Do you have any reputable estimates as to how much helium has been out
gassed over the last 4 billion years
Reputable estimates - no. Bad guesses, maybe.
Short answer: only trace amounts. It's impossible to have enough rare radioactive material to produce a real helium atmosphere. Even if the planet retains all it's helium from radioactive decay it should be measurable in parts per million, not a real atmosphere.
Long answer below:
The number of alpha particles that a radioactive element creates is pretty straight forward, just subtract the Baryons in the final state from the initial state. Beta decay changes the type of hadron, but not the number, so, from Uranium 238 into Lead 206 - loss of 32 baryons, and all of them through Alpha decay - Alpha particles become helium atoms, so each U-238 decays into 8 helium.
Thorium 232 into Lead 208 - 6 helium.
Uranium 235 into Lead 207 - 7 helium.
Potassium 40, the other common radioactive element only undergoes beta decay into CA40 (89.3%) and AR40 (10.7%), so it's irrelevant in the helium calculation. There are smaller amounts of other elements that emit alpha particles, like PT-190, but they're rare enough to be ignored. The bigger unknown is how many elements with shorter half-life were common when the Earth was young but are no longer present. I have no idea how to calculate that so that will be a bit of an unknown. One possible approach would be to say that the forming planet would be sufficiently hot to lose much of it's helium, but that's a bit of a cheat. I think, early on, predictions are rough due to unknown quantities of shorter-half life elements.
U-238 is easiest to calculate with a half-life of about the age of the Earth, so about half of Earth's U-238 has completed its decay change and produced 8 helium per U-238.
Thorium-232, with a half life of about 14 billion years, so only about 20% of the initial thorium-232 has decayed over the age of the Earth.
and U-235, with a half-life of about 700 million years, means that almost 99% of it has decayed over the age of the Earth. Current estimates say that about 0.72% of Uranium is U-235
I touched on this above, but the big unknown for this scenario is in the range of 100 million year half-life. That is, elements that are long-lasting enough to survive the Earth's formation but wouldn't remain in high concentrations today. That's in this range, bottom of the 10^12 seconds, top of the 10^15 seconds.
Curium-247 has a half-life of about 15 million years and it decays into lead-207 releasing 10 alpha particles in the process. If planet formation happens relatively quickly, that could supply some helium to a planet, but I'm going to ignore it as having slightly too short a half-life.
Uranium-236 has a half life of about 23 million years and it decays into Thorium, emitting just 1 alpha particle. Presumably much of it would become thorium by the time of planet formation, so I'm going to ignore it as well.
Plutonium-244, mostly synthetic, but it is the rarest primordial element on Earth. From the Wikipedia article:
Accurate measurements, beginning in the early 1970s, have detected
primordial plutonium-244, making it the second-shortest-lived
primordial nuclide after 146Sm. The amount of 244Pu in the pre-Solar
nebula (4.57×109 years ago) was estimated as 0.008 of amount of
238U.
If estimates put it at a bit less than 1% as common as U-238, I'm going to ignore as well, as 1% won't make a big difference.
Finally, Samarium-146 is worth considering with a half-life of about 100 million years (or 68 million depending on which source you use). It decays into Neodymium-142 releasing a single alpha particle. Samarium 147 and 148 also undergo alpha-decay but very slowly, half-life of 100 billion and 7 quadrillion respectively.
So these are the 4 primary helium "outgasing" isotopes relevant after planet formation. U-238 (8 helium), U-235 (7), Th-232 (6) and SM-146 (1). And as for how much, you have to ask how long the planet has been around.
SM-146 decays fairly quickly, half life 100 million years.
U-235 - 700 million years
U-238 - 4.5 billion years
T-232 - 14 billion years.
If we take a planet about the age of Earth
SM-146 - essentially all of it
U-235 - 99%
U-238 - 50%
T-232 - 20%
Now the trick is, how much of these elements is a terrestrial planet likely to have at formation.
If we go by radioactive decay and radiogenic heat estimates, TH-232 and U-238 combine to generate 20 trillion watts (first article). That' 6 x 10^20 joules per year or about 3.75 x 10^33 mEV per year, by radioactive decay of those 2 elements.
The Thorium decay chain emits about 42.6 MeV per decay for 7 alpha particles. (about 7 MeV per helium) and the U-238 decay chain, about 51.7 MeV for 8 alpha particles. (ballpark 6.5 MeV). That includes Neutrinos though and that energy needs to be factored out. I couldn't find precise numbers but about 10 MeV (or 20%) is lost into Neutrinos (technically anti-neutrinos) by this website. So, very ballpark, lets say 5 MeV heat creates one alpha particle.
current rates of decay of Thorium & U-238 should produce about 7.5 x 10^32 alpha particles, which corresponds to about 1.2-1.3 billion moles of Helium or about 5 million KG, between 5 and 6 tons of helium produced in the Earth's crust and mantle every year - based on radioactive measurements which are probably pretty accurate.
