A Analytical calculation of scattering rate

CERN-PH-TH/2013-091

SISSA 19/2013/FISI

On the Minimum Dark Matter Mass

[0.4cm] Testable by Neutrinos from the Sun

[1.5cm] Giorgio Busoni1, Andrea De Simone2, Wei-Chih Huang3

[1cm]

SISSA and INFN, Sezione di Trieste, via Bonomea 265, I-34136 Trieste, Italy

CERN, Theory Division, CH-1211 Geneva 23, Switzerland

Abstract

We discuss a limitation on extracting bounds on the scattering cross section of dark matter with nucleons, using neutrinos from the Sun. If the dark matter particle is sufficiently light (less than about 4 GeV), the effect of evaporation is not negligible and the capture process goes in equilibrium with the evaporation. In this regime, the flux of solar neutrinos of dark matter origin becomes independent of the scattering cross section and therefore no constraint can be placed on it. We find the minimum values of dark matter masses for which the scattering cross section on nucleons can be probed using neutrinos from the Sun. We also provide simple and accurate fitting functions for all the relevant processes of GeV-scale dark matter in the Sun.

## 1 Introduction

The indirect searches for the Dark Matter (DM) component of the Universe are primarily based on identifying excesses in fluxes of cosmic rays, such as positrons, anti-protons, neutrinos, etc; these stable Standard Model particles may be the end product of the annihilation (or decay) of DM in the galactic halo or in the Sun. Among the different ongoing search strategies, the search for the annihilation products of DM in the Sun is particularly interesting. In fact, the DM particles trapped in the core of the Sun may annihilate into anything, but only neutrinos would be able to escape the surface and reach the Earth. The role of neutrinos in DM searches of this type is then very special.

The indirect searches for DM in the Sun are tightly linked to direct detection searches, which are sensitive to the cross section for DM scattering off the nucleons of heavy nuclei (e.g. for protons). In fact, suppose that DM annihilates into several final states , with branching ratios BR, and producing a differential number of neutrinos per annihilation ; then, the flux of neutrinos of DM origin arriving at Earth is given by

 dΦνdEν=ΓA4πR2∑jBRjdNjdEν (1.1)

where is the Sun-Earth distance, is the rate of annihilations per unit time

 ΓA=12A⊙N2χ, (1.2)

is the number of DM particles in the Sun, and the annihilation coefficient will be defined and discussed later, see Eq. (2.3). Since depends on how many DM particles got trapped in the Sun, and hence generically depends on , observational limits on the flux translate into limits on , which can be competitive with those of direct detection searches.

This situation dramatically changes in the case of light DM, with mass around GeV. The number becomes independent of when the capture process goes in equilibrium with the evaporation, and annihilation is negligible. As a consequence, the experimental bounds on the neutrino flux from the Sun cannot be translated anymore into constraints on and the link between neutrino flux and DM-nucleon scattering cross section disappears. We also find simple and accurate fitting functions for all the relevant processes concerning DM in the Sun: annihilation, capture and evaporation.

The interest in (GeV) neutrinos as probes of DM has been recently reinvigorated by the proposal to consider the production in the Sun of muons and charged pions as products of DM annihilations, and their subsequent decay at rest [1, 2]. These neutrinos can be easily detected by neutrino telescopes based on water Cherenkov detectors, such as Super-Kamiokande [3]. One should also keep in mind that the energy to distinguish neutrinos originated by DM in the Sun is bounded from below; in fact, for DM masses below MeV, the detection process is based on inverse -decay , of which gets identified. The distribution of is mostly isotropic (see e.g. Ref. [4]), and the angular resolution is typically not good enough to extract information on the arrival direction. Therefore, it is not possible to distinguish neutrinos from DM annihilations in the Sun from those from the galactic halo, whose flux is much bigger [5]. While we will not commit ourselves to any specific model for GeV-scale DM, this situation can be realized in the context e.g. of asymmetric DM [6] or in explicit models such as the one in Ref. [7].

