Astron. Astrophys. 362, 762-773 (2000)

2. Basic formulae

The CGSE system of units will be used throughout. This means, in particular, that all wavelengths in the following formulae will be in centimeters rather than . The photon energy will be in ergs rather than KeV .

2.1. The refractive index and the extinction coefficient

2.1.1. Formulae for hydrogen or hydrogen-like gas

The essential point of the calculation of the spectrum of Cerenkov radiation is the evaluation of the refractive index n of the gaseous medium. This is easy to understand qualitatively from the necessary condition for producing Cerenkov radiation, . At a given wavelength , the larger the index n, the easier the condition to be satisfied, and the stronger the Cerenkov radiation at will be. Therefore, in order to get the theoretical spectrum of the Cerenkov radiation, it is necessary to calculate the refractive index and its dependence on , i.e. the dispersion curve . For a gaseous medium, the calculation of the curve is easy to do. Because our main interest is in the calculation of in the vicinity of , we must use the rigorous formula:

where

is the complex refractive index, the real part n is the refractive index of the gas, the imaginary part is the extinction coefficient. N is the number density of the atoms, and is the polarizability per atom. When the atoms are distributed over various energy levels with density in level a, will be replaced by . According to quantum theory, the atomic polarizability per atom in level a is given as (e.g. Handbook of Physics ed. Condon and Odishaw, 1958):

where e and m are the charge and the mass of an electron, and are the oscillation strength and the damping constant for the atomic line of frequency . Substituting Eq. (3) into Eq. (1), we get:

Because we are concerned with the neighborhood of a given atomic line , or the intrinsic frequency , (the subscripts u and l denote the upper and lower-levels corresponding to the intrinsic frequency ), it follows that we only need to keep the two largest terms in the above summation, Eq. (4) becomes:

where

and is obtained from Eq. (6) by replacing by . Note that and represent the absorption and emission oscillation strengths respectively, corresponding to the pair of energy levels of a given atom. Therefore, and are related to each other through the statistical weights and , . The absorption oscillation strength is related with Einstein's spontaneous emission coefficient as (Bethe & Salpeter 1957):

and the damping constant is also related with Einstein's coefficient as:

Using these relations, Eq. (1) becomes:

with

From Eq. (9) and Eq. (2), we obtain the refractive index and the extinction coefficient as:

where

Eq. (11) is the dispersion formula, which we need. The dispersion curve can be easily obtained by the digital calculation by use of Eq. (11).

However, as a good approximation, Eq. (11) can be replaced by a simple analytic formula as follows. For convenience in comparing with the observations, it is better to replace the frequency in Eq. (9) by wavelength , let represent the wavelength displacement, and let

denote the fractional displacement, because we are always interested in the calculation in the neighborhood of , . Therefore the small dimensionless quantity y will be very useful in theoretical analysis (e.g. the expansion series by use of the small quantity y). Replacing in Eq. (11), , and . Substituting this into Eq. (10) and Eq. (11), and keeping only the terms of the lowest order in y, we get:

Therefore, among the three quantities b, g and z, only z is -dependent. Because the Cerenkov line emission is not located in the exact position , it has a small redshift, so y is not an indefinitely small quantity (, in fact, at , i.e. , the Cerenkov radiation disappears). In the actually effective range of y, we always have:

For example, for , . Inserting these value in Eq. (13), we find that whenever (equivalently, ). On the other hand, under ordinary physical conditions, we can safely assume . Then even for as high as , will be true whenever is greater than (or ). A smaller N will give a still lower limit of . Therefore, the inequalities (14) always hold in the actually effective Cerenkov line-width (). For the other lines, we obtain the same conclusion (14), under similar considerations. Thus we need only retain the terms of the lowest order of the small quantities and in Eq. (11), and we obtain:

Substituting Eq. (13) into Eq. (15), finally we have the simplified approximate formulae:

Eqs. (16) are the approximate analytic formulae for n and respectively. From Eq. (16) we see that varies as . We shall see below that the Cerenkov spectral emissivity varies approximately as , so . However, the absorption varies as , i.e. the absorption decreases with more rapidly than the emissivity. That is why we have a net Cerenkov radiation at the position , if is not approaching to zero.

The calculated and curves are shown in Fig. 2. The shaded region (in Fig. 2) represents a narrow waveband , very near to the position of intrinsic wavelength , but with small redshift, where the extinction , and the net Cerenkov radiation survives. The waveband is so narrow(e.g. a few ), that looks more like a line-emission, rather than a continuum.

Fig. 2. Relation between refractive index n and wavelength , and relation between extinction coefficient and wavelength. The Cerenkov radiation survives in the shaded narrow region.

2.1.2. Extended formulae for gas of complex atoms(ions) in optical waveband

We emphasize that our derivation from Eq. (4) to Eq. (16) is limited to gas composed of hydrogen and/or hydrogen-like atoms(ions), e.g. , etc. However, a necessary extention to other multi-electron atoms (ions) can easily be done when we notice that Eq. (3) represents the contribution of an electron lying in the atomic energy level a to the atomic polarizability . Therefore, for a given emission line of a given multi-electron atom with frequency , the quantity in Eq. (1) and (5) has to be replaced by , i.e.

where N is the number density of complex atoms. (or ) represents the actual occupation number of the electrons at the lower level l (or upper level u) of complex atom. Obviously we have

where and are the degeneracy of level l and u respectively. From Eq. (1), () and equality , it is easy to see that, if the levels (l, u) have been fully occupied by electrons (`closed shell'), i.e. and , then we have , thus . In other words, for the multi-electron complex atoms, the Cerenkov line-like emission related to frequency can occur only when the upper and/or lower levels () are not fully occupied, i.e. , and/or  ¹. In this case, the formulae for quantities Z and g in Eq. (9) are same as in Eq. (10), but the representation of quantity b has to be changed. When is replaced by , we obtain

or

Thus the final forms of the Eq. (16) for gas composed of multi-electron complex atoms (ions) become

In the following sections of this paper, all formulae from Eq. (17) to Eq. (42) are used for hydrogen and hydrogen-like gas both for simplicity and practicality because hydrogen is the most abundant element in the cosmological plasmas. However, most of these equations are valid for the multi-electron complex atoms only if the following replacement is made:

2.1.3. Extended formulae for complex atoms(ions) gas in the X-ray waveband

Another important extension, which we would like to emphasize in this paper, is related to the possible application of Cerenkov line-like emission in X-ray astronomy because the waveband of some Cerenkov lines of complex atoms (ions), particularly the heavy elements, is in the X-ray range. A possible application of the Cerenkov line mechanism in the X-ray band may be a new explanation for the origin of the broad line of Fe ions observed in Seyfert AGNs in recent years. For convenience in comparing with the observations in the X-ray waveband, it is better to replace the wavelength in the formulae given above by the frequency , or equivalently, by the photon energy . This is easy to do without any remarkable changes in above formulae. We notice that the dimensionless small quantity y in Eq. (12) can also be written as

where represents the energy difference of the upper and lower levels (), and . i.e. the small quantity y also represents the fractional displacement of the frequency or photon-energy. Inserting Eq. () into the equations given above we obtain the following formulae which are suitable to the application to X-ray astronomy:

2.2. The Cerenkov spectral emissivity and the line width

2.2.1. Formulae for hydrogen or hydrogen-like gas

The Cerenkov spectral emissivity can be derived from the dispersion curve given above. It is known from the theory of Cerenkov radiation that the power emitted in a frequency interval () by an electron moving with velocity () is . Let be the number density of fast electrons in the energy interval (), ( is the Lorentz factor, which represents the dimensionless energy of the electron), then the power emitted in interval () by these electrons is . For an isotropic velocity distribution of the relativistic electrons as in normal astrophysical conditions, the Cerenkov radiation will also be isotropic(the definite angular distribution of the Cerenkov emission disappears when the relativistic electrons have an isotropic distribution of velocities). Hence, the spectral emissivity per unit volume and unit solid angle is:

where, are the lower and upper limit of the energy spectrum of the relativistic electrons respectively. For the relativistic electrons, we always have , , so , . Also we notice that in the actual effective emission range, the actual refractive index n of a gas is not far from unity, (see Eq. (16)). Therefore:

where, is the density of the relativistic electrons, and is the characteristic energy of the electrons (or the typical energy) in a given source. The definition of is , so . Hence:

Replacing to or y, we have and . Therefore:

Setting and inserting the expression for n (Eq. (16)) into Eq. (19), we get the Cerenkov line-width :

where

The Cerenkov radiation will be cut off at the wavelength displcaement . Inserting Eqs. (16) and (20) into Eq. (19), the spectral emissivity becomes:

where

It is obvious that the emissivity , when (or ). But for small wavelength displacement , , we have , i.e. decreases with slowly as . Therefore, in the actual effective emission range , we have a good approximation of Eq. (21)

We point out that there is a remarkable difference between the Cerenkov line emission and the usual spontaneous radiation transition, the later is determined by the population density in the upper level u, . But the Cerenkov emissivity is determined by the difference in populations between the lower and upper levels .

The original profile of the Cerenkov line is given by the calculated curve , as shown in Fig. 3a which is obtained from Eq. (22). However, for the small wavelength displacement , i.e. is very close to , the approximate Eq. (22) has to be replaced by the strict Eq. (11) and Eq. (17).

Fig. 3a-c. The Cerenkov emissivity  a , absorption coefficient  b , and the emergent intensity of the Cerenkov line (optically thick case) c . From c we see, the Cerenkov linewidth is large. The line profile is asymmetric with a small redshift.

2.2.2. Extended formulae of and for complex atoms(ions) or molecules gas

The basic formulae for Cerenkov spectral emissivity and maximum line-width , Eq. (20), Eq. (21) and Eq. (22) can be easily extended to the multi-electron complex atoms(ions) or molecular gas by use of the same simple replacement given by Eq. () in Sect. 2.1.2, i.e. , . This is sufficient for the Cerenkov line of complex atoms in the optical band. However, if the Cerenkov line is located in the X-ray band, e.g. the Cerenkov Fe line at , it is more convenient to replace the wavelength in Eqs. (20), (21) and (22) by the photon energy , . This is easy to do because of the simple relation and . Therefore, the Cerenkov spectral emissivity and line-width of complex atoms in X-ray band are still given by Eq. (20), (21) and (22) respectively, but the parameters are changed as

and

We emphasize again that in Eq. () and () is in unit of ergs in CGSE system, rather than in KeV as usually used in the X-ray astronomy.

2.3. The absorption coefficient

2.3.1. Formulae for hydrogen and hydrogen-like gas

For an optically thick dense gas for which the Cerenkov line mechanism is efficient, the final emergent intensity is determined by the competition between the emission and the absorption . Therefore it is necessary to consider the absorption of the gas at . For the optical and X-ray wavebands, there are two main absorption mechanisms that are relevant to the Cerenkov line emission. One is the line absorption in the vicinities of atomic lines, directly related to the extinction coefficient given in Eqs. (11) and (16) by the relation . Another is the photoelectric absorption , the free-free absorption is very small in the optical or X-ray band in which we are mainly interested, and can be neglected. Thus, for us, in the dust-free case, the total absorption is:

can be obtained from the well known formula in molecular optics, . Using Eq. (16) and retaining the lowest order of y, we have:

where

Therefore the resonance absorption decreases rapidly with .

Here, we point out that can be obtained in another way which is more familiar to astronomers. The well known formula for line absorption is:

where, is the Lorentz profile factor

Combining these two expressions, we obtain Eq. (24) again.

The photoionization absorption coefficient is , where the summation extends over all levels for which the photoionization potential is less than incident photon energy , . In normal astrophysical conditions, the most important photoionization absorber is the neutral hydrogen due to its great abundance. Therefore, in optical wavebands we only consider the absorption by neutral hydrogen in the calculation of ( is the next candidate in the detailed calculation). The photoelectric cross section of level s of the hydrogen atoms is:

or

(we keep the lowest order of y) Therefore

The last approximation step in Eq. (25) means that only the absorption of the lowest photoelectric level (i.e. the largest term in the summation) is taken into consideration. The calculated curve is shown in Fig. 3b. Comparing Eq. (25) with Eq. (24), we see that the line absorption decreases rapidly with increasing y, so is effective only in a very narrow range near to , while the photoelectric absorption is nearly independent of wavelength displacement (or y). Over the whole width of the Cerenkov line, is actually the main absorption that determines the integrated emergent intensity. The main effect of is to shift the line towords the red side of (i.e. the Cerenkov line redshift).

2.3.2. Extended formulae of absorption k for complex atoms(ions) gas

As for the complex atoms(ions), the absorption coefficient k is still given by Eq. (23), where the line absorption is still given by Eq. (24), . But the parameter is expressed as

i.e. we use the replacement , , where N is the density of the multi-electron complex atoms concerned. Futhermore, if the Cerenkov line is located in the X-ray band, is expressed as

where is in ergs . i.e. we have used the replacement in Eq. () to get ().

However, when the Cerenkov line is in the X-ray band, we must take care for the calculation of the second kind of absorption, the photoionization , because of the fact that the hydrogen is no longer an important species responsible for the photoelectric absorption due to its very small photoelectric absorption cross section in the X-ray band (, see Eq. (25)), in spite of the great abundance of hydrogen in cosmological plasma.

In fact, for a given emission line in the X-ray band, the dominant contributors to the photoelectric absorption are the heavy elements with high Z, including the relevant multi-electron atom itself, which produces the emission line concerned (see the frequency behaviour of the photoelectric absorption cross section shown in Eq. (26) below, which shows the cross section reaches a maximum at each absorption edge of the given atom, then decreases drastically as ). For example, for Fe line at , the dominant photoionization absorbers are approximately the iron atoms or ions themselves, rather than other elements, Ca, O, S, etc., due to the highest abundance of Fe among the heavy elements with high Z. Furthermore, the main energy levels of the electron of the complex atom which are responsible for the photoelectric absorption are the K, L, M shells of the heavy elements. For example, for Fe line , the most important contributors to the photoelctric absorption are the electrons in L shell of Fe ions. The ionization potential of the Fe K-shell is , higher than . Thus the electrons in the K-shell of the iron atom or ions can not be photoionized by the incident Fe photons . Therefore, for Fe line, we have

where is the density of Fe in the gas, is the occupation number of electrons at energy level, , is the photoelectric cross section of an electron at level. The last approximation in () means that we only keep the largest term in the summation due to the fact that is the largest one around , comparing with , , i.e. . For the Fe atom and/or ions, the hydrogen-like formula for the cross section is a good approximation, particularly for the low-lying levels

Therefore in Eq. () is (taking for level, i.e. for electron in the L-shell, the effective )

Inserting to Eq. (), we get

where in unit ergs .

2.4. The emergent spectral intensity (or ). The line profile

In this section, we present formulae describing the Cerenkov intensity and the line profile, emergent from a plane-parallel emission slab. Similarly, we first discuss the hydrogen or hydrogen-like gas. Using the and given above, the emergent Cerenkov line intensity from the surface of the slab can be calculated by the equation of radiative transfer:

where, is the elementary optical depth, is the source function.

2.4.1. (or ) and the line profile of the Cerenkov line of hydrogen or hydrogen-like gas

Case A. The uniform plane-parallel slab. For a uniform plane-parallel slab of an emitting gas with thickness L, the solution of Eq. (27) gives the emergent spectral intensity :

where is the optical thickness of the uniform slab of the emission gas. Replacing by , Eq. (28) can be expressed in an equivalent form:

However, we shall be particularly interested in the optically thick case, because our main interest is in the high energy astrophysical objects, such as QSOs, supernovae, solar flares, etc., and we know that these objects all have compact structures in which the gas density is high, and near the optically thick case. On the other hand, in Sec. 1, we have argued that the Cerenkov line radiation becomes important only when the gas is dense and optically thick for continuum, i.e. . In the optically thick case, Eq. (29) becomes or

where, , have been given in Eqs. (21), (24) and (25), in which the variable is , rather than . Therefore Eq. (30) becomes:

where, and are given in Eqs. (21), (24) and (25) respectively, and , is given in Eq. (20).

