You can do something similar to the Drake equation:
Nlife=Nstars∗Fplanet∗Tplanet∗Splanet∗Fsurface∗DTRNA∗VRNA∗NLRNAbase
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
log10Pabio∼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.
While cool, I didn't expect indefinite self-replication to be hard under these circumstances. The enzymes work by combining two halves of the other enzyme- i.e. they are not self-replicating using materials we would expect to ever naturally occur, they are self-replicating using bisected versions of themselves.
I've slightly downgraded my estimate for the minimum viable genome size for self-replicating RNA because I wasn't thinking about complicated groups of cross-catalyzing RNA.