How common are abiogenesis events?

You can do something similar to the Drake equation:

$N_{life}=\frac{N_{stars}\ast F_{planet}\ast T_{planet}\ast S_{planet}\ast F_{surface}\ast D}{T_{RNA}\ast V_{RNA}\ast N_{base}^{L_{RNA}}}$

where Nlife is how many stars with life there are in the Milky Way and it is assumed that a) once self-replicating molecule is evolved it produces life with 100% probability; b) there is an infinite supply of RNA monomers, and c) lifetime of RNA does not depend on its length. In addition:

• Nstars - how many stars capable of supporting life there are (between 100 and 400 billion),
• Fplanet - Number of planets and moons capable of supporting life per star - between 0.0006 (which is 0.2 of Earth-size planets per G2 star) and 20 (upper bound on planets, each having Enceladus/Europe-like moon)
• Tplanet - mean age of a planet capable of sustaining life (5-10 Gy)
• Splanet - typical surface area of a planet capable of sustaining life (can be obtained from radii of between 252 km for Enceladus and 2Rearth for Super Earths)
• Fsurface - fraction of surface where life can originate (between tectonically-active area fraction of about 0.3, and total area 1.0)
• D - typical depth of a layer above surface where life can originate (between 1m for surface-catalyzed RNA synthesis and 50 km for ocean depth on Enceladus or Europa)
• TRNA - typical time required to synthesize RNA molecule of typical size for replication, between 1s (from replication rate of 1000 nucleotides per second for RNA polymerases) and 30 min, a replication rate of E.coli
• VRNA - minimal volume where RNA synthesis can take place, between volume of a ribosome (20 nm in diameter) and size of eukaryotic cell (100 um in diameter)
• Rvolume - dilution of RNA replicators - between 1 (for tightly packed replicating units) and 10 million (which is calculated from a typical cell density for Earth' ocean of 5*10^4 cells/ml and a typical diameter of prokaryotic cell of 1.5 um)
• Nbase - number of bases in genetic code, equals to 4
• LRNA - minimal length of self-replicating RNA molecule.

You can combine everything except Nbase and LRNA into one factor Pabio, which would give you an approximation of "sampling power" of the galaxy: how many base pairs could have been sampled. If you take assumption that parameters are distributed log-normally with lower estimated range corresponding to mean minus 2 standard deviations and upper range to mean plus 2 standard deviations (and converting all to the same units), you will get the approximate sampling power of Milky Way of 

$lo{g}_{10}{P}_{abio}\sim Normal(55,4)$

Using this approximation you can see how long an RNA molecule should be to be found if you take top 5% of Pabio distribution: 102 bases.  Sequence of 122 bases could be found in at least one galaxy in the observable universe (with 5% probability).

In 2009 article https://www.science.org/doi/10.1126/science.1167856 the sequence of the RNA on the Fig. 1B contained 63 bases. Given the assumptions above, such an RNA molecule could have evolved 0.3 times - 300 trillion times per planet (for comparison, abiogenesis event on Earth' could have occurred 6-17 times in Earth's history, as calculated from the date of earliest evidence of life).

Small 16S ribosomal subunit of prokaryotes contains ~1500 nucleotides, there is no way such a complex machinery could have evolved in the observable universe by pure chance.

I read The Vital Question by Nick Lane a while ago and it was the most persuasive argument I've seen on abiogenesis. May be of interest to you. The argument made was that it could be fairly common based on proton gradients.

I'm not aware of an argument that there was only on abiogenesis event on Earth, just the observation that all known surviving lineages come from a universal common ancestor fairly early on. In principle that would be compatible with any number of initial events. It's just that once a given lineage evolved enough adaptions/improvements, it would spread and take over, and then no new lineage would be able to compete/get started.

Also, your scale for probability seems to be starting from assuming a single long self-replicating genome, but that isn't strictly necessary to bootstrap the evolution of a basic self-replicating metabolism. There are much shorter RNA strands (<200 base pairs) that have some catalytic activity including synthesizing additional RNA (though not copying themselves, AFAIK). Something like that could locally generate large numbers of shorter RNA strands, many with some form of catalytic activity of their own, collectively comprising some form of catalytic cycle that includes making more of all of them. Such a system would also be better able to cope with low copying fidelity b/c the individual strands that need to be copied correctly are shorter.

As far as going from bare RNA to a bacterium, I admit I don't know how this happen(ed? happens?). My naive initial thought is some RNA arising in this environment that could produce fatty acids, which could form a lipid layer around a cluster of RNA molecules spontaneously. Repeat and replicate enough times, and add in some endosymbiosis events and you're not too far off?

"Now, a pair of Scripps Research Institute scientists has taken a significant step toward answering that question. The scientists have synthesized for the first time RNA enzymes that can replicate themselves without the help of any proteins or other cellular components, and the process proceeds indefinitely." From https://www.sciencedaily.com/releases/2009/01/090109173205.htm