For convenience, we re-list the constants as follows:

(Note that the wavelength in Eqs. (32) is in centimeter rather than )

The last step approximation in Eqs. (32) is due to the fact that, in the normal conditions of a gaseous medium, at least for the lowest levels, we have , or . So , N is the total density of the hydrogen or hydrogenic atom (or ion) species concerned, e.g. for the calculation of Cerenkov lines of . is the fractional population in the level l of the concerned atom (ion) species. Similarly, , where is the number density of the neutral hydrogen atom in the uniform slab, is the fractional population of the neutral hydrogen in the lowest photoelectric level p.

Finally, for comparison with the observations, it is necessary to transform in Eq. (31) into . Using and , we have:

If the conventional unit () is adopted for , then Eq. (33) becomes . By use of Eqs. (31), (32) and (33), the calculated profile of the Cerenkov line in the optically thick case is shown in Fig. 3c, which is broad (line-width ); slightly red shifted; and red-asymmetric, as mentioned above. Here we emphasized that the formula of spectral intensity Eq. (31) is obtained by the use of the approximate formulae Eqs. (16), (21) and (24), which are valid only if the condition (14), e.g. , , is satisfied (That is, the fractional wavelength displacement y cannot be indefinitely small, . Otherwise and , which are obviously unacceptable). However, the derived formula of , i.e. Eq. (31) can be safely extended to without any divergence. Particularly we have at . Therefore, Eq. (31) is valid in the whole waveband of the Cerenkov line .

Case B. The non-uniform plane-parallel slab. For a non-uniform plane-parallel slab of emissive gas, Eq. (28), and hence Eqs. (29)-(31) are no longer valid. In this case, the solution of the equation of radiative transfer Eq. (27) will take the original form: . But we have pointed out that in the effecting Cerenkov emission range, the actual index . Therefore we have:

In the non-uniform case, the factor can not be taken outside the integral, because it is dependent on the position s, and , and so is the ratio . Therefore Eq. (34) can not be simplified to Eq. (28) (but neutral hydrogen is an exception, for , and , the ratio is s-independent, therefore Eq. (28)-(31) are still valid for hydrogen despite of the drastic nonuniform variation of near the front ). Eq. (34) can be re-written as:

If the effective interval where is rather than , then Eq. (35) can be rewritten as:

Eq. (36) gives the emergent spectral intensity of the nonuniform plan-parallel slab. The integral can be evaluated numerically if and are given at each point s along the ray. However, in some cases, a semi-quantitative estimation of is good enough. We suggest a simple expression with a similar form of Eq. (30) for the uniform case, to replace Eq. (36):

where, the integral region expresses the range where the atom (or ion) species concerned exists, hence , and .

The special importance of the non-uniform plane-parallel slab is in that, according to the conventional photoionization model of BLR of AGNs, most low-ionized ions particularly the hydrogenic ions, e.g. , exist in a layer near to the ionization front of , and have a non-uniform distribution, , . For example, it is known that the ionization thresholds of iron are , and the position of the front is at . So we infer that the low-ionized ions exist on both sides of the front , where the density of neutral hydrogen varies drastically with s, , the same as the density of . Fig. 4 shows the schematic distribution of , in the neighbourhood of the front of .

Fig. 4. The schematic distributions of and near the ionization front, i.e. the ionization structure of the surface layer of the cloud in BLR.

2.4.2. Extended formulae of intensity for complex atoms(ions) gas

For simplicity, we only give the formulae for the uniform and optically thick plane-parallel emission slab. The discussion is parallel to Sect. 2.4.1 for the hydrogenic atoms. Thus Eq. (31) is still valid for the heavy elements, only with a replacement of and (Eq. () in the expressions of in Eqs. (32), we then get the emergent intensity I and the line profile for the Cerenkov line of the complex atoms(ions). Furthermore, in the optical waveband, the dominant absorber of photoionization is still hydrogen due to the great abundance of H, thus is still given by the last equation in Eqs. (32).

However, if the Cerenkov line is located in the X-ray waveband, e.g. the Cerenkov Fe line , we must use the parameters and given by Eq. (), (), ) and (), () respectively. For convenience, we relist all of these parameters in Eq. () below. Therefore, in Eq. (31),

and is in unit ergs in CGSE system ().

Using Eq. (31), we calculated the Cerenkov line of the iron ion , as shown in Fig. 5. The calculation parameters are chosen as , (Fig. 5a) and , (Fig. 5b) respectively.

