Truth.You mis-spelled Babylon 5.
A transporter-less Trekverse
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I've taken a closer look at how much velocity change (delta V) is needed to get into low Earth orbit. I used the Saturn V rocket's performance, and I calculated a delta V of 9.5 km/s for getting into LEO. That's more than the 8.0 km/s Hohmann transfer orbit from the surface to 170 km, its LEO altitude. The difference is 1.5 km/s, and that's due to overcoming the Earth's gravity.
Let's now consider larger planets. Planet Models shows the smaller exoplanets with both mass and radius known. Even the lightest and smallest of them is still more massive and larger than the Earth. Up to about 3 to 3.5 Earth masses, they are all rock-iron, much like the Earth itself, but there are only 2 or 3 planets in that sample. But more than 3 or 3.5 Earth masses, some of the have lots of volatiles, especially hydrogen and helium, though some others still have densities consistent with Earthlike compositions.
I'll now use some structure calculations that exoplanet searcher Sara Seager and some others have worked on. For 3.5 Earth masses and an Earthlike composition, the radius is about 1.33 Earth radii (this model gives 0.97 Earth radii for 1 Earth mass). Its surface gravity is 1.98 times the Earth's, and its surface-satellite orbit velocity is 1.62 times the Earth's, or 12.8 km/s. That's 4.9 km/s more than the Earth's value of 7.9 km/s, and that increase is half the LEO delta V.
I estimate that the likes of the Saturn V and the Falcon 9 have close to a 20:1 mass ratio for (initial mass) / (LEO payload mass), and scaling it up means taking 20^(3/2) or 90. That means that one can put into low orbit only 1/4.5 times the mass that one can put into LEO, with the same initial rocket size. One will need an additional rocket stage, I think.
Let's now consider larger planets. Planet Models shows the smaller exoplanets with both mass and radius known. Even the lightest and smallest of them is still more massive and larger than the Earth. Up to about 3 to 3.5 Earth masses, they are all rock-iron, much like the Earth itself, but there are only 2 or 3 planets in that sample. But more than 3 or 3.5 Earth masses, some of the have lots of volatiles, especially hydrogen and helium, though some others still have densities consistent with Earthlike compositions.
I'll now use some structure calculations that exoplanet searcher Sara Seager and some others have worked on. For 3.5 Earth masses and an Earthlike composition, the radius is about 1.33 Earth radii (this model gives 0.97 Earth radii for 1 Earth mass). Its surface gravity is 1.98 times the Earth's, and its surface-satellite orbit velocity is 1.62 times the Earth's, or 12.8 km/s. That's 4.9 km/s more than the Earth's value of 7.9 km/s, and that increase is half the LEO delta V.
I estimate that the likes of the Saturn V and the Falcon 9 have close to a 20:1 mass ratio for (initial mass) / (LEO payload mass), and scaling it up means taking 20^(3/2) or 90. That means that one can put into low orbit only 1/4.5 times the mass that one can put into LEO, with the same initial rocket size. One will need an additional rocket stage, I think.
Habitable Planets for Man | RAND -- that's a very interesting book on planetary habitability published in 1964. Though it is very dated in some ways, it still has lots of valuable information. BTW, Janice Rand was named after the RAND Corporation on account of that book.
It has lots of stuff on it on what conditions that we can tolerate.
For gravity, anything more than 1.25 to 1.5 Earth gravities (g's) slows us down by a sizable fraction, and thus will not be very pleasant for us to live and work in. So that 3.5-Earth-mass planet, with its gravity twice the Earth's, would be torture for us.
About temperature, we have a maximum tolerable temperature of about 120 F / 49 C at 10% relative humidity, but only 80 F / 27 C at 100% relative humidity.
About light levels, the maximum illumination on our planet's surface is about 15 lumens/cm^2. Earth plants can photosynthesize in as much as 30 L/cm^2 about twice that, and we get uncomfortable glare at 50 L/cm^2, about 3 times that. A planet that received that much light from its star will likely be hot enough to boil water, so heat will be a problem before light becomes a problem. But even if we get a nice amount of light, if the star's photosphere is hotter than the Sun's, then we will get more ultraviolet light.
