To make the
asteroid window for primordial black hole masses (∼ 10**17--10**23 g)
somewhat intelligible, we can write a
fiducial value formula
for asteroid
mass:
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.
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 asteroid
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:
- Ceres
has actual mean density 2.162(8) g/cm**3.
This low value is probably because
Ceres is
1/4 water ice by
mass
(see Wikipedia: Ceres: Geology).
- Vesta has actual
mean density 3.456(35) g/cm**3.
This value a bit higher than our fiducial value
just because Vesta has somewhat
denser rock
on average than indicated by
our fiducial value.
- Itokawa
has actual mean density 1.95(14) g/cm**3.
This low value is primarily because Itokawa
is a rubble pile asteroid which
means there are cavities in its interior
(see Wikipedia:
25143 Itokawa: Physical characteristics).
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).
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 9.3835*10**23 1.30*10**24 1.39
4 Vesta 1807 525.4 2.59076*10**23 2.28*10**23 0.879
... M_upper ... 399.3 ... 1.0*10**23
... D = 100 km ... 100.0 ... 1.57*10**21
... M_lower ... 3.993 ... 1.0*10**17
25143 Itokawa 1998 0.330 3.35*10**13 5.64*0**13 1.59
... D = 1 m ... 0.001 ... 1.57*10**7
Notes:
- Reference:
Wikipedia:
List of exceptional asteroids: Largest asteroids by diameter;
Wikipedia:
List of exceptional asteroids: Spacecraft targets).
- 1 Ceres ⚳
is actually oblate
because of the centrifugal force
caused by its relatively fast rotation
with rotation period 9.074170(2)
hours.
From accurate/precise measurements,
Ceres
has mean equatorial
diameter
963.2 km and polar diameter 891.2 km.
- 4 Vesta ⚶
is approximately triaxial
with dimensions: 572.6 x 557.2 x 446.4 km.
Thus, even bodies larger than 300 km in size scale do NOT have to be
exactly spherical or oblate spherical.
The centrifugal force
and rigid body forces can partially withstand
gravity for
Vesta-size
astronomical objects.
- M_upper has the upper limit mass of the
asteroid window for primordial black hole masses (∼ 10**17--10**23 g).
- D = 100 km has the
fiducial value
diameter.
- M_lower has the lower limit mass of the
asteroid window for primordial black hole masses (∼ 10**17--10**23 g).
- 25143_Itokawa: An example
of relatively small asteroid.
It is a rubble pile asteroid.
- D = 1 m is has the diameter
of the smallest size
asteroid by definition
(see Wikipedia: Asteroid).