ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 7, с. 980-985
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ DARK MATTER BODIES IN STAR AND PLANET STRUCTURES
©2014 Yu. E. Pokrovsky*
National Research Center "Kurchatov Institute", Moscow, Russia Received April 3, 2012
The lowest frequency of the dipole f mode (surface gravity wave) of the Sun and some other stars is shown to be close to the orbital frequency of a trial body near the star surface, as well as the wave amplitude is shown to be resonantly increased to the values large enough to be observed. Therefore the Sun is considered to be a sensitive detector for hypothetical compact cosmic bodies made of dark matter particles. In this connection some possible characteristics of the dark matter bodies (DMB) are discussed, and DMB orbits in the Sun are calculated within a standard solar model in order to compare the wave amplitudes with data for the solar surface oscillations, and to estimate the masses and radii of the DMB. As well, some possible phenomena in star and planet structures are discussed with special attention on generation of flares of high X-ray classes, specific behavior of the Moon dust, formation of short-time vertical flows in deserts, oceans, and atmospheres on the Earth and other planets.
DOI: 10.7868/S0044002714070149
Experimental detection of dark matter (DM) due to interaction with ordinary matter and theoretical explanation of the nature of DM are the most challenging tasks in modern physics. According to the recent SNe Ia [1] and WMAP [2] data interpretation within the standard model of cosmology, the Universe is thought to contain about 5% of ordinary matter (OM), 25% of DM, and 70% of dark energy. All known candidates for the role of DM particles (the mirror particles [3—6], or particles of other extensions of the standard particle model, in particular, gravitino in the partial split supersymmetry with bilinear R-parity violation [7]) interact with OM at forces almost as weak as gravitational force. Therefore it is extremely hard to detect DM particles directly via their interactions with OM. Nevertheless, as a result of the cosmological evolution, interactions of DM particles may lead to a formation of compact astronomical DM Bodies (DMB - DM planets or DM stars). Then these DMB may be discovered by their reinforced gravitational interaction with OM. The question is which possible detectors are the most effective for this task?
The first simple estimations of the Sun surface oscillations as an example of manifestation of DMB orbiting under the solar surface has been done [8] in applications to the 160-minute oscillations first observed in Crimean Astrophysical Observatory (CrAO) [9], and independently in Pic-du-Midi observatory in France by Birmingham group [10]. Finally, period of these oscillations is measured as
E-mail: pokrovsky_ye@nrcki.ru
PCrAO = 159.9657 ± 0.0004 min [11]. The respective frequency is vCrAO = 104.1891 ± 0.0003 fiHz.
Nevertheless, there is no more attempts to consider the Sun as a detector of DMB, possibly because the simplest estimations of amplitudes of solar surface oscillations forced by the DMB are too small to be observed, or because the real existence of the 160-min oscillation is not generally accepted mostly because its characteristics are considered as too contradictive for a clear interpretation, and its period is too close to 9th diurnal harmonic (9 • 160 min = = 24 hour) to consider it naively as the diurnal artifact. The solar origin of the 160-min periodicity in the Doppler shift measurements was called into question by series of observations undertaken at Stanford and Tenerife [12, 13]. It was reported that the phase stability is no longer present and that the oscillation could be, in general, an artifact caused by atmospheric extinctions. Nevertheless, the potential influence of the Earth's atmosphere on the differential (i.e., made according to the principle "the Sun relative to the Sun") Doppler shift measurements of the Fe-1 line in the solar spectrum of (photospheric) velocity observation [11] is very different from the full-disk observations [13] based on the resonant-scattering spectroscopy using the Na (or K) line (which are therefore carried out for the low chromosphere of the Sun). As well neither the amplitude nor the phase of the 160-min oscillation observed over several years can be explained in terms of terrestrial atmospheric disturbances or by the data reduction procedure.
The classic f mode (f stands for fundamental) is recognized as a compressionless (9rV = 0) wave
of the matter velocity field V(r, t) (where r is a spatial coordinate, and t is time). In an inviscid solar atmosphere that is permeated by almost constant gravity field, its frequency u is given by the following deep water like dispersion relation [14, 15]
u2 = gk, (1)
where k = (l(l + 1))1/2/R q is the orbital wave vector, Rq = 695 700 km is the solar radius, and l is the orbital spherical degree. This dispersion relation shows that the classic f -mode frequency is independent of the internal structure of the Sun. One kind of the solar f mode is known as a surface gravity wave that resides at the chromospheres—corona transition region [16, 17] where the mass density of the gas rapidly drops, and the gas temperature rises. In another kind of solar f mode the wave propagates along the interface that divides the convection zone and the solar corona [ 18].
