Modelling the Lyman Beta Dayglow in the Jovian Atmosphere
Mathieu Barthélemy1, Chris Parkinson2, Jean Lilensten1, Renée Prangé3
1
Laboratoire de Planétologie de Grenoble. CNRS-UJF, France.
2
MS 150-21, Caltech,Pasadena, California, USA, 91125; MS 183-501, JPL, Pasadena,
California, USA, 9110.
3
Observatoire de Meudon. France.
Abstract
The Lyman Beta dayglow of Jupiter is an important source of information about its
atmosphere. However, it is affected by a coupling between the H Lyman beta and the H2 line
of the Lyman system. In this paper, we evaluate the effect of this overlapping via radiative
transfer modelling and show its effect both on the integrated intensity and the line profile. The
intensity increase around 1025 Å at the centre of the disc but tends to slightly decrease at the
limb as well as creating an asymmetry on the profile. This asymmetry is also present in the
Lyman alpha jovian Bulge region.
Introduction
The problem of the Lyman dayglow for the giant planets is very important to obtain
information about the atmosphere. Several previous studies have been done on the Lyman
alpha dayglow (Wallace et Hunten, 1973; Yung et Stroebel, 1980, Shemansky, 1985; Ben
Jaffel et al, 1988; Emerich et al, 1993; Griffioen, 2000), but the Lyman beta problem has been
virtually unstudied for Jupiter. One reason is that the H Lyman lines are close to some lines of
the H2 Lyman system (Electronic transition between the ground state X1g+ and the exited
state B1g+) : the wavelength gap with the H2(6-0 P(1)) is +0.21Å. This coincidence cannot be
neglected in the atmosphere of the giant planets because of the presence of both molecular
and atomic hydrogen in comparable quantities.
Until now, the Lyman beta problem has only been modelled as a single independent line for
Jupiter (Gladstone 1988) through a Feautrier technique. This study focuses on two atomic
oxygen lines for the Earth and the Lyman alpha and beta lines for Jupiter. The author predicts
an intensity of 10 R for the Lyman beta line at the centre of the disk with a subsolar angle of
20°. The centre to limb variation shows a very strong peak at the limb due to the very long
optical depth for this situation.
There have been other approaches to this problem. For instance, Feldman et al (1993) studied
the HUT (Hopkins University Telescope) data, but its resolution of about 3Å cannot give
direct information on the line profile. With a solar flux of 6.10+9 ph cm-2s-1 at 1AU, the
authors found a total emission of 307R at 1025 Å. The 30 R consisted of 10-12 R of the H2
Lyman (6-0) line and 18-20 R from Lyman beta.
Additionally, Liu and Dalgarno (1996) addressed the problem of the Lyman beta line with the
H2 coupling. They calculated the fluorescence on other molecular hydrogen lines due to the
pumping of Lyman beta and found that 59% of the total solar Lyman beta line is pumped,
giving fluorescence of the H2 lines. They obtain 17.5 R for the solar Lyman beta resonance
scattering and 10.4R for the (6-0 P(1)) H2 line. No information is given on the line profile.
In this study, we use a radiative transfer approach to address the computation of the Jovian H
Lyman beta emission taking into account the coupling with the molecular hydrogen lines.
This approach allows getting both the intensity and the line profile shape. We concentrate on
the dayside of the planet and avoid the auroral zones.
Model
Our method is based on the work by Gladstone (1983) and includes the additional lines
studied by Gladstone (1988) including Lyman beta. Vervack et al. (1995) used the same
method to study the He 584 Å line thus explaining the Voyager He 584 Å observations.
Griffioen (2000) and Parkinson (2002) extended this effort to include the case of several
overlapping lines in order to study the H-D case for Lyman alpha.
Amongst other advantages, this code has proven to be very fast a rapidly convergent lambda
operational method for solving resonance line scattering problems (Griffioen et al 1994). A
simple run (typically with 112 altitudes, more than 100 frequencies and 16 angles) takes about
4 minutes on a Intel Pentium 4 processor. This code also offers the possibility to keep the
information on the emission for each altitude and carefully considers the sphericity of the
planet for precise limb computations.
We have included updated neutral atmosphere profiles, spectroscopic and the solar flux
parameters.
Neutral atmosphere
The Jovian neutral atmospheric profiles of H, H2 and CH4 come from Parkinson (2002). The
eddy diffusion coefficient Kh has been calculated from the He (584 Å) airglow analysis of
Vervack et al (1995) and is 2. 106 cm2 s-1 at the homopause. The present case corresponds to
solar minimum conditions and is shown in figure 1
Figure 1 : Jovian neutral atmosphere
At low altitudes (between 500 km and 2000 km above the 1 bar level), molecular hydrogen is
the main species. At higher altitudes, the atomic hydrogen becomes more abundant because of
the H2 photodissociation. Finally, CH4 cannot be neglected above the  = 1 level at
approximately 500 km. (chris i have a question. Is it tau only for CH4 or global optical
depth? I think tau is much greater than 1 at around 500 km if we consider global optical
depth)
The exospheric temperature is obtained from the Galileo probe measurements (Seiff et al.,
1997) and is ~1100 K.
Solar flux
The solar input flux is modelled by a double gaussian profile so we may compare with
Gladstone results (1988).This kind of profiles have been suggested by Lemaire (1978) :
Error!
1
where x1 is the offset for each peak, and x0 is the width for each gaussian function.
The wavelength at the centre of the line for Lyman beta is 1025.72Å. In quiet solar
conditions, F is equal to 4.0x109ph cm-2s-1 at 1A.U. Considering Jupiter is at 5.2A.U., the
flux at the top of the atmosphere is 1.48x108ph cm-2s-1.
For Lyman beta we take x0=12.5 sdu (doppler unit) and x1=14.6 sdu (Gladstone 1988). For
this line 1 sdu=9.8 mÅ at a reference temperature of 500 K.
Spectroscopic parameters
We take the parameters of Gladstone(1988) for the atomic hydrogen Lyman beta line and
the parameters for the molecular hydrogen line are from Abgrall et al. (1993). However, we
must calculate the single scattering albedo  given by  =
Aij
kAkj
where i is the lower level, j the upper level, and k is all the levels with allowed transitions
starting from j. We consider rovibronic transitions, so we take into account all the vibrationnal
(v) and rotational (J) levels. The selection rules for the rovibronic transition are J=1.
J=0 is a forbidden transition, because for both electronic states of the Lyman system, the
electronic quantum number is =0.
There is no rule for vibronic transitions.
The spectroscopic parameters for H Lyman  and H2 P(1) 6-0 are given in Table 1.
Line
Wavelength
g ground
g upper
f
A(s-1)

H Lyman 
1025.72Å
2
6
0.0791
1.672x108
0.8816
H2 P(1) 6-0
1025.93 Å
3x3
1
0.912x10-02
1.7x108
0.13566
Table 1 : spectroscopic parameters for H Lyman  and H2 P(1) 6-0 lines
We can now compute the H2 populations in each vibrational and rotational level starting from
the total density No, considering 15 rotational levels for each vibrational level. We only
consider the two first vibrational levels since the higher vibrational levels are so high that
their populations are negligible. The energies of the levels come from Abgrall (1993) and
their populations are given by:
(2J+1)gsexp(-EvJ/kT)
N(vJ,z)=N0(z)
vJ(2J+1)gsexp(-EvJ/kT)
2
where:
EvJ is the energy of the level with vibrationnal number v and rotational level J.
gs is the spin degeneracy. It is equal to 3 for odd rotational level and to 1 for the others.
To take into account the frequency redistribution, we consider the angle averaged partial
redistribution function r(z, ,x,’,x’) given by Hummer (1962) and Mihalas (1978).
We assume that CH4 is independent of the frequency and its value is 3.2 10-17cm² at 1025 Å
(Berkowitz, 1979)
Results
We first validate our results by studying the uncoupled case and obtain a Lyman beta intensity
of 20.63R at the centre of the disk. This is slightly larger than previous studies: 18 to 20 R
measured by the HUT telescope by Feldman et al. (1993) and 17.5 by Liu and Dalgarno
(1996) with a different modelling method. At the limb, we obtain an integrated intensity of
49.6 R. This can only be compared to the Gladstone (1988) value of 25 R as there is no data
available. The large discrepancy between our result and Gladstone (1988) is mainly due to the
atmospheric model: our H column density is about ten times larger.
We now consider the coupled line case. In the non-bulge region, we obtain a line integrated
intensity of 23 R at the centre of the disk with a solar zenith angle of 20°. This is 10.3% larger
than in the uncoupled case. At the limb, with a point of sight at 1.01 Jovian radius we obtain
48.1 R with an angle of 20° between the line of sight and the solar ray. This value is smaller
than in the uncoupled case by 3%.
Examining figure 2, we see this result is due to geometric considerations. At high altitude, the
molecular hydrogen in the atmosphere is excited by an intense solar Lyman beta line, that is
scattered in all directions, including toward the observer. There is little pumping by the H
Lyman Beta because the (6-0 P(1)) H2 of the Lyman system is located at the wing of the H
Lyman Beta line profile, but there is a strong self-absorption due to fluorescence. A part of
the H2 absorption in emitted in other wavelength and then lost for the coupling with H. This
effect is about the same regardless of the geometry.
Further down in the atmosphere, the molecular hydrogen becomes more important and
absorbs more solar photons. A photon emitted by H2 at low altitude will have a larger
probability to be absorbed by H or H2 before it can leave the atmosphere. After 4 scatters by
H2, the probability that a photon has disappeared is 0.99966.
Therefore, the larger the atmospheric slab, the larger the frequency redistribution of the
photons toward wavelengths outside the H Lyman Beta region.
Figure 2: Jupiter / Sun / observer geometry. The bold lines represent the path of the lines of
sight through the atmosphere at the centre of disc and at the limb
We now examine the effect of the H – H2 coupling on the line profile.
Figure 3 shows how the presence of H2 has a strong effect on the line profile especially at the
limb. We can see on the red side of the line the peak due to the H2 line. However, an
interesting feature is the asymmetry of the two wings of the line at the limb. The red side is
pumped by the absorption of the H2 molecule, which is due to the same geometrical factor as
previously mentioned. We can understand this phenomena because, at the limb, the line of
sight passes through a very large layer of atmosphere. Hence, the effects of the absorption are
very important for this case. We can identify a self-reversal at the limb around the central
peak as calculated by Gladstone (1988).
Figure 3: Line profiles. The left panel is for the centre of disc case. The right panel is at the
limb. The red lines are the uncoupled case, and the blue ones are the coupled case.
At the centre of the disk, the strong asymmetry is not present. We note that the red side of the
line is slightly more important than the blue side. Comparing with Gladstone (1988), we
cannot identify any self-reversal at the centre of the disk. Close to the H2 line we still see an
emission peak due to this line.
For each case, we consider the intensity and the shape of the centre of the line are not affected
by the overlapping. This is due to the partial redistribution used in our calculation, which
supposes low interaction between the centre of the line and the wings.
Integrated intensity [R ]
The centre to limb variation is shown in figure 4: . on the disk, the shapes as before but the
Point of sight in
jovian radius
uncoupled case has a lower intensity. At the limb the peak is quite similar for both the
uncoupled and coupled cases. Compares favourably to previous results (Gladstone, 1988).
Figure 4: centre to limb variation. The green line corresponds to the uncoupled case and the
blue line to the coupled case.
The intensity of the emission computed here is smaller than the measurements of Feldman et
al (1993) using the HUT telescope (Feldman 1993) or those predicted by Liu and Dalgarno
(1996) without radiative transfer. Both authors found an intensity of 10 to 12 R for the
contribution of the H2 lines, which were averaged over the entire disc. Our upper value is 2.4
at the centre of the disc.
This discrepancy can be attributed to the H2 self-absorption. In order to understand the role of
this self-absorption, we compare the previous calculation to a new calculation with a single
scattering albedo H2=1. In this case, we find an intensity of 90.7 R at the centre of the disk
and 131.5 R at the limb. The line profile shows a very strong peak due to the H2 line and the
Lyman  contribution is not changed. Hence, we can estimate that the effect of the selfabsorption caused by the H2 line on the total integrated intensity to be around 70R at the
centre of the disk and 84R at the limb. This contribution is less than that calculated by Liu and
Dalgarno (1996) and by Feldman (1993).
Another interesting effect takes place in the Jovian bulge region. This bulge was first
observed in 1978 by a sounding rocket (Clarke et al, 1980) and confirmed by the Voyager
observations in 1979 (Sandel et al, 1980), exhibiting a strong asymmetry in the Lyman alpha
emission centred on the magnetic drift equator latitude, and fixed in magnetic longitude
(system III) at III=100°. This bulge is thought to be due to strong jets in the upper
atmosphere, explaining a strong heating in this region (Sommeria et al 1995). Hence, the
temperature between 2000km and 2500km increases drastically (more than 6000K) (figure 5).
This high temperature implies changes of the absorption and scattering cross sections and of
the width of the line profiles.
Figure 5: temperature profile in the bulge region (Parkinson, 2001)
We have calculated the Lyman beta line profile in the bulge region (figure 6). Our simulation
shows a very strong self-absorption around the central peak due to this hot layer. This
absorption is so strong that for some frequencies the emergent intensity is equal to zero.
Contrary to the normal case this reversal is present at the centre of the disk for short optical
paths.
The intensity at the centre of the disk is equal to 22 R. It is less than in non-bulge region. It
reveals that we cannot found any increasing of the intensity in the bulge with Lyman beta.
This feature is due to the self-absorption of the line.
Figure 6: Lyman beta line profiles. The bulge profile is in bleu and the non-bulge profile is in
green.
This means that this layer increases the optical depth and the effect of the absorption for each
peak of the incident solar line. For the coupled case, in the bulge region, the effects of the
coupling between the two lines are still present but there is no significant difference with the
non-bulge region.
Summary
In this paper, we have studied the influence of the coupling between the H Lyman beta and
the H2 line of the Lyman system. We found that the effect due to this overlapping is
important.
There is an effect both on the line-integrated intensity and on the line profile. We found these
effects increases the intensity around 1025 Å at the centre of the disc but tends to slightly
decrease it at the limb. Also, the coupling of lines creates an asymmetry on the profile. In
future work we hope extend our comparison to measurements made by FUSE (Far Ultraviolet
Spectroscopic Explorer).
Appendix
The formalism of our problem is based on the 1-D radiative transfer equation considering the
frequency redistribution. The one-dimensional, azimuthally averaged, non linear integrodifferential equation of radiative transfer can be written:

dI(z,,x)
+ E(x,z)I(z,,x) = E(x,z)S(z,,x)
dz
3
where:
z  height
  cosine of the angle measured from vertical
x  frequency in Doppler unit from the line centre.
I(z,,x)  specific intensity (photons cm-2s-1sr-1D-1)
E(x,z)  total extinction per unit path length and is defined by:
E(x,z) =na(z)a + ns(z)s(z,x)
4
where “n” stands for densities in cm-3 and  stands for cross section ; the index “a” represents
absorption while the index “s” represents scattering.
Assuming that the solar/external source has been separated from the diffused radiation, the
source function given by:
Error!
Error!
where:
  single scattering albedo
0cosine of solar zenith angle
5
r(z;0,x,’,x’)frequency redistribution function
F(x)solar flux (photons cm-2s-1)

dz'E(z',x') is the vertical optical depth;
(z,x)=
z
V(z,x)
 total volume production rate of an isotropic internal source.
4
The scattering cross sections are computed from:
e2f (a(z),x)
s(z,x) = 0(z)(a(z),x) = m c
e
D (z)
6
f is the oscillator strength.
(a(z),x)  Voigt profile, i.e. the convolution of a Gaussian profile (from Doppler width) and
a Lorenztian profile (from natural line braodening).
+
2a(z) 
e-y2
(a(z),x) = 3  a(z)2 + (x-y)2 dy
 