Thorium is currently about (ballpark) 4 times as abundant as U-238 but because it decays 3 times more slowly the helium production and heat generation in the Earth's core is similar (this chart suggests thorium's heat output is higher, but they are close).
The helium production from Thorium has been pretty consistent, as 80% of the Thorium at the formation of the planet is still here.
The production from U-238 was about twice as high when the Earth formed.
The production from U-235 is currently negligible compared to the other two, but was much higher when the Earth was young. working back from U-235 being about 0.7% of the total Uranium and it having gone through between 6 and 7 half-lifes since the formation of the Earth, (and accounting for U-238 being twice as abundant when the Earth formed), that means between 32 and 64 (lets say 50) x 0.7, so there was some 30-40% as much U-235 as U-238 when the Earth formed.
Bad math again:
Thorium, 2.5-3 tons per year now, 25% more than that when Earth formed. lets say 3 tons per year, average
U-238, 2.5 tons per year now, twice that when the earth formed. lets ballpark average that out to 3.5-4 tons per year on average.
I'm not going to average out U-235 because it starts out too high and decays too quickly but if there was about 30-40% as much U-235 as U-238 when the Earth formed I can just multiply U-238's numbers by 35% and by 7/8, working out to about 1 ton per year average.
I'm doing per year/on average just for quick bad math, so 3 + 3-4 + 1 = about 7.5 tons per year x 4.5 billion years - quick estimate for how much Helium was produced by those 3 elements - very bad estimate, 33 billion tons. Most of this Helium is produced and would be trapped in the Earth's interior until plate tectonics allows it to escape. The rate of escape is above my pay-grade.
Samarium-146 is the hardest one to estimate. It's essentially all decayed into Neodymium-142 but both Samarium (it has several stable isotopes) and Neodymium are abundant, and this chain produces a single helium. It likely doesn't make a huge difference in the total. Table of elemental abundance in Earth's crust.
Atmospherically speaking, 33 billion tons is nothing. Even if we adjust to to 40 or 60 or 300 billion tons, that's still essentially too little to be an atmosphere. Helium is currently about 5.2 ppm in the atmosphere and giving the Earth's atmosphere a mass of about 5,600 trillion tons, 5.2% by molecule, adjusted for the mass ratio of helium (4) to Nitrogen/Oxygen (29), there's about 4 billion tons of Helium in the Earth's atmosphere today. If we were to say that 400 billion tons of helium was produced on Earth by radioactive decay (about 10 times my estimate), that's still a trace amount, just 100 times the current amount or 520 PPM, which is less than the Argon (currently 9,300 PPM). Argon-40 which comes from the radioactive decay of Potassium-40 (about 10% of the time), is more common because Potassium-40 is much more common than the Uranium, Thorium and Samarium-146 that decay alpha-particles.
That's a very long answer that basically says there's not enough Alpha-particle emitting elements that last long enough to survive planetary formation, that produce anything more than a trace amount of helium.
Earth also retains some terrestrial helium, about 7% of Earth's helium is terrestrial, 93% formed by radioactive decay. Most of Earth's terrestrial helium is in it's mantle, very little in it's crust.
Some interesting and loosely related reading. Yellowstone is able to release more helium than the radioactive material produce because it's crust is unique, very old and it's trapped helium for over 2 billion years, only recently started to release it.
As to your original question. As I understand it, a planet needs to be very large to acquire significant amounts of helium. By particle, the universe is about 92% hydrogen and 8% helium (or 75%-25% by mass).
Jupiter's outer atmosphere is slightly helium richer than that, about 10% helium.
Uranus' uppermost atmosphere is about 15% helium, but only it's outer edge. Uranus, unlike Jupiter and Saturn, isn't primarily a hydrogen-helium planet, it's more of a water/ice planet, sometimes called an ice giant.
It seems that planets need to be very massive to obtain significant amounts of the most common elements, hydrogen and helium, then, as the Wikipedia article implies, they need to get hot to lose their hydrogen, so helium atmosphere planets probably need to be quite large, perhaps more massive than Neptune. It might be quite a it more difficult for a typical terrestrial planet to do that, unless if it was very massive and a terrestrial planet that massive might resemble a gas giant anyway due to a very thick atmosphere.
I have a tough time thinking of any other way a helium atmosphere is likely to happen, unless you got the planet below 20 degrees K and all the other gases turned to liquid or ice, but I suspect that's not the answer you're looking for. A planet probably needs to be quite large to retain significant amounts of helium I would think. outside of a very odd circumstance like a helium white dwarf colliding with another large object inside a stellar nursery, even then, helium gas doesn't clump like ices and rocky material, but maybe such a scenario could put enough helium into the planetary formation. (bit of a crazy what-if), and I'm not sure I buy it.
I tried to reason this one out so, corrections are welcome and encouraged. I'll try to clean up any grammar a little later.
