Energy and number spectra

The (energy) specific intensity has previously been defined (8.26) as follows:

(Energy specific (11.1)

intensity)

This quantity is appropriate for a diffuse (resolved) celestial source, one larger in angular size than the telescope resolution. It is often rewritten in terms of other combinations of units. For example, one could choose,

Ip(v): (photons s-1 m-2 eV-1 sr-1) (Number specific (11.2)

intensity)

which illustrates the two major variants: (i) the intensity is expressed as number of photons per second rather than power (energy per second), and (ii) the band is expressed in energy units (eV or J) rather than in frequency or wavelength units. A spectrum plotted with photons rather than energy in the numerator is often called a photon spectrum or a number spectrum in contrast to an energy spectrum. Note that the quantity "photons" is a dimensionless number,

(photons s-1 m-2 Hz-1 sr-1) = (s-1 m-2 Hz-1 sr-1) (11.3)

The choice between energy and photon spectra is quite arbitrary. The information content is the same in either case because at any frequency v, the energy E of a photon is known to be

where h = 6.63 x 10-34 J s. The units of Hz are inverse seconds (s-1) so the units balance.

The number specific intensity is converted to energy specific intensity simply by multiplying the number of photons in a narrow energy band Ip dv by the energy, E = h v, of each photon in that band,

In this conversion, one must use self-consistent units, for example hv in joules and I and Ip in the SI units of (1) and (3).

If the data represent the spectrum of a point (unresolved) source, the spectral flux density S describes the radiation. In energy units,

density)

or in terms of photons,

density)

Feminine Energy Frequency Number

Log frequency (Hz)

Figure 11.2. Spectrum of the Crab nebula and Crab pulsar from radio through gamma-ray frequencies. The log of the energy spectral flux density is plotted vs. log frequency. The straight-line segments are power-law spectra typical of the synchrotron emission process. The dashed lines show the spectrum that would be observed in the absence of interstellar dust and atoms. The time-average pulsar fluxes are shown as lines; typical peak fluxes of pulses are shown at two frequencies in the optical (dots). [Courtesy G. Fazio]

Log frequency (Hz)

Figure 11.2. Spectrum of the Crab nebula and Crab pulsar from radio through gamma-ray frequencies. The log of the energy spectral flux density is plotted vs. log frequency. The straight-line segments are power-law spectra typical of the synchrotron emission process. The dashed lines show the spectrum that would be observed in the absence of interstellar dust and atoms. The time-average pulsar fluxes are shown as lines; typical peak fluxes of pulses are shown at two frequencies in the optical (dots). [Courtesy G. Fazio]

An example is the spectrum of the Crab nebula (Fig. 2). The Crab is not really a point source; it is a nebula several arcminutes in angular extent. However, if the instrument beam encompasses the entire source, it appears as an unresolved point source. Stated otherwise, the flux plotted is the specific intensity integrated over the angular extent of the nebula.

Astronomers tend to choose units that yield numbers close to unity because it is easier to say them. Since radio sources are many orders of magnitude less intense than the SI unit of spectral flux density (W m-2 Hz-1), radio astronomers found it convenient to define a jansky, abbreviated as Jy,

The jansky was named after Karl Jansky, the discoverer of celestial radio waves (in 1932). This unit is used in Table 8.2.

Spectral reference band

Radio astronomers usually work in frequency units; optical and infrared astronomers think and work in terms of wavelength; while x-ray and gamma-ray astronomers typically use energy units of keV or MeV. Therefore one often sees the intensity or flux quoted with respect to (per unit) wavelength X or energy E rather than frequency v.

Frequency and wavelength

The conversion of the specific intensity between the frequency and wavelength units was quoted in (8.43). We now derive this conversion, but we do it for the flux density S (W m-2 Hz-1; frequency units). The flux density in wavelength units SX is power per unit area per unit wavelength interval:

Sx:(Wm-2 AX-1) or (W m-2 m-1) or simply (W/m3) (11.0)

The spectral flux density S in our usual frequency units may be written with subscript v to distinguish it from SX,

The energy flux F (W m-2) in a given band v 1 to v2 (or the corresponding X1 to X2) must be independent of the manner in which it is expressed, giving i 2 Sv dv = -/ 2 Sx dX (W/m2) (11.11)

The minus sign is necessary if both Sv and SX are to be positive quantities. (Convince yourself of this; note that X2 < X1 if v2 > v 1.) This equality (11) is valid over any arbitrary band pass, that is, for any arbitrary interval v 1,v2 (and the corresponding interval X1; X2). The only way this could be true is if the equality also holds for the integrands in (11),

The frequency and wavelength in a vacuum are related as

which yields the relation between the differential intervals dX and dv, by differentiation, dX/dv = -c/v2 (11.14)

Substitute into (12),

where (13) was invoked again.

This is the desired conversion from Si to Sv. It may be restated with our convention, S = Sv, and with the functional arguments,

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