source integrated over all wavelengths. It therefore goes beyond the magnitude at some particular wavelength. This must be added to the visual magnitude and its value in general is negative.

 

Brightness:

 

In case of point like sources like stars brightness is closely associated with the apparent magnitude of the star.

 

Effective temperature:

 

Effective temperature is used to characterize the overall emission of unit area of a radiation source. The effective temperature is that of Planckian radiation field having same integrated emission per unit area.

 

Extinction:

 

The light from stars first passes through absorbing or scattering medium like Earth’s atmosphere as well as interstellar medium and then reach the detector. During this passage the intensity of emitted light decreases. The extinction is the coefficient that depends on the medium  and  the  wavelength  of  the  transmission.  For earth’s  atmosphere  the  extinction coefficient at the observation wavelength is denoted by the k. While as for interstellar medium A is used.

 

Flux:

 

This measures the energy of the radiation field passing per unit area. This is more suitable unit than magnitude for general physical analysis.

 

Luminosity:

 

Luminosity is measure of the radiative power of a source. It relates to the entire output and therefore is related to the absolute bolometric magnitude as :

 

Mbol = 4.75 – 2.5 log ( L/L๏ )                                       (4)

 

Where L๏   is the solar luminosity .

 

Standard magnitude:

 

Apparent magnitude is too imprecise for science use. Over the last few decades, the magnitude system is more precisely defined that allows different workers to use their local measurements and advantageously to convert to a general recognizable scale eg standard V (visual) magnitude.

 

Standard Star:

 

Magnitude is defined in reference to standard star that is usually bright and well known such as Vega or the ten primary standards of the UBV system.

 

4.  METHODS OF  PHOTOMETRY:

 

We know the basic task of photometry is to gather light in a telescope from the star and extract the detected signal. The past measurements were based on eye-estimates, and even today all photometric measurements depend finally on the judgment of eye. With the help of instruments we can aid this judgment very much by limiting the point to be decided, whether two lights as seen are exactly same or not , or else making the decision depend on the visibility or non-visibility of some appearance. Even in some cases the unaided eye is quite as good as any photometric instrument.

 

In the sequences method, the observer only arranges the stars he is comparing in the order of their brightness by taking care that the stars in each sequence list are nearly at the same altitude, and seen under equally favorable circumstances. Then he arranges a second sequence by taking care such that some of the stars that were in the first should be included in it. Likewise he makes third, fourth and so on. Lastly, a list is formed from these sequences that include all the stars arranged in the order of brightness. However, this method does not give the number by which the light of the brightest exceeds that of the faintest.

 

In instrumental method the measurement are done by making the star to disappear by decreasing its light in some measurable way. This method is also called the method of extinctions. Secondly light of the star is made to be equal to some other standard light, by decreasing the brightness of the star or of the standard in some known ratio until they are perfectly equalized. This includes the photometers working on the principle of limiting apertures. The telescope is fitted such that the available aperture of the object-glass can be diminished whenever we want, and the observation consists in determining with what area of object-glass the star is just visible. The method has difficulty of constant errors related to the glass thick in the middle of the lens comes, and diffraction from very small apertures makes the image of star large and diffuse.

 

4.1  Aperture photometry:

 

Aperture photometry is done by measuring the brightness of a star using the aperture to collect the counts from the star. In this case the pixel values are summed within a circular region centered on the star, including light from the star as well as light from the sky. The latter is corrected, by measuring the sky brightness within an annular region surrounding the central aperture. Therefore, subtracting the sky measurement from the star plus sky measurement then yields target signal. This measurement can be taken continuously on consecutively using multiple aperture instruments.

Figure 4.1

 

The image detectors use the software apertures which can be optimized in number of ways to improve the photometric measurements and extract the clean signal.

  • Accurate focus on the target can be achieved
  • We can adopt the size and the shape of the aperture to each individual object,
  • The background disturbance can be reduced by reducing the aperture size
  • The clean region of the sky can be used that represents the  best background at the position of the target
  • The aperture geometry can be used to investigate the  impact of the selected apertures (target and background) on the photometric measurement
  • We can recognise the uncertainties due to other sources or instrumental effects in the image.

 

4.2  The Wedge Photometer:

 

In this method, the extinction is produced by a dark wedge neutral-tinted glass of six inches long, quarter of an inch wide, and at the thick end cuts of light. In the Pritchard for, the wedge is placed close to the eye at the eye-hole of the eye-piece; in some other forms it is placed at the principal focus of the object-glass, where micrometer wires would be put. In observation the wedge is pushed promptly until the star just disappears, and a graduation on the edge of the slider is read. The great simplicity of the instrument commends it, and if the wedge is a good one of really neutral glass (which is not easy to get), the results are remarkably accurate. These observations are however, very trying to the eyes on account of the straining to keep in sight an object just as it is becoming invisible. The constant of the wedge must be carefully determined in the laboratory, i.e., what length of the wedge corresponds to a diminution of the light of a star by just one magnitude (cutting of 0.602 of its light). It is convenient to have the slider graduated into inches or millimeters on the one edge and magnitudes on the other. The Uranometria Nova Oxoniensis” is a catalogue of the magnitudes of the naked-eye stars to the number of 2784,  between the pole and 10 degree south declination, observed with an instrument of this kind by Professor Pritchard, and published in 1885.

Figure. 4.2

 

The image detectors use the software apertures which can be optimized in number of ways to improve the photometric measurements and extract the clean signal.

  • Accurate focus on the target can be achieved
  • We can adopt the size and the shape of the aperture to each individual object,
  • The background disturbance can be reduced by reducing the aperture size
  • The clean region of the sky can be used that represents the  best background at the position of the target
  • The aperture geometry can be used to investigate the  impact of the selected apertures (target and background) on the photometric measurement
  • We can recognise the uncertainties due to other sources or instrumental effects in the image.

 

5.  Point Spread Function:

 

Real stars are not precisely point like, they have finite angular size. However the angular size of almost every star in the sky is too smaller than the diffraction limit of our optical telescopes, so we can treat stars as unresolved points. The stellar profiles are centrally concentrated and star counts fall off with increasing distance from the star’s centre: the Point Spread Function (PSF). In telescopes the dominant source of the PSF is caused by the passage of starlight through the Earth’s turbulent atmosphere. This makes images of the stars are like roundish disks of light. The shape of PSF is very complicated, it can be approximated by a central Gaussian “core” and a large “halo” approximated by power law. The angular size of the PSF can be approximated by the full width at half maximum (FWHM) i. e diameter where the flux falls to half its central value. More centrally concentrated PSF is, higher the sensitivity to detect faint stars will be to resolve closely-spaced stars. Poor telescope focusing and tracking errors leads to degradation of the PSF, therefore should be take care of with good observing techniques.

 

If telescope optics is used to determine the shape of the PSF, then the images are diffraction limited and appears as Airy disk surrounded by concentric minima and maxima. The size of the central disk depends on the wavelength of the light, and on the diameter of the object mirror of the telescope, D. Two equally bright stars are resolved if they are separated by at least the size of the Airy disk (Dawes’ criterion), which is given by the distance from the centre to the first minimum ( in radians):

 

Θ = ?. ?? λ/?                                                   (5)

 

Where λ is the wavelength of the radiation and diameter (D) of the primary mirror or lens.

 

For More Details ( on this topic and other topics discussed in Text Module) See

 

1. Astronomy Principles and practice by A E ROY and D CLARKE

2. Astronomical Photometry by E.F.Milone  nd  . Sterken

3. An Introduction to Astronomical Photometry Using CCDs by W. Romanishin.