EL SECADO IMPORTANTE EN EL PROCESAMIENTO DEL JENGIBRE

As already written above, the Fe is the element with the highest product of fluorescence yield and abundance in the ISM; these quantities are, respectively, 34% and 3.3× 10−5

(Bambynek et al., 1972). The energy and depth of the K-shell absorption edge, the energy of the fluorescent photon which results from the inner shell ionisation and the fluorescence yield all depend on the ionisation state of the element; for the case of Fe, the absorp- tion edge for the neutral (or low ionisation) Fe (FeI) is at an energy EK=7.1 keV, and

rises up to EK=9.3 keV for the H-like Fe ion (FeXXVI) (i.e. Morita & Fujita, 1983). The

cross section for photoelectric absorption of Fe decreases from σK,F eI=3.8×10−20cm2 to

σK,F eXXV I=3.3×10−20cm2 (the quoted values are per Fe atom). The energy of the fluores-

cent line also increases from 6.4 keV of the FeI to the 6.96 keV of the FeXXVI; this is clear if we think that the only electron in an H-like Fe ion is more bound to the nucleus because it is no more shielded by the other negative charge carried by the (now missing) 25 elec- trons. Moreover, also the fluorescence yield depends on the ionisation state of the atom; for Fe, it increases form the 34% of FeI to the 49% of FeII, and then is highly variable for the highest ionisation states, ranging values between 11% and 75% in the four final stages (FeXXIII-FeXXVI) (Bambynek et al., 1972). The branching ratio of the two components (6.404:6.391) of the Fe-Kα doublet is 2:1 (Bambynek et al., 1972); the natural width of the lines is ∼3.5 eV, and both this width and the energy difference between the two peaks are negligible for the work we will perform, since the current operating instrumentation is not able to spectrally resolve the doublet. Whereas the Fe-Kα line is produced by a 2p→1s transition, the 3p→1s could also occur; this produces the Fe-Kβ line whose energy

is 7.06 keV and its emission can be calculated to theoretically occur at ∼11% that of the Fe-Kα one (Kikoin et al., 1976). These factors are important because in all the spectra of the MCs studied in our work the signal to noise ratio is not high enough to have a good detection/statistics of the Fe-Kβ line; as a consequence, we were forced to fix both the energy of the line peak and its intensity compared with the much better detected Fe-Kα.

Here we want to briefly present the results of the simulations performed by George & Fabian (1991), since they are important for understanding the phenomenology of the X-ray illumination of cold and dense neutral material. These results, together with giving an exhaustive view of the theory and phenomenology of the physical processes involved in the photoionisation, will be important in the near future when X-ray calorimeters will allow us to achieve a better spectral resolution in the studies of the Fe-Kαline from cosmic sources. George & Fabian (1991) simulated the response of a homogeneous semi-infinite slab to X-ray illumination by an X-ray source with a power law spectrum. Moreover, the Fe line has been considered to be a single line peaked at 6.4 keV; the physical processes considered for the study of the opacity of the slab for both continuum and line radiation are photoelectric absorption and electron scattering. The number NK of fluorescent photons

per unit solid angle which could escape from the interacting region is a function of the source spectrum N0E0, the Fe abundance ZF e, the incident angle θ0 and the final direction

of the photon (θf,φf). NK can be written as (George & Fabian, 1991)

NK(θf, φf, Ef)dΩf = Z _∞

EK

N0(E0)PF e(E0)YF ePesc(θf, φf, Ef), (2.4)

where PF e ∝(σF e/σtot) is the absorption probability of the incident photon, and σtot

is the sum of absorption and scattering cross sections, i.e. σtot=σabs +σes. In the above

formula, YF e is the fluorescence yield for the Fe-Kα line production and Pesc is the escape

probability (into the solid angle dΩf) of the fluorescent photon, with an energy Ef ∼6.4

keV. PF e strongly varies as a function of the energy of the incident photon E0, since the

photoelectric cross section varies as E−3, whereas the scattering cross section is nearly constant up to about 100 keV. The effective fluorescence yield Yef f of the slab can be

written as the ratio between NK and the number of incident continuum photons which can

effectively produce inner shell ionisation of Fe, i.e. EK ≤E0 ≤30 keV; in formula (George