As to the atmosphere's composition, we need a partial pressure of oxygen at 0.1 to 0.5 bar; our atmosphere has 0.2 bar at sea level (total pressure = 1.013 bar, 1 bar = 10^5 pascal). But concentrations of oxygen above 0.2 bar may have another problem -- it may make it easier for vegetation fires to start. Nitrogen is at about 0.8 bar, and it causes nitrogen narcosis at about 3 bar. Carbon dioxide is at about 0.0004 bar, and it causes hypercapnia at about 0.01 bar. The book lists several other gases, like sulfur dioxide, carbon monoxide, and methane.
It has lots of stuff on it on what conditions that we can tolerate.
For gravity, anything more than 1.25 to 1.5 Earth gravities (g's) slows us down by a sizable fraction, and thus will not be very pleasant for us to live and work in. So that 3.5-Earth-mass planet, with its gravity twice the Earth's, would be torture for us.
About temperature, we have a maximum tolerable temperature of about 120 F / 49 C at 10% relative humidity, but only 80 F / 27 C at 100% relative humidity.
About light levels, the maximum illumination on our planet's surface is about 15 lumens/cm^2. Earth plants can photosynthesize in as much as 30 L/cm^2 about twice that, and we get uncomfortable glare at 50 L/cm^2, about 3 times that. A planet that received that much light from its star will likely be hot enough to boil water, so heat will be a problem before light becomes a problem. But even if we get a nice amount of light, if the star's photosphere is hotter than the Sun's, then we will get more ultraviolet light.
As to the atmosphere's composition, we need a partial pressure of oxygen at 0.1 to 0.5 bar; our atmosphere has 0.2 bar at sea level (total pressure = 1.013 bar, 1 bar = 10^5 pascal). But concentrations of oxygen above 0.2 bar may have another problem -- it may make it easier for vegetation fires to start. Nitrogen is at about 0.8 bar, and it causes nitrogen narcosis at about 3 bar. Carbon dioxide is at about 0.0004 bar, and it causes hypercapnia at about 0.01 bar. The book lists several other gases, like sulfur dioxide, carbon monoxide, and methane.
Are you taking into account changes in atmospheric density as well? Higher gravity means the atmosphere will be pulled in closer to the surface, making it more dense, meaning that you've got more drag to deal with in addition to more gravity to fight. That should reduce atmospheric delta-v fairly significantly as well.
It's all still perfectly dooable, but they'd have to put a lot more thought into how they did it. The shuttle would have to descend far away and then fly a long way at treetop level to avoid detection.
Well the Doomsday Machine one is pretty simple: instead of a transporter, they're trying to escape on Constellation's one remaining shuttlecraft, and it's so badly damaged that they can barely get the engine working in time to get off the ship. So the big dramatic moment, instead of scotty trying to get the transporter working, you have Kirk running into the hangar deck to find the shuttle's engine on fire and Scotty and/or Chekov desperately trying to get the thing working in the 30 seconds they have left before Constellation goes down the tube. They finally get the engine started and bolt through the shuttle bay at the last minute and only BARELY outrun the explosion.
And digging up Data... well, why not? It's risky as hell, but it would be interesting to see them have to dig up his body and then hightail it before the pissed-off locals had them all drawn and quartered.
That's not true at all. Actually, the best high-exhaust-velocity engines we have only work in an atmosphere. Ramjets and scramjets being the primary examples of these: they are capable of fairly high exhaust velocities but ONLY for a craft already traveling at very high speed through an atmosphere. Scramjets are (theoretically) even competitive with the best rockets we have, but again, hindered by the fact that they only work when they have air to suck down.
The low thrust from ion engines is purely a function of their power input. The extent to which ion engines are scalable to higher power outputs is not VERY well understood, but we know there are some limitations that good design can account for.
Other things like magnetoplasma rockets are VERY scalable; they will maintain their exhaust velocity under a variety of conditions and you can get them to produce a hell of a lot more thrust with more power being fed to them. As an example, a conventional magnetoplasmadynamic thruster can produce something like 1 newton of thrust for every 10kW of power fed into it. Most satellites only produce around 500 watts of electrical power from their solar panels, so this isn't a lot of thrust, but with exhaust velocities around 25km/s they're pretty fuel efficient. But if you could feed the same thruster something like 150 megawatts (about the output of a nuclear reactor) you'd have about 15kN of thrust. Similar to a modern OMS engine, but probably terribly small for something with a 150MW reactor.
Something like a VASIMR or a similar design is likely to have a much better power-to-thrust ratio, maybe 1 newton for every 4 or 5 KW. That 150MW would get you 30KN easily. Give it another 200 years of development and replacing electromagnets with forcefield generators and feed it a highly compact power source like a fusion reactor, (say, 800MW per reactor) and you could have a plasma rocket with an exhaust velocity of 80km/s and still produce a few hundred kilowatts of thrust like any other rocket.
That's only if it has the same column density, mass per unit area. Since (pressure) = (column density) * (gravity), it will go up. But if a planet has less atmosphere, then its surface pressure and density could be around ours.
The higher gravity has an interesting consequence: the planet's atmosphere would fall off more rapidly with height: (fall-off distance) ~ 1/(gravity).
I note that a higher acceleration of gravity has additional effects, like making mountains lower and inhabitants shorter -- (height) ~ 1/(gravity).
But considering the source of the atmosphere, a larger planet should, if anything, have more atmosphere than Earth, not less, shouldn't it? More total mass to begin with, so there'd be more vapor to outgas during planetary formation.
The square-cube law? That might work, though it may be hard to outgas from anything deeper than the crust and uppermost mantle.
But it must be noted that that is a big unknown. We don't know for sure whether Earthlike planets tend to form with about as much volatiles as the Earth has, whether they tend to form with much more, making ocean planets, or whether they tend to form with much less, making desert planets -- or whether they can form with a big range of volatiles.
Fair points; that is starting to get out in the weeds of planetary formation, and it's probably the sort of thing we can't really answer well without more observation of extrasolar planets.
I'll look at some numbers. The Earth's oceans have about 96.54% of the Earth's water in its crust, on it, and above it (How much water is there on Earth, from the USGS Water Science School). The total mass of "crust water" is about 1.4*10^(21) kg. However, the Earth's mass is about 5.972*10^(24) kg, meaning that the crust water is about 2.4*10^(-4) of the Earth's total mass. I've seen some estimates of how much water there is in the Earth's mantle rocks, but such estimates seem to me to be a lot of hand waving.
The Earth's crust has a mass of about 2.77*10^(22) kg (4.6*10^(-3) of the total), and the Earth's atmosphere a mass of 5.1480*10^(18) kg (8.62*10^(-7) of the total). Our atmosphere thus contains about 3.89*10^(18) kg of nitrogen, 1.19*10^(18) kg of oxygen, 6.63*10^(16) kg of argon, and 3.13*10^(15) of carbon dioxide, giving 8.54*10^(14) kg of carbon.
Abundances of the elements (data page) - Wikipedia has a collection of estimates of the mass fractions of many of them for the Earth's crust. For hydrogen, I calculate (3.88 - 4.21)*10^(19) kg, giving a possible (3.47 - 3.76)*10^(20) kg of water. This is (0.25 - 0.27) of the mass of the oceans. For carbon, it has (4.99*10^(18), 5.41*10^(18), 1.01*10^(20)) kg of carbon (estimates varying by a factor of 20). The latter figure would be enough to give the Earth's atmosphere a column density of CO2 not far from what Venus's atmosphere has (partial pressure 20 bar, Venus scaled to the Earth's gravity: 100 bar). Finally, nitrogen has (5.3, 5.5)*10^(17) kg and argon 9.7*10^(16) kg.
From that data page, seawater contains 3.9*10^(16) kg of carbon, (7.0*10^(14), 2.24*10^(16)) kg of nitrogen, and 6.3*10^(14) kg of argon.
If the Earth had Venus's atmosphere's relative amount of carbon in its crust, it would have 8.5*10^(-5) of its total mass as carbon. That's about 1/3 of the mass of the Earth's crust water, most of it in the oceans.
The Earth's crust has a mass of about 2.77*10^(22) kg (4.6*10^(-3) of the total), and the Earth's atmosphere a mass of 5.1480*10^(18) kg (8.62*10^(-7) of the total). Our atmosphere thus contains about 3.89*10^(18) kg of nitrogen, 1.19*10^(18) kg of oxygen, 6.63*10^(16) kg of argon, and 3.13*10^(15) of carbon dioxide, giving 8.54*10^(14) kg of carbon.
Abundances of the elements (data page) - Wikipedia has a collection of estimates of the mass fractions of many of them for the Earth's crust. For hydrogen, I calculate (3.88 - 4.21)*10^(19) kg, giving a possible (3.47 - 3.76)*10^(20) kg of water. This is (0.25 - 0.27) of the mass of the oceans. For carbon, it has (4.99*10^(18), 5.41*10^(18), 1.01*10^(20)) kg of carbon (estimates varying by a factor of 20). The latter figure would be enough to give the Earth's atmosphere a column density of CO2 not far from what Venus's atmosphere has (partial pressure 20 bar, Venus scaled to the Earth's gravity: 100 bar). Finally, nitrogen has (5.3, 5.5)*10^(17) kg and argon 9.7*10^(16) kg.
From that data page, seawater contains 3.9*10^(16) kg of carbon, (7.0*10^(14), 2.24*10^(16)) kg of nitrogen, and 6.3*10^(14) kg of argon.
If the Earth had Venus's atmosphere's relative amount of carbon in its crust, it would have 8.5*10^(-5) of its total mass as carbon. That's about 1/3 of the mass of the Earth's crust water, most of it in the oceans.
Earth-interior water:
Scientists Detect Evidence of 'Oceans Worth' of Water in Earth's Mantle - Astrobiology Magazine
Massive 'ocean' discovered towards Earth's core | New Scientist
Found! Hidden Ocean Locked Up Deep in Earth's Mantle
The Earth's mantle has a "transition zone" that is between 440 and 610 km below the surface. Some diamonds that came from there have some small mineral grains with trapped water (about 1% by mass from the Astrobiology article), trapped by being chemically bound as hydroxide ions. From that Astrobiology article,
Using a density of 4 g/cm^3, transition zone has a mass of about 3*10^(23) kg, giving a water mass of about 3*10^(21) kg. That's about 2 Earth oceans of water.
Scientists Detect Evidence of 'Oceans Worth' of Water in Earth's Mantle - Astrobiology Magazine
Massive 'ocean' discovered towards Earth's core | New Scientist
Found! Hidden Ocean Locked Up Deep in Earth's Mantle
The Earth's mantle has a "transition zone" that is between 440 and 610 km below the surface. Some diamonds that came from there have some small mineral grains with trapped water (about 1% by mass from the Astrobiology article), trapped by being chemically bound as hydroxide ions. From that Astrobiology article,
Using a density of 4 g/cm^3, transition zone has a mass of about 3*10^(23) kg, giving a water mass of about 3*10^(21) kg. That's about 2 Earth oceans of water.
That 20-bar estimate earlier was for carbon alone. If that carbon is combined with oxygen to make CO2, then I get 70 bar, not much less than the Earth-scaled value for Venus's atmosphere, 100 bar.
That aside, there is reason to suspect that the amount of volatiles can vary wildly in Earthlike planets. The variation can come from a rather odd source: inspiraling Jovian planets. Numerous "warm Jupiters" and "hot Jupiters" are now known, planets that spiraled in after forming in colder spots, spiraling in because of interaction with the protoplanetary disk. One may expect that they would keep planets from forming, but something odd happens instead: much of the material that gets scattered by passing close to them nevertheless remains, and stays available to form planets. Furthermore, an inspiraling Jovian planet will mix the protoplanetary disk, bringing watery material inward, and making possible Earthlike planets with superdeep oceans.
At arxiv.org:
[astro-ph/0610314] On the formation of terrestrial planets in hot-Jupiter systems
[astro-ph/0701048] Formation of Earth-like Planets During and After Giant Planet Migration
[1209.5125] The Compositional Diversity of Extrasolar Terrestrial Planets: II. Migration Simulations
The authors of that last one considered what would happen in the absence of giant-planet mixing: [1004.0971] The Compositional Diversity of Extrasolar Terrestrial Planets: I. In-Situ Simulations -- the protoplanetary disk does not get very mixed, and a habitable-zone Earthlike planet would likely be a desert planet with a thin atmosphere. The second paper used a composition of 10^(-5) water by mass. For the Earth, this is equivalent to an average depth over the entire planet's surface of 120 m. The corresponding figure for the Earth's oceans is around 2700 m. Their actual average depth is about 3800 m, but they cover 71% of the Earth's surface, thus the difference.
For a 120-m average ocean depth, the other volatiles may scale similarly, giving an atmosphere pressure for the Earth's gravity of only 0.04 - 0.05 times ours. Also, there may not be much free water on the surface -- most of it may be chemically bound to the crustal rocks, meaning much less surface and atmospheric water.
