O N T H E D E R I V A T I O N O F E M P I R I C A L L I M I T S O N T H E

H E L I U M A B U N D A N C E I N C O R O N A L H O L E S B E L O W 1.5 Rs

SHADIA RIFAI HABBAL and RUTH ESSER Harvard-Smithsonian Center for Astrophysics

A b s t r a c t . We present a simple technique describing how limits on the helium abundance, a , the ratio of helium to proton number density, can be inferred from measurements of the electron density, t empera ture and their gradients below 1.5 ~ . As an illustration, we apply this technique to emission line intensities in the extreme ultraviolet, measured in polar coronal holes. The example indicates tha t a can be significantly large in the inner corona. This technique could be applicable to the more extensive da ta to be obtained from coordinated ground and space-based observations during the Ulysses south polar passage and the Spar tan flight, and subsequently during the SOHO mission. Limits on the helium abundance in the solar wind can thus be derived from its source region and compared to interplanetary values.

K e y words-" - Helium Abundance - Solar Wind

1. I n t r o d u c t i o n

The helium abundance, defined as the ratio of the helium to proton densities, as observed at 1 AU, is relatively constant in high speed solar wind streams, averaging around 4 to 5%. On the other hand, it is much more variable in low speed winds (see, e.g., Neugebauer and Snyder 1966). Studies of three fluid models of the solar wind show that observations both in the corona and at 1AU do not preclude the possibility that very large helium abundances, for example > 100%, could be present at the lower corona (see, e.g. Joselyn and Holzer 1978; Biirgi and Geiss 1986; and references therein). Furthermore, recent studies have shown that the presence of helium in the solar wind might reduce the sensitivity of the solar wind mass flux to the temperature at the coronal base (Biirgi 1992, Leer, Holzer and Shoub 1992).

So far, however, at tempts to derive constrMnts on the helium abundance in the inner corona from observations have not been successful (see, e.g., Parkinson and Gabriel 1986). We present in this paper a simple technique illustrating how such limits can be derived from knowledge of the electron temperature, density and their gradients in the lower corona.

2. T h e o r e t i c a l C o n c e p t

For a multispecies solar wind plasma composed of electrons, e, protons, p, and fully ionized helium, (or alpha particles) c~, the force balance equation in the very inner corona (below 1.5 R,) where the flow speeds are negligibly

Space Science Reviews 72: 39-44. �9 I995 KluwerAcademic Publishers. Printed in the Netherlands,

40 S.R. HABBAL AND R. ESSER

small, is given by

0 = dp GM, dr p r2 , (1)

where G is the solar constant, Ms the solar mass, and r the heliocentric distance. The mass density, p, is given by

p ~ r~p(~p + 4 n . ) , (2)

The total pressure, p, is given by

p + np + .)kB T , (3)

with the assumption that Te = Tp = Ta = T is valid very close to the coronal base where the coupling between the species is expected to be strong. Charge neutrality dictates that

n. = n v + 2n,~. (4)

Using the definition of the helium abundance, a, as the ratio of the helium ion to proton densities, i.e.

nc~ = - - , (5)

np

and eliminating up from the expressions for p and p, Eq. (1) can be rewritten a s :

1 dna = ( l + 4 a ) m p G M a 1 . 2 + 3 a . 1 dT 2(1+2(~) 1 dne n~ dr ~ kBr 2 T § (--"~---)T-~r § ~ n~ dr .(6)

Eq. (6) can be used to derive limits on the helium abundance in the inner corona using empirical inferences of electron density and temperature and their gradients there, together with the condition that dn~/dr < 0. The procedure is as follows: For a given heliocentric distance, r, we substitute the variables and their gradients in Eq. (6) by their empirical values, then search for values of a for which the right hand side is negative. We note here that the temperature gradient in the inner corona plays an important role. If dT/dr < 0, Eq. (6) yields a lower limit to the helium abundance. For dT/dr > O, an upper limit to the helium abundance is inferred.

3. E x a m p l e : I n f e r e n c e s f r o m E U V D a t a

There are two types of data sets which are expected to be acquired with the SPARTAN 201 (Kohl et al 1994) and SOHO missions (e.g., Domingo 1989), and from which inferences of electron density and temperature gradients in the inner corona will be derived. These consist of white light polarization

THE DERIVATION OF EMPIRICAL LIMITS 41

brightness measurements and emission line intensities. As an iUustration. we apply the procedure outlined above to the more complex case of extreme ultraviolet (EUV) line intensities.

We show first how the density gradient in Eq. (6) can be substituted by the gradient of the intensity of emission of an optically thin line. The intensity of an emission line is given by

Ix = /L f ( )~ ,T) ne np ds , (7)

where the emissivity, f ( )~ ,T) , depends on the atomic data, the elemental abundance, the ionization balance, and the electron temperature (see, e.g. Raymond 1988, and references therein). We assume that the temperature and density are constant along the pathlength L which contributes to the emission along the line of sight. Expressing np in terms of ne and a, the above equation can be rewritten as

Ix _ rSe 2 I~ = f ( ~ , T ) (1 + 2a)L" (8)

The literature commonly refers to I~ as emission measure. While I~ is the measured quantity, f(A, T) can be computed theoretically. It then follows that Eq. (8) can be replaced by (for details, see Habbal and Esser 1994)

1 dn~ _ (1 + a) [(1 + 4a) m~GMs 1 (2 + 3a) 1 dT

n~ dr ( l + 3 a ) a kBr ~ T + a T dr

( l + 2 a ) 1 dI'~ l dL

+.(i+.)(ii----- dr L (9)

In what follows we assume that L is constant. Eq. (9) then replaces Eq. (6), and is solved as outlined in section 2, by substituting I~, dI~/dr, T and dT/d r by their empirically inferred values.

