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According to the University of California Museum of Palaontology webpage The Ordovician Period, the late Ordovician period witnessed

when Gondwana finally settled on the South Pole during the Upper Ordovician, massive glaciers formed, causing shallow seas to drain and sea levels to drop. This likely caused the mass extinctions that characterize the end of the Ordovician in which 60% of all marine invertebrate genera and 25% of all families went extinct.

NASA's webpage Explosions in Space May Have Initiated Ancient Extinction on Earth and the paper Did a gamma-ray burst initiate the late Ordovician mass extinction? suggest that a gamma ray burst may have initiated the cooling at approximately 440 million years ago.

What palaeoclimatological and geochemical evidence is there to support or refute the gamma ray burst cause of the late Ordovician mass extinction?

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As a preamble, let me say that I don't know remotely enough on galactic dynamic to know if a supernova could have possibly been close enough during the Ordovician for Earth to be affected by a gamma ray burst, nor do I know enough about geochemistry to know if there are ways to detect such an event in the fossil record.

That being said, the questions that I'll try to answer are: do we need to invoke an extraterrestrial event to explain the Late Ordovician glaciation? does the taxa selectivity of the extinction event support the GRB hypothesis?

So let's look at the different hypotheses on how the glaciation occurred, the original conundrum being: how could a glaciation occur when mid-Ordovician $p_{CO_2}$ is believed to be up to 14 times Quaternary level? Brenchley et al. 1994 suggested that the Ordovician ocean circulation changed from a system having warm saline bottom waters to a system with cold bottom waters (i. e. a thermohaline instead of an haline circulation), which triggered a phytoplanktonic bloom, which then decreased atmospheric $p_{CO_2}$ and provoked an eutrophization of the ocean. Kump et al. 1999 argued in favor of a tectonically derived $p_{CO_2}$ decline: with the Appalachian and Caledonides orogenesis came an increased weathering which decreased $p_{CO_2}$. Of course the two hypotheses are not mutually exclusive since increased weathering also means increasing nutrients in seawater and therefore increasing primary productivity.

Another question is which groups went extinct during the glaciation: recent studies show that acritarchs (the most diverse group of phytoplankton known for that period) did not suffer from these events (Vecoli 2008); oppositely benthic group such as crinoids or brachiopods did (see Sheehan 2001 for a review); some pelagics groups such as graptolites and nautiloids did suffer as well during the event while some other didn't (radiolarians for instance). In summary there is no clear pattern, but there is also no reason to think that there was a preferential extinction of surface organisms over deeper-living ones.

So, does that answer the question: no; but it seems to me that GRB doesn't explain better the event that unfolded during the late Ordovician than other hypotheses (namely the ocean eutrophization and the continental weathering). Does it discard the hypothesis of a GRB: no; but since there is no direct evidence that it indeed happened at that time, and that we can explain the Ordovician events without, it seems, to me, more parsimonious to think that it didn't.

References:
Brenchley et al., 1994. Bathymetric and isotopic evidence for a short-lived Late Ordovician glaciation in a greenhouse period. Geology, 22: 292-298.
Kump et al., 1999. A weathering hypothesis for glaciation at high atmospheric $p_{CO_2}$ during the Late Ordovician. Palaeogeography, Palaeoclimatology, Palaeoecology, 152: 173-187.
Sheehan, 2001. The Late Ordovician Mass Extinction. Annual review of Earth and Planetary Science, 29: 331-364.
Vecoli, 2008. Fossil microphytoplankton dynamics across the Ordovician–Silurian boundary. Review of Paleobotany and Palynology, 148: 91-107.

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The answer that you're asking is within the paper that you cited in the question:
Did a gamma-ray burst initiate the late Ordovician mass extinction? (Melot et al).

Specifically the section titled "ASPECTS OF THE LATE ORDOVICIAN MASS EXTINCTION POTENTIALLY COMPATIBLE WITH A GRB".

But if you're looking for an article summary, here is the evidence they mention that may point to a Gamma Ray Burst (GRB) as a possible source of the extinction:

Rapid global cooling followed by rapid global warming
Specifically, they say that a GRB may cause the production of NO2, which will increase the opaqueness of the atmosphere, causing cooling.

Some species impacted more than others
They mention that this may have happened, but that they would expect:

planktonic organisms or larvae would more likely be affected than benthic organisms because the former are less shielded from radiation.

Similarly, epifaunal organisms would be more affected than infaunal organisms buried and shielded in sediments.

They go on to say that both of these patterns "may exist".

Differential extinction
They probably say this best:

longer planktonic larval phases, when such can be inferred, are associated with increased extinction probability

The mention that species whose adults live within plankton were completely wiped out. This differential extinction is somewhat suggestive of a GRB.

Summary
Ultimately, there's not a clear and definitive link to a Gamma Ray Burst as the sole or primary source of this extinction event. However, there are some correlations. In the author's words,

clearly additional tests are required


source: Did a gamma-ray burst initiate the late Ordovician mass extinction? (Melot et al).

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