The rest of the paper is organized as follows. In Section 2 we will briefly discuss the relevant processes for DM inside the Sun, and then turn to compute the total number of the DM inside the Sun in Section 3. Our concluding remarks are in Section 4.

## 2 Relevant processes of DM in the Sun

The DM inside the Sun undergoes several processes: it gets captured, via the energy losses from scattering with the nuclei; it annihilates, whenever two DM particles meet; or it can even evaporate, if the collisions with nuclei make it escape the Sun. The total number of DM inside the Sun is thus determined by the interplay of these three processes. Let us discuss them in more detail (see also Ref. [8] for a previous analysis of these processes, and Ref. [9] for a recent update in the regime where evaporation is not important).

### 2.1 Annihilation

The first important process to consider is the annihilation of two DM particles inside the Sun, and we want to compute the rate for this process (we follow closely the discussion in Ref. [10]). We approximate the phase space distribution of the DM trapped in the Sun by a global temperature and the local gravitational potential , defined with respect to the solar core, as

 ϕ(r)=∫r0GNM⊙(r′)r′2dr′, (2.1)

where is Newton’s constant and is the solar mass within radius . Throughout the paper we use the density profile from the solar model AGSS09 [11]. The DM number density is determined by solar gravitational potential and scales as

 nχ(r)=n0e−mχϕ(r)/Tχ, (2.2)

where is the density at the core. The annihilation coefficient in Eq. (1.2) is defined as

 A⊙≡⟨σvrel⟩⊙∫Sunnχ(r)2d3r[∫Sunnχ(r)% d3r]2, (2.3)

where the thermally-averaged annihilation cross section is assumed to be independent on the DM position in the Sun, and we assume the number density of DM particles equal to that of antiparticles. The factor of 1/2 in Eq. (1.2) simply avoids double counting of pairs in the annihilation.

To compute the annihilation coefficient , we need to know , which is obtained as follows. The average DM orbit radius is the mean value of the DM distance from the center of the Sun,

 ¯r(mχ)=∫Sunrnχ(r)d3r∫Sunnχ(r)d3r, (2.4)

and it depends on the DM mass (see Fig. 1, left panel). The temperature of the population of DM particles trapped in the Sun, or DM temperature for brevity, is taken to be the local solar temperature at the DM mean orbit:

 Tχ=T⊙(¯r), (2.5)

and it depends on through . The dependence of on the DM mass is shown in the right panel of Fig. 1.

If the DM particles are heavier than a few GeV, they get trapped near the solar core and the corresponding will be very small. As a consequence, the DM temperature will be close to the central solar temperature. In the limit where the DM is much heavier than the nucleon mass , the DM temperature will approach the solar temperature at the center. The determination of , and hence the annihilation coefficient, for DM masses of a few GeV (or less) requires taking into account the full solar density profile, as the DM orbit can span a wide region inside the Sun and the approximation of constant solar density is no longer valid.

For the annihilation coefficient we find the following fitting function

 A⊙≃2.91e−1.34[log(20GeVmχ)]1.14(⟨σannv⟩⊙3×10−26cm3/s)×10−55s−1, (2.6)

valid in the range GeV, to better then 9%. In Fig. 2, we show the comparison between our numerical results with Eqs. (16) of Ref. [12] for . They are consistent with each other up to TeV, except for GeV, which is due to the breakdown of the constant density approximation.

In the following, we will only consider the case where the annihilation cross section is velocity-independent (-wave annihilations). As a reference value for the thermally average cross section in the Sun today we take , although the actual value depends on the effective degrees of freedom at the freeze-out temperature, which in turn depends on the DM mass (see e.g. Ref. [13]). The case of pure -wave annihilations results in a smaller annihilation cross section today than at freeze-out. We will not explore this case thoroughly, although in our analysis we will vary the annihilation cross section with respect to its reference value.