The data selected for the application of this technique were taken from the Harvard EUV experiment on Skylab (see Reeves et al. 1977). Measurements were made out to 1.08 Rs off the limb in a south polar coronal hole where two different radial directions, referred to as panels I and II in Figure 1, were selected for this study. Three emission lines were used: Mg X (625 A), Ne VII (465/~) and O VI (1032 ~) , which have a maximum ionization fraction temperature of 1.12 • 106, 5 • 10 5 and 2.8 x 105 K respectively. The emission measure for each line, I~, was derived from its measured intensity, Ix, and its corresponding fx was computed using a code provided by J. l~aymond. The average, I', of the emission measures from all three lines, is shown in panels Ia and IIa of Figure 1. The temperature profiles, shown in panels Ib and IIb of Figure 1, were inferred from the ratios of the line intensities. (See Habbal, Esser and Arndt, 1993, for details about the technique and

42 S.R. HABBAL AND R. ESSER

I I I I I I I I l I I I I I I

|a

300

2OO

I'

IO0 - I

0 I 1 I I I I I I I 1 I I 1

1"0 I I I ' I I - Ib I 1 ' I ' I l ' I 1 l i b

-~ 0.9 -

0.8 I 1 I

I 1 I I I 1 I I I 0 .4 - - lc llc -

-- y 0 . 2 - - -

0 I r I I 1 1.02 1.04 1.06 1.02 1.04 1.06

r/R 0 r/R 0

Fig. 1. Labels I and II refer to measurements and inferences along two different radial directions in a polar coronal hole. (a) Average emission measure, I', for the range of empirical values derived from emission llne intensities~ (b) average temperature profiles inferred from emission line ratios, (c) amin inferred from the data in (a) and (b) using the technique described in the text.

uncer ta in t ies in this inference). Subs t i t u t ing for the values of I ' and T, thus inferred, in Eq. (9), we derive a lower l imit for a , amin, as expected , since d T / d r > 0 (see panels Ib and IIb). The values of a , , i , for these two different radia l d i rect ions , are shown in panels Ic a n d IIc. T h e y indica te t h a t a can be signif icantly large a t the coronal base, i.e. a t least 0.20, bu t drops very sharp ly to i n t e r p l ane t a ry values wi th in a hel iocentr ic dis tance of 1.05 Rs.

THE DERIVATION OF EMPIRICAL LIMITS 43

4. Discuss ion and Conclusion

The technique outlined in this paper shows that limits on the helium abun- dance, a , in the lower corona within coronal holes, where the flow speed is negligibly small, can be determined from the momentum balance equa- tion using simultaneous empirical inferences of the coronal temperature and density as a function of heliocentric distance. Application of this technique to EUV data currently available indicates that the helium abundance, a, is at least 0.20 at the coronal base. This lower limit drops very rapidly to interplanetary values within a heliocentric distance of 1.1 1~. In the exam- ple given, a~an carries at least the same uncertainty entering the empirical derivation of the profiles for I' and T, which are typically 25%.

In the derivation provided we neglected flow speeds and external body forces, as expected, for example, from MHD waves. We also made the simpli- fying assumption that electrons, protons and a 's have the same temperature. For the distances considered in the example given, model computations indi- cate that such assumptions are appropriate. We also neglected the effects of thermal diffusion which might be important in the subsonic region (cf. Geiss et al 1970; Jokipii 1966). In the absence of any direct measurements, however, we cannot asses their importance except from numerical models. Ideally, self-consistent results can be obtained if model computations are carried out using the empirical derivations described above.

The other simplifying assumption used in our approach is the constan- cy of the density and temperature along the pathlength contributing to the emission along the line of sight. Such an assumption implies an approximate- ly homogeneous atmosphere. To include the effects of local inhomogeneities along the line of sight would require extensive modeling. The relative impor- tance of local inhomegeneities will be easier to evaluate when tomographic observations become available from the different instruments onboard the upcoming SOHO mission, scheduled for launch in 1995. More accurate tem- perature inferences are also expected from this mission, since they will be derived from ratios of emission lines produced by the same atomic species (such as with SUMER). We note, however, that the assumptions entering our computations are the same as those used to date in all inferences of plasma parameters. These assumptions will also be used in the inference of helium abundance from the measurements of helium lines if these become possi- ble, for example, with the SOHO mission. With the very large data base to be acquired with the different instruments onboard SOHO, we expect that the technique presented here will yield reliable constraints on the helium abundance in the lower corona.

44 S.R. HABBAL AND R. ESSER

5. Acknowledgements

This work was supported by NASA grant NAGW-249, and Air Force grant AFOSR-91-0244. The authors extend their thanks to the referee for her/his useful comments.

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