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Jay Wacker1 (professor of physics at the SLAC National Accelerator Laboratory) stated:

The first stars (known as Pop III) were made out of hydrogen and helium. They had no planets.

Why couldn't they have had gas planets?

[1] Requires login at LinkedIn

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The earliest stars did not have planets primarily due to a lack of metals. Metals in this sense is an element (with some extra properties that are not relevant in this context) heavier than helium. The very article that you linked to references this. This leads to the following:

  1. Stars without metals tend to not last very long. Metals in a star act to slow down the reaction speed of the fusion. Without metals, the stars quickly get to a state where they will explode. Short time scales do not allow for enough time to form planets.
  2. Metals seem to be the initial building block of planets. This wikipedia article discusses the current leading theories for rocky and gas planets. Basically, they both start with a rock forming that's big enough, leading to a chain effect which ends up to be a planet. Rocks can't form from hydrogen and helium, making planet formation difficult.
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  • $\begingroup$ There may be a subtle difference here between planets forming around Population III stars and planets being found around Population III stars. Since Population III and Population II stars had some overlap, is it possible that one of the last Pop III stars to form captured a rogue planet ejected from its young Pop II star orbit? Perhaps it is more precise to say, "No planets formed around Population III stars." $\endgroup$ – Dave Jarvis Oct 3 '19 at 7:06
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Several answers occurred to me.

  1. Timescale. Population III stars only last a few million years. This article, which is billed as as evidence for a much shorter timescale of planetary formation than previously thought, still quotes 10 million years (caveat: for terrestrial planets, but I didn't find a source right off the bat for gas giants). However, if I'm wrong there, there's always these other answers:

  2. In continuation of the discussion of How can a Population III star be so massive?, the metallicity may be a factor. The original stars had virtually no metals, which made their cool down much, much less efficient. For the same reason that you can get larger stars that way, perhaps there won't be much material left over for planet formation?

  3. The upper layers of very large, fluffy stars are barely gravitationally bound, and they have enormous stellar winds. Maybe such massive winds and/or the enormous luminosity of a Pop III star either prevent planets from forming in the first place, or ablate embryonic planets into oblivion?

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