### 2.2 Capture

The other relevant processes occurring in the Sun are capture and evaporation. A DM particle can collide with nuclei and lose energy when it traverses the Sun. If the final velocity of the DM particle after the collision is less than the local escape velocity , then it gets gravitationally trapped. This capture process makes the popoulation of DM particles in the Sun grow. However, the captured DM particles may scatter off energetic nuclei and be ejected, whenever the DM velocity after the collision is larger than the local escape velocity. This process is called evaporation. The formalism to describe capture and evaporation is the same, apart from the requirement on the final velocity to be larger or smaller than .

The local escape velocity is defined as , where is the local gravitational potential in Eq. (2.1). The basic quantity is the rate per unit time at which a single DM particle of velocity scatters to a final velocity between and , off a thermal distribution of nuclei with number density , mass and temperature . The plus (minus) sign refers to whether the final velocity is larger (smaller) than the initial one. This quantity has been first computed in Ref. [14], under the assumption of isotropic, velocity-independent DM-nucleus cross section , and we provide the details of the calculation in Appendix A. The scattering rate per unit time results to be (see Eqs. (A.12)-(A.13))

 R±i(w→v′)dv′=σinNiwμ2+,iμi⎡⎣[Erf(α+,i)−% Erf(±α−,i)]+e−mχ(v′2−w2)2TN,i[Erf(β+,i)−Erf(±β−,i)]⎤⎦v′dv′ (2.7)

where Erf is the error function and

 α±,i ≡ √mNi2TNi(μ+,iv′±μ−,iw), β±,i ≡ √mNi2TNi(μ−,iv′±μ+,iw), μi ≡ mχmNi, μ±,i ≡ μi±12.
(2.8)

The rate per unit time is simply related to the differential scattering cross section by . The rates per unit time are simply obtained by appropriate integrations over the final DM velocity

 Ω−ve,i(w) = ∫ve|μ−,i|μ+,iwR−i(w→v′)dv′, (2.9) Ω+ve,i(w) = ∫+∞veR+i(w→v′)dv′. (2.10)

The lower integration limit in Eq. (2.9) is the minimal final velocity simply set by kinematics. The rate is what controls capture, while controls evaporation. We discuss here the capture process and defer evaporation to the next subsection.

The local capture rate of DM per unit volume at radius , due to nucleus of mass , can be written as [15, 16]

 dC⊙,idV=∫umaxi0dufv⊙(u)uwΩ−ve,i(w), (2.11)

where is the DM velocity at infinity, is the local DM velocity inside the Sun before the scattering, and corresponds to a DM scattering with a final velocity equal to .

The function is the velocity distribution of DM particles seen by an observer moving at the velocity of the Sun km/s, with respect to the DM rest frame. The velocity distribution of DM particles in the galactic halo, in their rest frame, is approximated by a Maxwell-Boltzmann with a velocity dispersion

 Missing or unrecognized delimiter for \right (2.12)

where is the average mass density of DM in the halo. We will set km/s. By making a Galilean transformation of velocity , it is straightforward to derive the distribution

 fv⊙(u)=ρχmχ√32πuv⊙vd[exp(−3(u−v⊙)22v2d)−exp(−3(u+v⊙)22v2d)]. (2.13)

In the Sun, the solar temperature is much smaller than the escape energy of a DM particle, so for capture it suffices to deal with the zero-temperature limit. We checked that taking into account the finite-temperature corrections would reduce the capture rate by less than 10% with respect to the one computed for .

In the limit , and for elastic isospin-invariant contact interactions between DM and nuclei, simple analytical formulae can be derived. The scattering rate per unit time for nucleus is

 R−i(w→v′)dv′=2nNiσiwμ2+,iμiv′dv′. (2.14)

and the total rate (2.9) becomes

 Ω−ve,H(w)=σHnNHw⎛⎝v2e−μ2−,HμHu2⎞⎠, (2.15)

which is valid only for Hydrogen (H). In fact, for scatterings with heavier elements one should take into account the decoherence effect. One simple way to do so is to multiply the scattering rate by a form factor , depending on the recoil energy, which is the difference between the energies of the DM particle before and after the collision . So for Hydrogen , while for heavier elements we consider the simple exponential form factor [15, 17, 16]:

 |Fi(ER)|2=exp(−ER/Ei),withEi=3/(2mNiR2i),Ri=[0.91(mNi/GeV)1/3+0.3]fm, (2.16)

which has the advantage of making possible a simple analytical integration of Eq. (2.9), to get

