In article <
mt2.0-14757...@hydra.herts.ac.uk>,
Robert L. Oldershaw <
rlold...@amherst.edu> wrote:
>"If our 3.02 GHz P(D) smoothness constraint and the Fixsen et al.
>(2011) 3.3 GHz excess brightness are both correct, then a very
>numerous (N > 10^13 over the whole sky) and unexpected population of
>radio sources not associated with galaxies has been discovered."
>
>(1) N > 10^13 source objects
>
>(2) Not preferentially associates with external galaxies
>
>Could this be a vast population of previously undetected isolated
>stellar-mass black holes populating the highly extended halo of the
>Milky Way Galaxy?
OK, let's look at this.
First of all, the masses would have to be at the low end of
'stellar mass' -- less than about 0.1 solar mass if they provide all
the dark matter (
http://adsabs.harvard.edu/abs/2005MNRAS.364..433B);
but we know that's actually ruled out by microlensing constraints, so
more likely less than about 0.01 solar masses
(
http://arxiv.org/abs/astro-ph/0001272). It's not clear that there's
any mechanism for formation of such things, but that probably bothers
me more than it bothers you.
Secondly, some mechanism is needed to make them radiate in the radio.
These must be isolated black holes, so they won't be accreting
significant amounts of matter. Their Hawking temperature is way too
low to be interesting, so accretion has to be the answer. Now there is
not a lot of stuff for them to accrete out in the halo. I think the
most generous numbers for halo matter density are likely to come from
papers involving ram pressure stripping of dwarf galaxies, like this
one (
http://adsabs.harvard.edu/abs/2004ApJ...603L..77P) which gives a
density of the order 10^-4 cm^-3. That gives total baryonic masses in
the halo of the order of a few x 10^10 solar masses, so it doesn't
seem too crazy.
Now let's estimate the accretion rate. Bondi accretion should be good
enough for this, since there's no reason to expect lots of angular
momentum to be involved. From
http://adsabs.harvard.edu/abs/2007MNRAS.376.1849H I take the formula
Mdot = pi rho G^2 M_BH^2/c_s^3
In SI units,
M_BH = 0.01 * 2e30 kg
G = 6.67e-11 m^3 kg^-2 s^-1
rho = 1e-4 * 1e6 * 1.67e-27 kg m^-3 (i.e. we assume hydrogen)
The thing we don't know is the sound speed, but it might be reasonable
to assume that this is something like the velocity dispersion on these
scales, ~ 100 km/s. (Sanity check -- this gives temperatures ~ 10^6 K
which is (a) a reasonable virial temperature and (b) conveniently hard
to see gas at.)
Then Mdot works out at about 1e-3 kg/s. In other words, if we can
convert the accreted material to radiation at about 10% efficiency,
which is about the maximum possible, each of these things might be
able to radiate at about 1e-3 * 0.1 * 9e16 = 1e14 W. (Less than a
trillionth of a solar luminosity, so they're certainly 'dark'...)
Let's check the Eddington luminosity for such a BH.
L_Edd = 4*Pi*G*M_BH*m_p*c/sigma_T ~ 1e29 W.
So the accretion is certainly very sub-Eddington, which almost
certainly means that we'd have something like an ADAF and that the
radiative efficiency of 10% I've used above is a gross overestimate.
But let's push on with this for now.
Fixsen et al quote the excess radio background as a power law in
antenna temperature. We can translate this to a power law in received
power: flux density S is given by
S = 8 Pi kT_b / lambda^2
(if we assume integration over the whole sky) so, as T_b has the form
T_b = T_0 (nu/nu_0)^-2.6
we have
S = 8 Pi kT_0 nu^-0.6 / (c^2 nu_0^-2.6)
where S is measured in W Hz^-1 m^-2. We can integrate this over the
region of the observed excess to get a total flux in W m^2:
S_tot = int S dnu = (20 Pi kT_0 / c^2 nu_0^-2.6)) [nu_upper^0.4 - nu_lower^0.4]
Let's take the range to be 22 MHz to 10 GHz. Then we can put in some
numbers:
S_tot = 2.5e-11 W m^(-2)
(sanity check: from the Condon et al paper, this should be of the same
order of magnitude as 34e-9*1e-26*1e13*3e9, which it is.)
Now, can our black holes provide this power? Let's very generously
assume that *all* their radiated power comes out in the band of the
ARCADE-2 excess. Let's further assume to get a number out that we have
a uniform distribution of these things out to some cutoff radius R,
and that we are in the centre of the distribution; the latter is
almost true, the former not so true, but we should get the right order
of magnitude. Then, trivially, the total flux received from all the
BHs is
S_tot_BH = n_BH L_BH R (W m^-2 as expected)
But n_BH = N_BH/((4/3)*Pi*R^3), where N_BH is the total number of BHs, so
S_tot_BH = 3 L_BH N_BH / (4*Pi*R^2)
L_BH = 1e14 W, N_BH = 1e13, and let's say R = 100 kpc; then
S_tot_BH = 2.5e-17 W
This is a factor of a million below the ARCADE-2 power. Conclusion: it
is very difficult to get this population to do what we want.
*Very* favourable assumptions I've made include that all the accretion
power from the BH comes out in the radio and that the conversion from
accretion to radiation is 10%. Slightly less favourable ones include
the temperature of the accreted medium and the mass of the BHs (but
note that the maximum possible, 0.1 M_solar, gets you only an extra
two orders of magnitude, which is not enough). The biggest uncertainty
is in the assumptions for the accretion mechanism and the densities
available for accretion. Playing around with these numbers and
checking for any mistakes is left as an exercise for the reader!
>Might Spektr-R be able to contribute important information here?
No. Its sensitivity is way, way too low to see objects of the flux
density estimated by Condon et al, independent of your model for what
they are.
Martin
--
Martin Hardcastle
School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK