Historically, one of the driving motivations behind attempts to measure the diffuse background radiation at far-ultraviolet wavelengths has been the realization that such observations could potentially reveal the existence of a cosmologically significant (b 1), "lukewarm" (103 K < T < 106 K) intergalactic medium (cf. Kurt & Sunyaev 1967; Davidsen, Bowyer & Lampton 1974; Paresce & Jakobsen 1980).
If a baryonic intergalactic medium (IGM) does exist, standard Big Bang nucleosynthesis calculations predict that it must consist primarily of a mixture of 90% hydrogen and 10% helium atoms. The dominant emission from such a mixture at temperatures between T 103 K and T 106 K is recombination and collisionally excited line radiation in the HI and HeII Ly lines at l = 1216Å and l = 304Å, respectively. This line radiation, smeared by the redshift, will give rise to a diffuse background at wavelengths l of intensity (1)
where 0 l is the observed wavelength, H0 is the Hubble constant and l(z) is the line volume emissivity (in units of photons s-1 cm-3) evaluated at the appropriate redshift, = 0 / l - 1 (see the Appendix for a derivation of this expression). In the case of a smooth IGM emitting through recombination or collisional excitation, where the emission goes as density squared, the line emissivity can be written
where n0H = 7.8 x 10-6 b h2 cm-3 is the IGM hydrogen density at z = 0, b the baryonic IGM contribution to the cosmological density parameter, H0 = 100h km s-1 Mpc-1, and l(T) is a suitably normalized emission coefficient.
Figure 1 shows the run of l(T) as a function of temperature for a cosmological mixture of hydrogen and helium in collisional and thermal equilibrium. It is seen that collisionally excited emission in the HI and HeII Ly lines is especially intense near the two "thermostat" temperatures T 2 x 104 K and T 8 x 104 K, i.e. at the temperatures at which the dominant ionization states change from HI to HII and HeII to HeIII, respectively. If, following recombination, the IGM was re-heated and re-ionized by dissipative processes, such as shock heating, the gas would have had to pass through these two temperatures. Depending on the redshift and duration of the re-heating process, the IGM emission could give rise to observable redshifted Ly and HeII 304Å spectral signatures in the ultraviolet background. Several detailed IGM models giving rise to such features can be found in the literature (Weymann 1967; Sherman 1979, 1982).
Figure 1. Lower frame: Ionization structure of a cosmological mixture of hydrogen and helium in collisional equilibrium as a function of temperature. Upper frame: Corresponding emissivity per hydrogen atom of collisionally excited HI and HeII Ly emission. The dotted lines show the equivalent line emission due to recombination in a fully photoionized gas. Emission from HeI is negligible compared to that of HI and HeII.
Of course, the intergalactic medium is not only constrained by limits on its possible emission, but also by its possible absorption. By far the most stringent limit on the IGM comes from the classical Gunn & Peterson (1965) test, which severely constrains the density of intergalactic neutral hydrogen at high redshift, nHI(z), from the observed lack of an intense redshift-smeared Ly absorption trough seen just shortward of emitted Ly in the spectra of high redshift quasars:
In this expression, (z) is the optical depth of the absorption and l = l ( e2) / (me c2) fij = 4.5 x 10-18 cm2 is the integrated Ly absorption cross section. Both ground-based (Steidel & Sargent 1987) and IUE observations (Kinney et al. 1991) of the spectra of quasars show no signs of absorption troughs over the redshift range 0 < z < 4 at the (z) 0.1 level. The corresponding upper limit on the intergalactic neutral hydrogen density is of order nHI(z) 10-12 cm-3. This stringent limit on the residual neutral hydrogen component of the IGM (and its HeI equivalent; Green et al. 1980; Reimers et al. 1989, 1992; Tripp, Green & Bechtold 1990; Beaver et al. 1991) implies that the IGM - if it exists at all - must be highly ionized and have a temperature greater than T 105 K (compare lower frame of Figure 1).
Alternatively, if photoionization, due for example to the integrated ionizing flux of quasars, is the process responsible for re-ionizing the IGM, then the IGM will still emit primarily in the HI and HeII Ly lines, albeit somewhat less efficiently (Figure 1). On the other hand, since the Gunn-Peterson test suggests that the IGM is transparent, the redshifted ionizing radiation source itself should be directly observable in the ultraviolet background. Absolute measurements of the ultraviolet background intensity can therefore in principle be used to derive the maximum IGM density that can be photoionized by the observed flux to the level required by the Gunn-Peterson test. However, the available observational limits on the extragalactic UV background intensity are generally far brighter than anticipated for realistic intergalactic fluxes and the corresponding constraints placed on the IGM density from the UV background observations are therefore not particularly confining, i.e. typically 2b h3 102.
The dramatic developments that have taken place during the last decade in the new field of quasar absorption lines - which can be regarded as the "high resolution" refinement of the classical Gunn-Peterson test - call for the above considerations to be revisited. The discovery of the dense "Lyman forest" of weak Ly absorption lines seen in the spectra of all quasars has revealed the existence of a highly clumped component of the IGM consisting of an abundant and evolving population of intergalactic clouds of possibly primordial gas. Surveys of the more massive metal-containing absorption systems associated with galaxies have also led to new understandings concerning the transparency of the universe in the ultraviolet out to large redshift.
2.1 Quasar Absorption Lines in Brief
In the following, a few key results of quasar absorption line studies of particular relevance for the topic of the diffuse ultraviolet background are briefly highlighted. A series of excellent and more comprehensive reviews that do the far-reaching topic far better justice can be found in the compilation of Blades, Turnshek & Norman (1988).
If classified according to their HI column density, there are two classes of intervening quasar absorption line systems: the numerous "Lyman forest" systems, whose column densities fall in the range NHI 1013 - 1017 cm-2 and the scarcer, but denser, "Lyman limit" systems having column densities NHI 1017 - 1022 cm-2. Lyman limit systems nearly always show matching absorption from heavy elements, and are therefore thought to be associated with the gaseous halos of galaxies. The Lyman forest systems, on the other hand, show little or no evidence for heavy elements and are therefore believed to be due to intergalactic clouds of possibly primordial material.
The Lyman forest systems are extremely numerous and evolve rapidly with redshift. Their line-of-sight density evolution is usually parameterized in the form (Sargent et al. 1980; Murdoch et al. 1986; Hunstead 1988)
with A 10 and 2 - 3. Lyman limit absorbers are roughly ten times scarcer (A 1) and evolve less rapidly ( 1) than Lyman forest systems (Tytler 1982; Bechtold et al. 1984; Lanzetta 1988; Sargent, Steidel & Boksenberg 1989; Bahcall et al. 1993). The column density spectrum of both classes of absorber is approximately a power law, dP / dNH N-sH, with index s 1.2 - 1.6 (Tytler 1988). The fact that the detailed statistics of the various types of HI containing quasar absorption systems are now reasonably well known permits a re-assessment of the question of the overall transparency of the UV universe out to high redshift (Section 2.3). These results have important implications for the interpretation of the UV background.
Of special relevance for the topic at hand is that the Lyman forest clouds are believed to be kept highly photoionized by a metagalactic flux of ionizing radiation. One key piece of evidence for this is found in the so-called "proximity effect" (Carswell et al. 1982; Murdoch et al. 1986; Bajtlik, Duncan & Ostriker 1988; Lu, Wolfe & Turnshek 1991). As the emission redshift is approached, the Lyman forest absorbers in a given quasar show a gradual under-density of absorbers with respect to the global density given by equation (4). This effect is interpreted as being caused by the radiation field of the background quasar enhancing the total ionizing flux above the metagalactic level. Since the quasar flux can be estimated from the magnitude and spectrum of the quasar, an estimate of the background metagalactic ionizing background intensity can be derived from the measured contrast of the proximity effect. Through this technique one infers that the intergalactic ionizing background in the redshift range 1.7 z 3.8 is consistent with an I -, 0.5 power law with intensity at the Lyman limit of order IH 10-21 ergs s-1 cm-2 sr-1 Hz-1.
The origin of this ionizing flux is a topic of some debate (Bechtold et al. 1987; Miralda-Escudé & Ostriker 1990, 1992; Madau 1992). One obvious candidate is the integrated flux from quasars. However, only by pushing current uncertainties in our knowledge of the quasar luminosity function and evolution can the intensity required by the proximity effect barely be approached (Bajtlik et al. 1988; Lu et al. 1991; Meiksin & Madau 1993). Moreover, taking into account absorption from the Lyman forest and Lyman limit systems themselves (Section 2.3), worsens the discrepancy further by a factor of 4-5 (Figure 2). Several authors (Bechtold et al. 1987; Miralda-Escudé & Ostriker 1990, 1992; Songaila, Cowie & Lilly 1990) have suggested that an alternative candidate may be radiation from primordial galaxies. Regardless of its origin, this estimate of the ionizing flux at high redshift has important implications for the possible contributions to the extragalactic UV background from a photoionized IGM.
A final relevant constraint stemming indirectly from the quasar absorption line studies concerns the information on the properties any pervasive IGM gas surrounding the Lyman forest clouds that result from considerations of Lyman forest cloud confinement and survival (Ostriker & Ikeuchi 1983; Ikeuchi & Ostriker 1986). The classical Gunn-Peterson test limits, combined with constraints on cloud/IGM pressure and cloud lifetimes to evaporation, generally constrain the ambient IGM to have at density corresponding to b 10-1 and a temperature T 105 K. These constraints also translate into limits on possible IGM emission contributions to the UV background.
Figure 2. Predicted integrated ionizing background due to quasars for several quasar evolution models. The upper dashed curves show the integrated background intensity at the Lyman limit as a function of redshift according to the model of Bajtlik, Duncan & Ostriker (1988). The lower full curves show the same fluxes attenuated by the accumulated Lyman continuum absorption from the Lyman forest and Lyman limit classes of quasar absorption line systems (see Section 2.4). The background level required to explain the proximity effect is also indicated.
2.2 Redshifted Lyman Alpha Emission from the IGM Revisited
Diffuse emission in Ly 1216Å from the IGM at redshifts 0 z 0.7 will give rise to diffuse background radiation at far-UV wavelengths 1200 - 2000 Å. Two emission processes could produce such diffuse Ly radiation: radiative recombination in case of a photoionized IGM, and collisional excitation in the case of a shock-heated IGM.
As mentioned in Section 2.1, the Lyman forest clouds and the smooth ambient IGM are believed to be held highly photoionized by a metagalactic ionizing radiation field, presumably due to the integrated light of quasars and primeval galaxies. What is the possible contribution to the UV background from redshifted Ly recombination emission from this gas?
Consider first the case of recombination radiation from a smooth photoionized IGM. The line emissivity to be inserted into equation (1) is (Osterbrock 1989)
where l 2 x 10-13 cm3 s-1 is the effective Ly recombination coefficient, and nHII and ne are the proton and electron density in the gas. With an I - ionizing background of average intensity IH(z) at the Lyman limit, the equation for hydrogen photoionization equilibrium is
where H2 3 x 10-13 cm3 s-1 is the total recombination coefficient and 0H = 6.3 x 10-18 cm2 is the HI cross section at the Lyman limit. Combining equations (5) and (6) with the expression (3) for nHI from the Gunn-Peterson test and inserting into equation (1) leads to the following expression for the intensity of the resulting redshift-smeared Ly background due to recombination emission
In this expression, the last factor in brackets is simply the intensity of the ionizing flux, IH(z) converted to suitable units and redshifted to zero redshift; that is, the background intensity that would be seen today if the ionizing flux propagated freely from redshift z. Since the product of the remaining factors in equation (7) is of order unity or less, this equation is merely reminding us that since l / H 0.7 Ly photons are emitted per photoionization event, the intensity of the resulting redshift-smeared Ly recombination background can never be greater than that of the ionizing input flux itself. This principle of photon conservation, of course, applies equally well in the clumped case. The equivalent expression for the redshift-smeared recombination emission from the Ly forest clouds can be obtained through the substitution
Note that since equation (6) implicitly assumes optically thin conditions, equations (7) and (8) are as they stand only valid in that limit. However, the fact that the redshifted recombination line emission is bounded by the intensity of the ionizing input flux is inherent to the nature of the recombination process and true regardless of the amount and detailed spatial distribution of the matter being photoionized and the recombination line in question. In other words, since the photoionized intergalactic gas is simply acting as a simple photon down-converter, the task of estimating the possible contribution to the UV background from line emission from photoionized intergalactic matter boils down to the task of estimating the intensity of the ionizing input flux at high redshift, IH(z).
The proximity effect displayed by the Ly forest implies that IH 10-21 ergs s-1 cm-2 sr-1 Hz-1 within the 1.7 z 3.8 redshift range that can be probed with ground-based telescopes. Judging from Figure 2, one anticipates an even lower ionizing background of intensity IH(z) 10-22 ergs s-1 cm-2 sr-1 Hz-1 at the redshifts z 1 of interest here. This expectation borne out observationally by recent observations of the proximity effect at low redshift carried out with HST (Kulkarni & Fall 1993) and by limits on the local extragalactic ionizing flux derived from H observations of high-latitude and extragalactic HI clouds (Reynolds et al. 1986; Songaila, Bryant & Cowie 1989; Kutyrev & Reynolds 1989). This ionizing flux, if redshifted from z 0.5, corresponds to an equivalent far-UV background of I 0.4 photons s-1 cm-2 sr-1 Å-1 - an intensity an order of magnitude below the observational limits on a possible extragalactic contribution. It follows that redshifted Ly recombination emission from the Lyman forest clouds or a smooth photoionized IGM is not likely to be a significant contributor to the far-UV background.
A similar conclusion can be reached in the alternative scenario of Ly emission from a shock-heated IGM. In the case of collisionally excited emission from a smooth IGM component, the line emissivity can be written
where l(T) is the Ly collisional excitation rate, and nHI and ne are the neutral hydrogen and electron densities in the IGM. The HI density is constrained observationally by the Gunn-Peterson limit, equation (3), while the electron density is constrained by the total baryonic density of the nearly fully ionized IGM. Combining, equations (1), (3) and (9), and introducing ne(z) = n0e(1 + z)3 where n0e 1.2n0H = 9.3 x 10-6 b h2 cm-3 is the present epoch electron density, one obtains the following expression for the redshift-smeared Ly background
For an ambient IGM temperature of T 105 K as inferred from considerations of quasar absorption line cloud survival, the Ly collisional excitation rate is l 4 x 10-9 cm3 s-1. This value, together with the Gunn-Peterson limit of (z) < 0.1, yields a predicted smeared Ly intensity of I 4 x 10-2 b h2 photons s-1 cm-2 sr-1 Å-1 in the far-UV. It follows that collisionally excited Ly emission from a smooth IGM component at 0 z 0.7 is a negligible contributor to the far-UV background for any reasonable value of b h2. The reason for this firm conclusion is simply that it takes intergalactic HI atoms to produce collisionally excited Ly emission, and nHI(z) is severely constrained by the Gunn-Peterson test.
Similar conclusions are reached concerning collisionally excited emission from the Lyman forest clouds. The equivalent expression to equation (10) for the redshift-smeared Ly background in the clumped case can be obtained through the substitutions (8) and ne(z) -> ne. This leads to
where ne is now the in situ electron density in the Lyman forest clouds. Based on studies of correlated Lyman forest absorption in quasar pairs combined with considerations of reasonable ionization levels (Carswell 1988; Sargent 1988), the gas density in Lyman forest absorbers is believed to be of order ne 10-3 cm-3. The observed line widths of the Lyman forest systems limit their temperatures to T 6 x 104 K, in which case l 3 x 10-10 cm3 s-1. With these numbers and the values E[dn / dz(z)] 30 and < NHI > 1015 cm-2, appropriate to the Lyman forest at z 0.3, equation (11) yields I 0.3 photons s-1 cm-2 sr-1 Å-1. Again, we conclude that collisionally excited Ly emission from the Lyman forest cannot a significant contributor to the far-UV background.
2.3 Redshifted HeII 304 Å Emission from the IGM Revisited
As illustrated in Figure 1, the second most important emission line from a luke warm photoionized or shock-heated intergalactic primordial plasma is HeII Ly emission at 304Å. The discussion of the possible far-UV background contribution due to this source from very high redshifts (3 z 5) is slightly more complicated with respect to that of HI Ly for several reasons. For one very little is currently known the intergalactic abundance of the HeII ion, since the HeII 304Å equivalent of the Gunn-Peterson test has yet to be carried out in the far-UV with HST or Lyman/FUSE (cf. Jakobsen et al. 1993). The expectation is that if the Lyman forest clouds and an ambient IGM are indeed photoionized by an I -0.5 power law, then the HeII ion should be an order of magnitude more abundant than HI, in which case the HeII Gunn-Peterson effect should be extremely strong (and HeI absorption very weak). A second important difference with respect to HI Ly is that HeII 304Å emission falls below the photoionization edge of neutral hydrogen and is therefore subject to absorption by intergalactic HI in the Lyman forest and especially the Lyman limit classes of quasar absorption systems. This last topic is addressed in detail in the following section. In spite of these complications, it is still possible to draw several reasonably firm conclusions concerning the possible contribution of HeII 304Å emission to the far-UV background.
In the previous section, the intensity of redshifted far-UV Ly recombination radiation from photoionized intergalactic gas was constrained on the basis of estimates on the metagalactic ionizing background derived from the proximity effect and more local observations of diffuse H emission from high-latitude and extragalactic HI clouds. The fundamental constraint expressed by equation (7), namely that the redshifted line background from a photoionized IGM can never be greater than that of the input ionizing flux, obviously applies equally well in the case of redshifted recombination HeII 304Å emission. In particular, with a flat I -0.5 spectrum for the metagalactic ionizing background, the intensity of the ionizing flux at the HeII ionization edge at 228Å is of the same order of magnitude as the flux at the Lyman limit at 912Å of IH(z) 10-21 ergs s-1 cm-2 sr-1 Hz-1 derived from the proximity effect at 1.7 z 3.8. This intensity, if redshifted from z 4 will give rise to a far-UV background of intensity I 1 photons s-1 cm-2 sr-1 Å-1. Since this background limit falls far below the observational limits - even without including the effects of intervening Lyman continuum absorption - redshifted recombination HeII 304Å emission from photoionized intergalactic gas can also be ruled out as a significant source of far-UV background radiation.
Collisionally excited HeII 304Å from a shock-heated IGM on the other hand is not quite as easily dismissed. Since the amount of HeII present in intergalactic space has not yet been measured by observations of the HeII version of the Gunn-Peterson test and the anticipated "helium forest" matching that seen in Ly, the HeII equivalents of equations (10) and (11) cannot be used to bracket the possible background contribution from this source. Instead, we are forced back to the more theoretical predictions described by equation (2). As shown in Figure 1, the net HeII 304Å emissivity per HI atom peaks at l 1.2 x 10-12 cm3 s-1 at a temperature T 8 x 104 K. Inserting this maximum emissivity into equations (1) and (2) yields a predicted far-UV background intensity at 1600Å of I 400 2b h3 photons s-1 cm-2 sr-1 Å-1. Hence depending on the values of b and H0, redshifted collisionally excited HeII 304Å radiation from z 3-5 could in principle yield a significant far-UV background flux. On the other hand, considerations of the survival of quasar absorption line clouds and primordial nucleosynthesis both point toward b 0.1, in which case the HeII line flux is insignificant. In any event, as discussed in the following section, the census of absorbing HI gas present in the Universe represented by the statistics of quasar absorption lines implies that the far-UV Universe is opaque in the Lyman continuum out to high redshift. This absorption will attenuate any diffuse HeII 304Å radiation emitted at z 3 by about two orders of magnitude, thereby reducing even the most optimistic HeII background flux to an unobservable level.
2.4 Accumulated Lyman Continuum Opacity of the Universe
Any contribution to the far-UV background at observed wavelength 0 emitted originally at a wavelength below the Lyman limit, H = 912Å, at a redshift ze 0 / H - 1 will be subject to photoelectric absorption by neutral hydrogen encountered along at least part of its path. Although it has been known for some time that the classical Gunn-Peterson test demonstrates that the Lyman continuum opacity of any smoothly distributed IGM is negligible, it has only recently been fully appreciated that the statistics of quasar absorption lines imply the that the accumulated absorption out to moderate and high redshift from the clumped component is quite substantial.
The character and magnitude of the accumulated Lyman continuum absorption from the Lyman forest and Lyman limit classes of quasar absorption lines has been discussed in detail by Møller & Jakobsen (1990). The general expression for the average transmission through a clumpy medium experienced by a photon emitted at wavelength, ze, and received at wavelength, 0, is (Paresce, Bowyer & McKee 1980)
where < qc(0, z) > = < exp(-NH H (0 / (1 + z))) > is the average individual cloud transmission and H() is the HI photoelectric cross section given by
where 0H = 6.3 x 10-18 cm2 is the photoionization cross section of neutral hydrogen at the Lyman limit.
