3

He(4 He, γ)7 Be 106 17

Be(e’ , ν)7 Li(p,4 He)4 He 10’1 ’1/2

7

7

Be(p, γ)8 B 102 13

B(β + ν)8 Be— (4 He)4 He 10’8

8

0

Considering the 3 He-destroying reactions, we ¬nd that

ppI ∝ X3 T 16 ppII + ppIII ∝ X3 (1 ’ X)T 17 ,

2

and

and from the 7 Be-destroying reactions,

1

ppII ∝ X7 (1 + X)T ’ 2 ppIII ∝ X7 XT 13 ;

and

here X, X3 and X7 are the abundances by mass of H, 3 He, and 7 Be, respec-

tively, It follows that the two ratios we wish to lower are

’1

R2 ∝ (1 + X ’1 )T ’13.5 .

R1 ∝ (1 ’ X)X3 T and (13.1)

In a standard solar model without motion in the core, species are destroyed

in the same place that they are created. Approximately 80% of the 3 He

is destroyed the ppI reaction, so its abundance is roughly determined by

balancing its creation by the p + p reaction with its destruction by the

3

He + 3 He reaction, giving

X 2 T 4 ∝ X3 T 16 ’ X3 ∝ XT ’6 ,

2

and so the two ratios become

R1 ∝ (X ’1 ’ 1)T 7 R2 ∝ (1 + X ’1 )T ’13.5 .

and

Clearly within the framework of a standard solar model there is no way of

lowering both of these ratios by changing the core temperature, as reducing

the central temperature lowers R1 but increases R2 . Altering the hydrogen

3

He transport and the solar neutrino problem 197

Fig. 13.3. Standard solar neutrino ¬‚uxes (left) and 3 He pro¬le (right).

abundance su¬ciently would be inconsistent with the amount of helium

expected from big bang-nucleosynthesis.

13.3 Cumming and Haxton™s model

Cumming and Haxton (1996) proposed a mechanism by which both ratios

could be reduced, reproducing something like the best-¬t astrophysical so-

lution. To get around the restrictive temperature dependencies above, they

suggested burning 3 He out of equilibrium by transporting it around the core.

Unlike Dilke and Gough (1972), where 3 He is burnt out of equilibrium by

periodic mixing events, they considered a steady-state model.

The contribution to the neutrino ¬‚uxes as a function of radius, and the

3

He abundance for a standard solar model are given in Figure 13.3. The

ppII and ppIII reactions occur predominantly at very small radii, where the

3

He abundance is very low. The ratio of the two rates (R2 ) increases with

radius, owing to the decrease in temperature.

Cumming and Haxton assumed that 3 He was transported quickly from

larger radii, where its equilibrium abundance is high but the temperature

is too low for the ppII and ppIII reactions to occur, to smaller radii, where

the temperature is high enough to suppress ppII terminations in favour of

ppIII terminations. Ratio R1 is reduced, as X3 has been increased, and

ratio R2 is reduced, as the 3 He is being burnt at low radii and therefore high

temperature (e.g. equation 13.1). They adjusted the 3 He pro¬le arti¬cially

whilst maintaining global equilibrium. Their modi¬ed 3 He pro¬le is shown in

Figure 13.4 and a schematic of their suggested ¬‚ow is shown in Figure 13.5.

Jordinson

198

Fig. 13.4. Modi¬ed 3 He pro¬le (from Cumming and Haxton, 1996).

Fig. 13.5. Schematic of ¬‚ow (from Cumming and Haxton, 1996).

13.4 Modelling the ¬‚ow

To see if such a scheme could really work, I modelled the kinematic e¬ect

of imposing a steady ¬‚ow on the solar core. The timescale of the ¬‚ow is

of the order of the timescale for 3 He destruction in the core, i.e. about

107 years. This is much less than the hydrogen burning timescale, so the

hydrogen abundance should be approximately constant along streamlines.

For simplicity I assume that the hydrogen abundance is constant within the

region mixed by the ¬‚ow, and equal to the SSM abundance elsewhere. The

3

He transport and the solar neutrino problem 199

Fig. 13.6. Latitudinal dependence of the vertical mass ¬‚ux, normalized to one at

the equator, as a function of colatitude.

¬‚ow timescale is much longer than the lifetime of 7 Be in the core (which is

less than a second), so I assume that the 7 Be abundance is equal, everywhere,

to the local equilibrium abundance. The ¬‚ow timescale is also many orders

of magnitude larger than the dynamical time for the solar core, so I assume

hydrostatic equilibrium. I am not considering how the ¬‚ow is driven, only

its e¬ect on the neutrino ¬‚uxes.

Since we want to reproduce a valid model for the sun, I matched my model

to the radius, mass, luminosity and temperature of a standard solar model

at the base of the convection zone. I assume that things are approximately

spherically symmetrical in the convection zone, so I set the temperature to

be uniform at its base and I match the non-spherically-symmetric parts of

the gravitational ¬eld to a vacuum solution.

I describe my ¬‚ow velocity u by a stream function ψ:

ψ(r, θ) ˆ

ρu = ∇ — φ,

r sin θ

ˆ

where r is distance to the centre, θ is colatitude and φ is a unit vector in

the longitude direction; also ρ is density. Note that the continuity equation

is satis¬ed implicitly. The ¬‚ow needs to have a sharp down¬‚ow to carry 3 He

into the core without it getting burnt too soon, and a slow up¬‚ow to allow

the 3 He to be replenished. For the form of my stream function I choose

ψ(r, θ) = Ψ(r) cosN +1 θ ’ cos θ

with N even, which gives the following form for the vertical mass ¬‚ux:

Jordinson

200

Fig. 13.7. Cross section of the ¬‚ow.

Ψ(r)

1 ’ (N + 1) cosN θ .

ρur = 2

r

The latitudinal dependence is shown in Figure 13.6. There is a sharp down-

¬‚ow at the pole and a slower up¬‚ow elsewhere. N is the ratio of the maxi-

mum down¬‚ux to the maximum up¬‚ux, and also determines the solid angle

occupied by the down¬‚ow. The following parameters can be adjusted to

create di¬erent ¬‚ows:

• N , the ratio of the downward ¬‚ux at the pole to the upward ¬‚ux at the