-
7
where “a” is the ratio between the Doppler width D(z) and the natural line width.
This transfer equation includes for an isotropic scattering phase function with an arbitrary
frequency redistribution and both external and internal sources.
The classical solving procedure uses the differential equation approach (Cannon 85). The
present integral equation approach first calculates the source function, S(z;,x). The
conversion of equation 5 to an integral equation is desirable to facilitate the manipulation of
this problem to generate physical solutions satisfying the equation, and is done as follows:
rearrange equation 5 so as to leave S(z;,x) only on the right handside. Then we can
immediately write:
Error!
8
The upwelling radiation flux is defined by (Cannon ,1985):
Error!
Error!
9
The downwelling radiation flux is defined by
Error!Error!
10
where 0  <1. For the majority of cases I+(zmin;-,x) and I-(zmax;,x) will be set to zero (i.e.
zero diffuse incident radiation flux at the top of the atmosphere and no radiation flux
downwelling at the bottom of the atmosphere). Incorporating this assumption, integral
equations 9 and 10 are now substituted into 3. This results in an integral equation in S(z;,x)
only:
S ( z ,  , x) 
 ( z , x)ns ( z ) 
2 E ( z , x)

1
z max
0
z
  d r ( z ,  , x,  , x )
1
 dx s ( z, x)[  d r ( z,  , x,  , x)

dz 
0
z
 dz 
z min
E ( z , x )
S ( z ,  , x ).e [ ( z, x) ( z , x)] /  

E ( z , x)
S ( z ,  , x ).e [ ( z , x) ( z, x)] /   ]  S 0

where
 ( z , x)nts ( z ) 
 ( z , x)  ( z , x)
S 0 ( z ,  , x)  [
dx s ( z , x) R ( z , x,  , x,  0 )F ( x) exp(
)

4
0
4

  ( z , x)
V ( z , x)
] / E ( x, z )
4
11
The Feautrier technique has been described in detail by Mihalas (1978), Gladstone (1983),
Cannon (1985) and Griffoen (2000). The technique overcomes the exponential error
increasing with optical depth by transforming the radiative transfer equation into second order
integro-differential form. Expanding the two boundary conditions through the atmosphere, we
constraint these errors.
Our most difficult problem is that of the overlapping of different lines, which has to be
included in the formulation of this problem. To this purpose, we use a different formulation of
the source term (Griffoen 2000, Parkinson 2002).
Error!
+
1
H nH(z)

+
dx'H(z,x') 
d'rH(z,,x,',x')I(z,',x')
2
-1
-
+
1
dx'(H2 nH2(z)H2(z,x')rH2(z,,x,-0',x')+ H nH(z)H(z,x')rH(z,,x,-0',x'))
+
4 
-
.F(x')e
[-(z,x')/0]
}
12
where
E(z,x) = nH(z)H(z,x) + nCH4(z)CH4(z)+ nH2(z)H2(z,x)
and
 = NH(z)H(z,x) + NCH4(z)CH4(z)+ NH2(z)H2(z,x)
Ni is the column amount of species i. Note that we have taken into account the absorption by
the methane and since there is no resonant emission line of CH4, we include CH4 only in the
absorption term.
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