Image 1 Caption:
"The formation of the
observable universe
without and with
primordial black holes (PBHs)."
(Slightly edited.)
Features:
Such astroidal-mass PBHs (∼ 10**17--10**23 g) have very small Schwarzschild radii: note,
R_sch = 2GM/c**2 in general
= 1.4851 Å [M/(10**20 g)] for middle of the asteroidal mass range for PBHs
= 0.14851 μm [M/(10**23 g)] for high end of the asteroidal mass range for PBHs
= 2.9532 km (M/M_☉)
≅ 3 km (M/M_☉)
= 0.019741 AU [M/(M_☉**6)]
≅ 0.02 AU [M/(M_☉*10**6)]
= 19.741 AU [M/(M_☉**9)]
≅ 20 AU [M/(M_☉*10**9)]
= 9.5741*10**(-5) pc [M/(M_☉*10**9)]
≅ 10**(-4) pc [M/(M_☉*10**9)] .
Note,
astroidal-mass PBHs (∼ 10**17--10**23 g)
have microscopic
event horizons.
Given that they are so small individually, there must be a lot of them if they make up all dark matter. If they all have mass 10**19 g (implying R_sch = 0.14851 Å which is sub-atomic), then their mean separation in the Milky Way is ∼ 15 astronomical units (AU). See What is the space density of astroidal black holes in the Solar System?.
There may be one astroidal-mass PBH (∼ 10**17--10**23 g) in the Solar System out to the Jupiter orbit (mean orbital radius = 5.2038 AU) at any time, but this estimate is very uncertain. See What is the space density of astroidal black holes in the Solar System?.
In order to make the
asteroid window for primordial black hole masses (∼ 10**17--10**23 g)
somewhat intelligible, we consider the
masses of
asteroids below with
the masses of
Ceres
(NOT shown in Image 2),
Vesta,
and Itokawa
as examples.
For rocky asteroids,
the mean density does NOT
vary much from 3 g/cm**3.
For rocky-icy
asteroids
with ∼ 50% rock
and ∼ 50% water ice by
mass,
the mean density does NOT
vary much from 2 g/cm**3.
The mean diameters
and hence the masses
of asteroids
do vary widely.
Table: Asteroid Mass
below gives some example asteroid
masses from
established measurements (the M values) and those calculated
from the fiducial value formula
(the M_fid values).
We see that the
fiducial value formula is
better than
factor of
2 accurate when
compared to the masses
of real asteroids.
The main reason for the disagreement between the
M values and M_fid values is density.
Recall, all M_fid values were calculated with the
fiducial value ρ = 3 g/cm**3.
To explicate:
We also see that our
fiducial values
for mean diameter
and mean density give
a fiducial value
mass 1.57*10**21 g
that is larger than
the logarithmic mean
mass
of the
asteroid window for primordial black hole masses (∼ 10**17--10**23 g) which is (17+23)/2 = 20 (i.e., in
antilogarithm form
mass
10**20 g) by more than 1
dex
(i.e., in
antilogarithm form
by more than a factor
of 10).
M_asteroid = [(4π/3)*R**3]*ρ
= [(1.57 ...)*10**21 g]*[(D/100 km)**3]*[ρ/(3 g/cm**3)] ,
where R is mean radius,
D is mean diameter,
ρ is mean density,
100 km is a fiducial value
for an asteroid
mean diameter,
and 3 g/cm**3 is a fiducial value
for a rocky asteroid
mean density.
Aste- Name Year of Mean Mass M Mass M_fid from the fiducial Ratio
roid Discov- Diameter value formula with fiducial Mass_fid
No. ery (km) (g) density ρ = 3 g/cm**3: (g) /Mass
1 Ceres 1801 939.4 9.38392*10**23 1.302*10**24 1.39
4 Vesta 1807 525.4 2.59076*10**23 2.278*10**23 0.88
... PBH_max ... 399.3 ... 1.0*10**23
... D = 100 km ... 100.0 ... 1.57*10**21
... PBH_min ... 3.993 ... 1.0*10**17
25143 Itokawa 1998 0.330 3.51*10**13 5.64*0**13 1.61
... D = 1 m ... 0.001 ... 1.57*10**6
Notes: