Take 1E-13 solar masse as target regime. Einstein radius is 3E=7 smaller than solar mass MACHO.
...
diameter of a white dwarf star is 5000 km or 5e6m or 5e8 cm. But they're faint! There are HST-detected white dwarf stars in the LMC. What's Einstein radius for PBH in our halo?
Assume 10 kpc distance, and mass of 1E-12 solar masses. MACHO project pooped out at 1E-7 solar masses for solar type stars. white dwarf is 3.5E-3 times smaller.
That should gain us 5 orders of magnitude in mass, going down to 1E-12 solar masses. OK, that's in the right regime, finally.
Event duration for Jupiter mass MW halo lenses is around 3 days, at 1E-3 solar masses. If we drop nine orders in mass to 10^-12 M_solar, then time scale is 3E4 shorter, or 1E-4 days or 10 seconds.
For an optical depth of 5E-7 need to look at a single source for 2E7 seconds, or 200 days. If, as this paper says, there are 10 of them in the cluster then we need to observe
for 20 days, continuously.
...
what about pulsars? They're only 20 km or so in diameter! That's 2e4 m or 2e6 cm! yup, found a good paper on that....
If diffraction becomes important on Schwarzchild radius rather than Einstein radius scale, what is that? GM/c^2 for M=10^20 gm = 10^17 kg = 6.67E-11*1E17/(3e8)^2 = 7 E-11 m
Sure enough, that's way less than a micron. This might be because the entire Einstein radius does not make a thin lens, light only arrives from the marginal rays.
How many hit the Earth? If R_E for LMC distance is 100 km, surface area of the microlensing sausage is 2piR*L = 600km * 1.5E18 km ~ 10^21 km^2. We get about one PBH per year
through that, at 10^12 solar masses. Cross section of the Earth is pi*(6500km)^2 = 1.3e8 km^2. So we get hit once in every 10^21/10^8 = 10^13 years, or >> a Hubble time. rats.