The purposes of this paper are (i) to show that for a normal solar model (for example, the standard solar model (S) [19]) the ground mode of the quadropole surface gravity wave can be almost resonantly excited by DMB orbiting near the Sun surface which can be considered as a sensitive detector for hypothetical compact cosmic DMB, and (ii) to compare frequencies (v) and amplitudes of surface velocity oscillations (V) of the quadrupole surface gravity wave with CrAO [11] data (vCrAO = 104.1891 ± 0.0003 ¡Hz, VCrAO = 0.27 ± 0.05 m/s), and the Solar and He-liospheric Observatory (SoHO) [20] data (vsoHo = = 220.7 ¡Hz, VSoHO = 0.0045 ± 0.0015 m/s) in order (iii) to conclude that at least two DMB (DMBCrAO and DMBsoho) are orbiting the Sun near its surface and to estimate their masses and radii, and (iv) to discuss briefly a possible existence of another DMB (DMB-3) which may be orbit the Sun just under the surface and appeares in major flares with frequency vflarei9 = 103.72 ¡Hz observed in solar cycle 19, and with a little higher frequency vflare21 = 103.96 ¡Hz observed in cycle 21 [21].
Figure 1 illustrates adiabatic oscillation eigenfre-quencies computed for a normal solar model [22]. Modes of given radial order n are connected by curved lines. The observed solar oscillations have frequencies in excess of the fundamental dynamical frequency
vdyn = (GMq/RQ)1/2/(2n) = 99.9178 ¡Hz (2)
(here G is the gravitational constant, and MQ is the solar mass), and correspond to the surface gravity waves (f modes) effectively localized at the solar surface. At relatively high orbital or radial degrees (l, n) surface eigenmodes are labeled as p modes — the so-called acoustic waves (Fig. 1). The modes labeled as g modes are internal (effectively localized
30 20 10 0
l
Fig. 1. Frequencies v as functions of orbital degree l, computed for a normal solar model [19]. Modes of given radial order n are connected. The quadrupole surface gravity f mode (n = 0, l = 2) corresponds to the end of the curve with the eigenfrequency v « 100 ^Hz.
at r < 0.55Rq ) gravity waves. It is conventional to assign positive and negative radial orders n to p and g modes, respectively, with n = 0 for f modes. With this definition, frequency is an increasing function of n forgiven l; also, in most cases \n\ corresponds to the number of radial nodes in the radial component of the solar matter displacement, excluding the node at the center. At last it should be noted that the frequency of the quadrupole (n = 0, l = 2) f mode v02 & vdyn is close enough to vcrAO = 104.1891 ¡Hz for significant resonant increase of its amplitude.
This is a reason to consider solar surface oscillations observed in CrAO and SoHO as a result of almost resonant tidal excitation of quadrupole f mode by the gravitational field of a small mass DMB (MDMB < MQ) which orbits the Sun at the radii RorbCrAO = 1.54Rq and RorbSoHO = 0.94Rq with the DMB orbital frequencies:
vorbCrAO = vCrAO/2 = (3)
= (GMqRbCrAO)1/2/(2n) = 52.0945 ¡Hz,
vorbSoHO = vSoHO/2 =
= (GMq/R03rbSoHO)1/2/(2n) = 110.35 |Hz,
and their velocities
VcrAO = (GMq/RorbCrAO)1/2 - 350 km/s, (4)
Vsoho = (GMq/RorbSoHO)1/2 - 450 km/s
are supersonic (the respective sound speed at the solar surface in the solar model S [19] is cs — 7 km/s).
The forced oscillations of the solar matter are described by the following equations:
dtp + dr(pV) = 0, (5)
dr V = 0, (6)
p[dt V + (Vdr) V] = -drp + (7)
+ p(g - GmoMBlr - roMB(i)r°(r - rDMe(i)),
where p is the solar mass density, V is the velocity, p is the pressure, g is the gravitational acceleration, and rDMB(t) is the DMB trajectory. Further simplifications are possible if radius of DMB (RDMB) is large enough (Rdmb » RA, where RA is the accretion radius) to consider only potential flows in the solar matter without a significant accretion or strong shock waves. In the supersonic case
Ra = 2GMdmb/V2mb = 2Rq(Mdmb/Mq), (8)
and if MDMB = 10"8Mq, then RA — 14 m.
Value of Rdmb can be estimated only under different assumptions on the DM structure. For the case of mirror DM matter with mass density close to the mass density of OM (pom — 1—10 g/c m ) Rdmb can be estimated as
Rdmb — (3Mdmb/(4^om ))1/0, (9)
and if MDMB - 10"8Mq, then RDMB - 8001700 km > Ra .
Alternative estimation can be done for DMB from gravitino. A mass and radius of this DMB can be estimated for a known gravitino mass. The radius of the ground state of the nonrelativistic system from gravitino can be calculated [23] as
Rdmb — 2.84/p (mg/Mdmb)1/0 (mp/mg)3, (10)
where lP and mP are the P
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