& Fabian, 1991) Yef f(θf, φf, Ef)dΩf = N_RK(θf, φf, Ef)dΩf ∞ EK N0(E0)dE0 (2.5) The effective fluorescence yield slightly increases towards higher values of the spec- tral index Γ of the incident spectrum, since a softer spectrum has more photons with energy close to the EK value (7.1 keV for Fe). Substituting equation 2.4 into 2.5 it is

possible to calculate Yef f integrating over E0; as a result, this quantity lies in the range

10−3 .Yef f .10−1 sr−1 (George & Fabian, 1991). In the left panel of Fig.2.3 we show the

results of the simulations for the value of Yef f averaged over all the possible final direc-

2.2 Photoionisation: theory and phenomenology 47

Figure 2.3: Left: the effective fluorescence yield of a neutral slab averaged over the solid angle Ωf=2π, plotted against the incident angle of the ionising photon θ0. The different

curves represent photons which have been scattered zero, one and two times. Middle: fraction of the fluorescent photons produced which are able to escape form the slab for unit depth plotted as a function of the Thomson depth for electron scattering in the slab for different incident angle θ0=0, 60 and 85 degrees. Right: the G(E0,θ0) function as a

function of the energy of the incident photon for different values of the angle θ0=15 and

75 degrees (all the plots are from George & Fabian, 1991).

spectrum has been assumed to be a power law with a spectral index Γ=1.7. The different curves display the effect of Compton scattering (the photon collides with a free electron and transmits some of its energy to the ’rest’ particle) on the fluorescent photons, and particularly the ones which suffered no, one and two Compton scatterings on the ambient medium.

In the plot we can see that Yef f slightly increases towards higher values of θ0 for the

unscattered Fe-Kα photons, with its value rising from 1.8×10−3 sr−1 at θ0=0 (direction

normal to the surface of the slab) up to 5.5×10−3 _sr−1 _{at θ}

0=90 deg (George & Fabian,

1991). This behaviour is a consequence of the Pesc dependence on θ0; for θ0 values close to

90 degrees the fluorescent photons are produced at a smaller vertical depth within the slab, and therefore have more chances to escape. This behaviour can be seen in the middle panel of Fig.2.3, which shows the number of fluorescent photons per unit Thomson depth which escape the slab as a function of the vertical Thomson depth crossed by these photons; given a certain vertical depth, the fraction of fluorescent photons that can escape is lower towards higher θ0. Following George & Fabian (1991), it is possible to define the function

G(E0,θ0) as the ratio between the number of photons incident on the slab with an angleθ0

and energy E0 which produce 6.4-keV photons which are then able to leave the absorption

region, and the total number of incident photons with energy E0; this function, which

represents the probability of photons at different energies to induce Fe fluorescence which can be observed (the photon leaves the slab), is shown in the right panel of Fig.2.3, plotted for different values of the incident angleθ0. Definingµ=cosθ0, we write (George & Fabian,

1991) G(E0, θ0) = g(EK, θ0)×f(), (2.6) where g(EK, θ0) = 10−2 ×(6.5−5.6µ0+ 2.2µ20), (2.7) and f(E) = 7.4×10−2+ 2.5 exp−(E−1.8)/5.7 (2.8) Looking at the plot, we can see that there is a region next to the the threshold energy (7.1 keV) where G is quite hard (G∝E−1.2_{), whereas for E&10keV the G function becomes}

steeper, being proportional to E−2_{. At higher energies, above} ∼_{50 keV, the G function}

gets flatter because of the Compton down-scattering of high energy photons (photons lose energy), which therefore have a lower energy and increase their probability of being able to photoionise Fe atoms (George & Fabian, 1991).