As outlined in Section 2.1, the statistics of quasar absorption systems are today sufficiently well known to permit a reasonably accurate evaluation of equation (12). Figure 3 shows, as a function of wavelength, the resulting accumulated average residual transmission out to various redshifts from the combined total absorption due to the Lyman forest and Lyman limit systems. The characteristic "Lyman valley" shape of the accumulated absorption spectrum is caused by the interplay between the 3 dependence of the HI photoelectric cross section and redshift evolution and pathlength effects.
The main point to be read from Figure 3 is that the anticipated net absorption out to high redshift in the ultraviolet is rather high. As a specific example, the lower panel of Figure 4 shows the average residual absorption at received wavelength 0 1600Å as a function of emission redshift. From this figure it is seen that any HeII 304Å radiation emitted at ze 4.3 will be attenuated by a factor of order 10-2. This high opacity effectively implies that even if the IGM did go through a phase of intense HeII emission during re-heating, the resulting far-UV background radiation will in all likelihood remain forever hidden from our view.
It is important to stress that the accumulated intergalactic absorption given in Figures 3 and 4 refers to the average transmission. Since the dominant contributor to the opacity is the scarcer but optically thick Lyman limit systems, large fluctuations
Figure 3. Average opacity of the UV universe out to high redshifts as a function of wavelength. The shown "Lyman valley" absorption spectra include the accumulated Lyman continuum opacity from both the Lyman forest and Lyman limit classes of quasar absorption systems. The same absorber parameters as used by Møller & Jakobsen (1990) are assumed.
around this average are expected along any given line of sight (i.e. in any given quasar spectrum). The magnitude of the fluctuations can be calculated from the expression for the second moment of the accumulated transmission
The upper panel of Figure 4 shows the predicted relative transmission fluctuations, ( q / q) = (E[q2] - E[q]2)1/2 / E[q], as a function of ze, again for 0 = 1600Å. In the example of HeII Ly emission from ze 4.3 quoted above, the 1 level fluctuations amount to a factor 5. In other words, a characteristic signature of any contributor to the far-UV background originating at high redshift should be an extremely patchy background component.
Figure 4. Lower frame: Average transmission of the Universe at received wavelength 1600Å as a function of emission redshift. Upper frame: Predicted relative fluctuations around the average transmission for an infinitely narrow pencil beam due to fluctuations in the composition and location of the absorbing HI clouds along the line of sight.
This leads to the interesting prospect of detecting or constraining a high-z component to the far-UV background - regardless of its origin - through measurements of background intensity fluctuations. In fact, from the observations of Martin and Bowyer (1989), it is known that the far-UV background at 0 1600Å is very smooth on angular scales of = 8' and larger: ( I / I) 6%. If it is assumed that the background consists of the sum of a smooth local component and an attenuated distant high redshift component of average intensity E[q(z)] Iz, the fractional contribution to the total average emission from the distant component, z, can be estimated from the observed dilution of the opacity fluctuations
Taken at face value, the Martin and Bowyer (1989) fluctuation limit implies that less than z 6% / 5 1% of the far-UV background can originate from z 4.3. However, a slightly subtle point has been overlooked; namely that the ( q / q) values in Figure 4 refer statistically to the absorption sampled along an infinitely narrow pencil beam (namely the line of sight to a quasar), whereas the UV background observations have been obtained with a finite = 8' beam size. Since the opacity is dominated by the Lyman limit absorbers, which are assumed to be associated with galaxy halos of, say, D 50 kpc size, the absorption-generated background fluctuations on the sky are expected to be correlated only on very small scales of order c cD / H0 10". Such small scale fluctuations would only appear in the finite beam measurements of Martin and Bowyer (1989) diluted by a factor of order / c 50. The upper limit on z derived above therefore has to be relaxed by the same amount to z 50%. Given the severe practical problems of measuring UV background fluctuations on such small scales as 10", it is unlikely that this constraint will be tightened much further in the foreseeable future. On the other hand, the above considerations do serve to demonstrate that, because of the very small size of the anticipated absorption coherence patch on the sky, use of the average attenuation given by equation (12) is well justified when dealing with the radiative transfer of diffuse background light.
1 Throughout this paper I refers to the specific intensity of the background expressed in the units preferred by the observers; namely photons s-1 cm-2 sr-1 Å-1. Back.