Fig. 5a-c. The emergent intensity of ions. a  shows the line profile of in gas of higher density , ; b  shows a symmetric profile in gas of lower density , ; c  combines the line-profiles of a and b for comparason.

2.5. The Cerenkov line redshift (or )

2.5.1. Formulae of (or ) for hydrogen or hydrogen-like gas

Using Eq. (31), the small "Cerenkov line redshift" can be obtained from the equation . Thus we obtained:

If the gas is very dense (e.g. ), we have a line width (, but is density-independent, see Eqs. (20), (32). Therefore from Eqs. (32) and (38) we get a simplified redshift formula:

where represents the "abundance" of the concerned hydrogenic atom (ions) species. For the gas of neutral hydrogen . It is easy to show that the wavelength displacement in Eq. (39) is just at the wavelength position where . For the wavelength region which is close to , we have which means that the Cerenkov line redshift is just caused by the great line absorption at vicinity of . From Eq. (39) we see, the redshift for the dense gas is N-independent, is only dependent on the atomic parameters () and the temperature T of the gas, and the "abundance" .

Another limiting case is for the gas with lower density, which makes . In this case (but still optically thick for continuum, ), Eq. (38) is simplified as:

where, is given by Eq. (20) (). The critical point of "higher" and "lower" densities is determined by a critical equality , from which we get a critical density of the concerned species in a gas:

Therefore, Eq. (39) and Eq. (40) are valid for and respectively. Fig. 3c shows the profile and the redshift of the Cerenkov line for the case . The red-asymmetry of the profile is remarkable. But for the cases and (see Fig. 5b), the profile becomes more symmetric. And the Cerenkov line redshift becomes very small.

We emphasize again, both Eq. (39) for and Eq. (40) for are derived from Eq. (31). Both Eq. (39) and Eq. (40) are valid only for the optically thick case, .

2.5.2. Extended formulae of (or ) for complex atoms (ions) gas

The formula of the Cerenkov line redshift, Eq. (38) is also valid for a gas composed of complex atoms (ions). The related parameters in (38), i.e. and are still given by Eq. (32) only with replacement , or equivalently, , (see Eq. (), with unchanged because the dominant photoionization absorber in the optical wavelength is still hydrogen.

In parallel, in the limiting cases, and , or equivalently, and , the simplified redshift formulae are still given by Eq. (39) and (40) respectively.

If the Cerenkov line of complex atoms or ions is in the X-ray band, it is better to change the wavelength in (32) into the photon energy, , , i.e. the parameters and in (38) have to be expressed by Eq. (). given in Sect. 2.4.2, where is in unit ergs in the CGSE system, rather than KeV (1KeV =).

2.6. The emergent intensity of the Cerenkov line

2.6.1. Formulae of for hydrogen of hydrogen-like gas

Integrating Eq. (31), we obtained the total intensity of the Cerenkov line, for the optically thich uniform plane-parallel layer Thus

where, , is the density of relativistic electrons, , , , and are given by Eq. (32), is given by Eq. (20).

Eq. (42) is in principle valid for a uniform plane-parallel slab of emissive gas. But as an exception, it can be used to calculate the line intensity of the neutral hydrogen without any problem. Although the density of atoms, varies with depth s drastically near the front , , we point out that the ratio is constant, despite the nonuniform ionization structure in the neighbourhood of the front . Therefore can be taken out of the integral , which ensures the validity of Eq. (42).

For the non-uniform case, the low-ionization line, such as , , the emergent intensity of Cerenkov line , is also obtained from , and is given by Eq. (35). The calculation is somewhat complicate. However, in the semi-quantitative estimation of , we can still invoke Eq. (42) only regarding the factors , , particularly, etc. as the average value , in the interval , where the concerned atoms or ions exist.

2.6.2. Extended formulae of for complex atoms (ions) gas

The emergent total intensity of the Cerenkov line of the complex atoms (ions) in the optically thick case is also given by Eq. (42), where and are given by Eq. (32) replacing by and by . Note that, when the Cerenkov line is in the X-ray band, Eq. () has to be used to replace Eq. (32).

© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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