As shown above, the number of fluorescent photons created (and hence Yef f) also varies

with the final direction of the emitted photon. The fluorescent emission is isotropic (it emits equally in all directions), so that Yef f does not depend on φf for a certain θf. On the

other hand, the fluorescence yield of a semi-infinite absorbing slab strongly depends on the escaping angle of the fluorescent photon. The left panel of Fig.2.4 shows the φef f averaged

value of the slab yield Yef f plotted against the emerging angle (θef f) of the 6.4 keV photon

for different incident angles (θ0) of ionising radiation; the first noticeable effect is the sharp

decrease of the fluorescence yield towards high values of θf; this behaviour reflects the

higher optical depth that the fluorescent photon has to cross in order to leave the absorption region (George & Fabian, 1991). Moreover, the efficiency of Fe-Kα production increases with increasing incident angle of the incoming photons. A further interesting feature of the Fe-Kα line is related to its shape; fluorescent photons which are created inside the slab can suffer Compton scattering with the ambient electrons and be significantly down-scattered; depending on how many times a certain photon is scattered, the final recorded energy associated to this fluorescent radiation will be lower than the unscattered 6.4 keV. As a consequence, a red wing is created in the spectrum, i.e. some line flux is distributed over energies lower than 6.4 keV (see Fig.2.4, right panel); the original 6.4 keV line spreads over two Compton wavelengths, suffering an energy down shifting of ∆E=2E2/m2c2 after each

scattering (George & Fabian, 1991). The line profile which emerges after these additional electron-photon interactions consists of an unscattered core, which represents the bulk of the emission, and a series of red (lower energy) shoulders; most of the contribution to the first shoulder comes from the photons which undergo one scattering before escaping the interaction region, whereas the second shoulder is dominated by the ones which undergo two scatterings and so on. We also note that the exact shape of this red wing depends on the abundance of elements lighter than Fe, which dominate the photoabsorption cross section at 6.4 keV (George & Fabian, 1991).

2.2 Photoionisation: theory and phenomenology 49

Figure 2.4: Left: directionality of the Fe fluorescent line. Here we plot the averaged (over

φf) fluorescent yield as a function of the final direction of the photon for different values

of the incident angle θ0=0, 60 and 85 degrees. Right: line profile of the Fe-Kα fluorescent

line at 6.4 keV for different illumination angles. The low energy wing/s are created by the scattering of the monoenergetic 6.4 keV photons produced by fluorescence within the slab (all the plots are from George & Fabian, 1991).

So far we have only discussed the 6.4 keV line production and the physics of its inter- actions with the neutral medium. What happens to the X-ray illuminating continuum? To answer these questions, we write the expression for the albedo A of the slab as the ratio between the number of continuum photons Nc with a certain energy Ef which after

interaction with the slab escape in a solid angle dΩf in the direction given by the angles

θf and φf, and the number of incident photons with energy Ef. In formula (George &

Fabian, 1991)

A(θf, φf, Ef)dΩf =

Nc(θf, φf, Ef)dΩf

N0Ef

, (2.9)

where Nc also accounts for the contribution by those photons with E≥Ef which suffered

downscattering to Ef within the slab. This albedo, the probability of a continuum photon

to survive the passage through the slab, is strongly related to the probability to undergo electron scattering rather than being absorbed, i.e. the relative strenght of these two cross sections. To summarise the results by George & Fabian (1991), the function A averaged over all the possible escaping directions (θf and φf) (1) increases with Ef for a given θ0,

(2) it increases with θ0 for an assumed Ef and (3) the fractional increase of A between

θ0=0 and θ0=90 depends on Ef. The first point is clear since towards the higher energies

the photoelectric cross section decreases sharply as E−3, and therefore Compton scattering dominates over absorption. If a continuum photon enters the slab with an higherθ0 (more

probability of being absorbed (second point). The third point is also a consequence of the decreasing of σabs with increasing photon energy.