 Ω−ve,i(w)=σinNiw(μi+1)22mχμiEi[e−mχu2/(2Ei)−e−mχw2μi/(2μ2+,iEi)]. (2.17)

We checked that using the more accurate Helm-Lewin-Smith form factor [18, 19], the capture rate would differ by less than 2% in the mass range considered, and the corresponding number of DM particles (to be discussed in the next section) by less than 1%, for GeV.

Finally, the total capture rate inside the Sun is obtained by integrating Eq. (2.11), with (2.13), (2.15) and (2.17), over the solar volume and summing over the different nuclear species in the Sun

 C⊙=∑i∫SundC⊙,idVd3r, (2.18)

where refers to the nucleus . The quantity of phenomenological interest is the DM-proton scattering cross section , which is related to the cross section on the nucleus (with mass number ) by

 σi=σpA2im2Nim2p(mχ+mp)2(mχ+mNi)2, (2.19)

and we assume equal couplings of the DM to protons and neutrons. The generalization to account for different DM-nucleon couplings is straightforward. For spin-independent (SI) DM-nucleus interactions, we have an enhancement of the cross section from constructive interference between nucleons inside the nucleus . Therefore, we have included contributions from the most important elements up to Ni. On the other hand, for spin-dependent (SD) interactions, only Hydrogen is considered since another dominant element, Helium, has spin zero. So, the capture rate for SD interactions is computed using Eqs. (2.11) and (2.15) with and unit form factor . The capture rates we obtained are shown in Fig. 3, for the SD and SI cases. Our results are in very good agreement with those of Ref. [12].

We find the following simple fitting functions for the capture rate corresponding to SD and SI DM-nucleus interactions

 C⊙ ≃ Missing or unrecognized delimiter for \right (2.20) C⊙ ≃ 5.27e3.73×10−2[log(20GeVmχ)]2.23(σp10−40cm2)×1025s−1,(SI) (2.21)

valid in the range GeV, with an accuracy better than 3% and 6%, respectively.

### 2.3 Evaporation

As highlighted in the previous subsection, the formalism for describing evaporation is identical to that for capture. However, contrarily to capture, the evaporation is highly sensitive to the temperature of the distribution of nuclei in the Sun, and therefore we now need to work in the finite temperature regime . Also, we willl work in the regime where the Sun is optically thin with respect to the DM particles, and we do not consider the refinements of the calculations in the optically thick regime [20, 21]

The evaporation rate per unit volume at radius is given by

 dE⊙,idV=∫ve0f⊙(w)Ω+ve,i(w)dw, (2.22)

with given by Eq. (2.10). We will approximate the velocity distribution of the population of DM particles trapped in the Sun, as a Maxwell-Boltzmann distribution, depending on the DM mass and temperature

 f⊙(w)=nχ4√π(mχ2Tχ)3/2w2e−mχw2/(2Tχ). (2.23)

The approximation of thermal distribution is valid for DM mass GeV, while the actual DM distribution deviates from the thermal distribution for larger masses. We use the results of [14] to account for the corrections due to a non-thermal distribution. The total evaporation rate per DM particle is obtained by integrating Eq. (2.22) over the solar volume and divide by the total number of DM particles in the Sun

 E⊙=∑i∫SundE⊙,idVd3r∫Sunnχ(r)d3r (2.24)

Again, for SD interactions, only Hydrogen is considered but for SI interactions, we include all the elements up to Nickel, using the solar model AGSS09 [11].

There is a simple analytical approximation of Eq. (2.24) [14, 20], which is valid for

 Eapprox⊙≃8π3√2mχπT⊙(¯r)ve(0)2¯r3e−mχve(0)22T⊙(¯r)Σevap, (2.25)

where is the escape velocity at the solar center. The quantity is the sum of the scattering cross sections of all the nuclei within a radius , where the solar temperature has dropped to 95% of the DM temperature. The derivation of Eq. (2.25) is sketched in Appendix B. We present our numerical results for in Fig. 4. Notice that for GeV the evaporation rate drops rapidly. We found that the approximated formula of Eq. (2.25) is off by a factor with respect to the full numerical result, in the relevant region GeV, in agreement with what stated in Ref. [20].

For the evaporation rate for SI and SD DM-nucleus interactions, we find the following simple fitting functions

 E⊙ ≃ 1.09e−34.97(1GeVmχ)0.0467−9.25(mχ1GeV)0.95(σp10−40cm2)×109s−1,(% SD) (2.26) E⊙ ≃ 5.13e−39.6(1GeVmχ)0.077−8.92(mχ1GeV)0.97(σp10−40cm2)×1011s−1,(% SI) (2.27)

which reproduce the full numerical results with an accuracy better than and , respectively, in the range GeV. For heavier DM masses, the evaporation is completely negligible.

## 3 Results

### 3.1 The number of DM particles in the Sun

We have now all the tools to determine the number of DM particles in the Sun, which depends on the DM mass , its annihilation cross section and its scattering cross section with proton . The time evolution of the number is described by the simple differential equation [8]

 dN(t)dt=C⊙−E⊙N(t)−A⊙N(t)2, (3.1)

whose solution, evaluated at the age of the Sun , is

 N(t⊙)=√C⊙A⊙⋅tanh(kt⊙/τ)k+12E⊙τtanh(kt⊙/τ)≡Nχ, (3.2)

where and .

Depending on the DM mass and cross sections, the different processes have different relevances, and ultimately two regimes are possible: capture and annihilation are in equilibrium, or capture and evaporation are in equilibrium. For the cross sections of interest, , the quantity is always bigger than one meaning that the equilibrium condition is always fulfilled. When evaporation is negligible, , then and the number simply reduces to

 Nχ≃√C⊙A⊙tanh(t⊙/τ)≃√C⊙A⊙. (3.3)

In this situation the capture and annihilation processes are in equilibrium. On the other hand, in the opposite regime , the annihilation becomes negligible and the equilibrium is attained by capture and evaporation and the number of DM particles becomes

 Nχ≃C⊙E⊙ (3.4)

becomes independent of the DM-nucleus cross section.

The main parameter determining whether evaporation is relevant or not is the DM mass. Since the evaporation drops rapidly for about GeV, so we expect that in this regime annihilation and capture are in equilibrium; however, for lighter DM, the capture goes in equilibrium with evaporation.

In Fig. 5, we show as a function of for different values of , in the range . Notice that tends to the curve , corresponding to when the equilibrium between capture and evaporation is attained, and the number of DM particles does not depend on anymore. Notice also that the maximum of occurs around GeV because below this value the evaporation is important, yielding fewer , and above that the number of DM particles passing through the Sun decreases as .

### 3.2 The minimum testable DM mass

In order to characterize how evaporation affects , one can define two quantities with dimension of a mass: the “evaporation” mass and the “minimum” mass . First, the evaporation mass is defined as the mass for which the inverse of the evaporation rate is equal to the age of the Sun yrs, i.e., [20]. Second, we introduce the mass , corresponding to the DM mass for which approaches the equilibrium value , which is independent of . Quantitatively,

 ∣∣∣Nχ(mmin)−C⊙E⊙∣∣∣≡0.1Nχ(mmin), (3.5)

where we arbitrarily chose 10% as a satisfactory level of approaching . The standard lore regarding is that the evaporation rate becomes negligible when ; on the other hand, for , the annihilation rate becomes negligible. As a consequence, one would expect that for the capture and evaporation processes are in equilibrium and the number of DM particles in the Sun can be approximated by the equilibrium value in Eq. (3.4), which does not depend on . What we want to point out here is that it is actually (and not ) which qualifies the inability of extracting constraints on , since the number of DM particles is not sensitive to anymore, for .

We have found some simple fits of of as a function of and

 mmin ≃ [2.5+0.15log10(σp10−40cm2)−0.15log10(⟨σannv⟩⊙3⋅10−26cm3/s)]GeV(SD), (3.6) mmin ≃ [2.7+0.15log10(σp10−40cm2)−0.15log10(⟨σannv⟩⊙3⋅10−26cm3/s)]GeV(SI), (3.7)

which are valid to better than , in the interval: , . In these intervals, the evaporation mass is always greater than .

A simple argument to understand the positive correlation between and goes as follows. First of all, in the regime where capture and evaporation are the relevant processes, the larger , the more difficult is for nuclei to expel DM particles, so is larger. Then, increasing leads to more DM particles captured by the Sun, so larger . Therefore, turns out to be larger.

In Fig. 6, we plot in the plane. The plot shows the region of parameter space where it is not possible to contrain with neutrino data from the Sun. For comparison, we also show some of the exclusion curves obtained in the analysis of Super-K data of Refs. [2, 12]. For instance, for , data on neutrinos from the Sun are not able to provide information on the DM-proton scattering cross section below . On the other hand, for a given value of the scattering cross section there is a minimum DM mass (see Eqs. (3.6)-(3.7)) which can be probed by neutrino fluxes from the Sun. Increasing (decreasing) the annihilation cross section leads to a smaller (bigger) at fixed , as confirmed by Eqs. (3.6)-(3.7); the effect of varying the annihilation cross section by a factor of 10 with respect to its reference value is also shown in Fig. 6.

## 4 Conclusions

In this paper we have considered the implications of the presence of GeV-scale DM in the Sun, the relevant processes it is subject to, and the constraints which can be placed on its properties, namely mass and cross sections, using neutrino data. We can summarize our main results as follows:

• for DM masses below about 4 GeV the effect of evaporation cannot be neglected, and we provide handy and accurate fitting functions for all the relevant processes of light DM in the Sun: annihilation Eq. (2.6), capture Eqs. (2.20)-(2.21) and evaporation Eqs. (2.26)-(2.27);

• we point out a limitation on extracting cross section bounds when evaporation is important; we provide expressions for the minimum DM mass below which the number of DM particles in the Sun does not depend on , Eqs. (3.6)-(3.7), and the link with DM direct detection bounds disappears;

• we identify the region of the parameter space () (see Fig. 6) which is not accessible by data on neutrino fluxes from the Sun.

## Acknowledgments

We thank P. Machado, S. Palomares-Ruiz and S. Petcov for useful discussions and comments on the manuscript. We also thank J. Edsjö for interesting discussions. ADS acknowledges partial support from the European Union FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). GB is partially supported by the Swiss National Science Foundation (SNSF), project N. 200021140236.

## Appendix A Analytical calculation of scattering rate

In this appendix we present the calculation of the scattering rate per unit time at which a single DM particle of velocity scatters to a final velocity between and , off a thermal distribution of nuclei with number density , mass and temperature (for simplicity, throughout this appendix we drop the index referring to a particular nucleus). We will use this result to compute the rates for capture and evaporation. This calculation was first performed by Gould in Ref. [14] and we reproduce it here, although in a different form. The differential scattering rate of a DM particle of initial speed (in the lab frame) and final speed on a nucleus of speed is

 Ct2se−κ2(2μμ+t2+2μ+s2)θ(w−|s−t|)θ(s+t−w)δ[v−(s2+t2−2zst)1/2]dzdsdt, (A.1)

where , and are defined as

 (1+μ)s=|→u+μ→w|,(1+μ)t=|→w−→u|,κ=√mN2TN, (A.2)

are defined in (2.8) and is a multiplicative factor

 C=16μ4+√πκ3nNσeκ2μw2w. (A.3)

The momentum conservation in the lab frame is obtained by integrating the following expression

 ∫1−1δ[v−(s2+t2−2zst)1/2]dz=vstθ(v−|s−t|)θ(s+t−v). (A.4)

The integration domain is determined by the 4 -functions, that give us 4 inequalities

 v−|s−t|≥0,w−|s−t|≥0, (A.5) s+t−v≥0,s+t−w≥0, (A.6)

and we get that the new variables are subject to the constrains

 x1=|v−w|2,x2=v+w2, (A.7)
 {x1≤t≤x2,max[v,w]−t≤s≤min[v,w]+t,x2≤t≤∞,t−min[v,w]≤s≤min[v,w]+t. (A.8)

The integral is gaussian, thus we obtain (case )

 R+(w→v) = Cv∫x2x1dt∫w+tv−tdste−κ2(2μμ+t2+2μ+s2)+Cv∫∞x2dt∫w+tt−wdste−κ2(2μμ+t2+2μ+s2) (A.9) = Cvκ√2μ+∫x2x1dtχ(κ√2μ+(v−t),κ√2μ+(w+t))te−κ2(2μμ+t2) +Cvκ√2μ+∫∞x2dtχ(κ√2μ+(t−w),κ√2μ+(w+t))te−κ2(2μμ+t2),

where

Using the fact that, for any real numbers ,

 ∫dtχ(bt+c,dt+e)te−A2t2 = −e−A2t2χ(bt+c,dt+e)2A2−√πb4A2e−A2c2A2+b2Erf[bc+A2t+b2t√A2+b2]√A2+b2 (A.11) +√πd4A2e−A2e2A2+d2Erf[de+A2t+d2t√A2+d2]√A2+d2,

and defining as in (2.8), we get the final result

 R+(w→v)=2√πnNσvwμ2+μ[χ(α−,α+)+e−k2μ(v2−w2)χ(β−,β+)]. (A.12)

The case can be done in the same way, and we get

 R−(w→v)=2√πnNσvwμ2+μ[χ(−α−,α+)+e−k2μ(v2−w2)χ(−β−,β+)]. (A.13)

The results in Eqs. (A.12)-(A.13) reproduce the expression in Eq. (2.7).

## Appendix B Analytical approximation of the evaporation rate

Using the identities in Ref. [14] to evaluate the integrals (2.9)-(2.10), one finds (for simplicity, throughout this appendix we drop the index referring to a particular nucleus)

 Ω±ve(w) = ±12√π2TNmN1μ2σnNw[μ(±α+e−α2−−α−e−α2+) (B.1) +(μ−2μα+α−−2μ+μ−)χ(±α−,α+)+2μ2+e−mχ(v2e−w2)2TNχ(±β−,β+)⎤⎦,

where is defined as in (A.10), and the evaporation rate per unit volume is defined as in Eq. (2.22)

 dE⊙dV=∫ve0f0(w)Ω+ve(w)dw. (B.2)

This is a function of . The analytical evaluation of this integral is possible (although lengthy) when is a thermal Maxwell-Boltzmann distribution as in (2.23), and the result is

 dE⊙dV=σA(r,mχ)nN(r)n0e−mχϕ(r)/Tχe−(Eesc(r)−Eesc(0))/TχEesc(r)Eesc(0)˜R(mχ), (B.3)

where is the escape energy at radius , and

 ˜R(mχ) = 2√π√2TχmχEesc(0)Tχe−Eesc(0)/Tχ, (B.4) A(r,mχ) = 1√π(TNTχ)3/2⎧⎪ ⎪⎨⎪ ⎪⎩e−Eesc(r)TχμTN/Tχμ2−+μ(TN/Tχ)⎡⎢ ⎢⎣TχTNμ−√μ2−+μTN/Tχ(1+μ2−μTN/Tχ−μ2−μ) (B.5) +μ3+μ√μ2−+μTN/Tχ(TNTχ−1)⎤⎥ ⎥⎦χ(γ−,γ+) +TχTN⎡⎢ ⎢⎣(Eesc(r)TN−12μ+μ2−