Going in the other direction, some mixing simulations have a strong likelihood of producing huge oceans, with a median mass of about 100 times the Earth's oceans. Imagine 270-km-deep oceans. Even the tallest mountains would have heights only a small fraction of the ocean's depth.
That aside, there is reason to suspect that the amount of volatiles can vary wildly in Earthlike planets. The variation can come from a rather odd source: inspiraling Jovian planets. Numerous "warm Jupiters" and "hot Jupiters" are now known, planets that spiraled in after forming in colder spots, spiraling in because of interaction with the protoplanetary disk. One may expect that they would keep planets from forming, but something odd happens instead: much of the material that gets scattered by passing close to them nevertheless remains, and stays available to form planets. Furthermore, an inspiraling Jovian planet will mix the protoplanetary disk, bringing watery material inward, and making possible Earthlike planets with superdeep oceans.
At arxiv.org:
[astro-ph/0610314] On the formation of terrestrial planets in hot-Jupiter systems
[astro-ph/0701048] Formation of Earth-like Planets During and After Giant Planet Migration
[1209.5125] The Compositional Diversity of Extrasolar Terrestrial Planets: II. Migration Simulations
The authors of that last one considered what would happen in the absence of giant-planet mixing: [1004.0971] The Compositional Diversity of Extrasolar Terrestrial Planets: I. In-Situ Simulations -- the protoplanetary disk does not get very mixed, and a habitable-zone Earthlike planet would likely be a desert planet with a thin atmosphere. The second paper used a composition of 10^(-5) water by mass. For the Earth, this is equivalent to an average depth over the entire planet's surface of 120 m. The corresponding figure for the Earth's oceans is around 2700 m. Their actual average depth is about 3800 m, but they cover 71% of the Earth's surface, thus the difference.
For a 120-m average ocean depth, the other volatiles may scale similarly, giving an atmosphere pressure for the Earth's gravity of only 0.04 - 0.05 times ours. Also, there may not be much free water on the surface -- most of it may be chemically bound to the crustal rocks, meaning much less surface and atmospheric water.
Going in the other direction, some mixing simulations have a strong likelihood of producing huge oceans, with a median mass of about 100 times the Earth's oceans. Imagine 270-km-deep oceans. Even the tallest mountains would have heights only a small fraction of the ocean's depth.
^ Yeah, but that's hydrated minerals, not "water" per se. Conditions that would result in those minerals breaking down and actually producing liquid water in the sense you're describing aren't even that common to Earth, let alone to the other planets in the solar system. A highly geologically active mantle might release some water vapor over time, but how much and how fast and for how long it does so will depend largely on the amount of radioactive material in the lower mantle and the core, and to a certain extent the extent of tidal forces on that planet caused by its parent body.
Put that another way: if Earth had formed with fewer radioactives, those hydrates would have stayed in the mantle and this planet would be as dry and as desolate as the moon. If, on the other hand, Earth had been captured into a stable orbit by a hot jupiter (even one still in the habitable zone) tidal forces would have made the planet so geologically active that we'd have an insanely thick and exotic atmosphere rich with water, methane and sulfuric compounds.
Put that another way: if Earth had formed with fewer radioactives, those hydrates would have stayed in the mantle and this planet would be as dry and as desolate as the moon. If, on the other hand, Earth had been captured into a stable orbit by a hot jupiter (even one still in the habitable zone) tidal forces would have made the planet so geologically active that we'd have an insanely thick and exotic atmosphere rich with water, methane and sulfuric compounds.
There is a weird issue with uranium. U-238 has a half-life of 4.468 billion years, about the Earth's age. U-235 has a half-life of 703.8 million years, much less. U-235 is about 0.72% of the uranium in the present-day Earth's crust, but when the Earth formed, about 4.55 billion years ago, its amount of U-235 was about 0.31 times its amount of U-238.
Looking at another long-lived radioisotope, thorium-232, there is about 3.6 times as much of it in the Earth's crust as there is of uranium. For the Solar System overall, likely from meteorites, it's 2.5 - 2.7, and I'll use 2.6. Its half-life is 14.05 billion years, and looking back to the formation of the Solar System, I find 2.2 for the Earth's crust and 1.6 for the Solar System.
Judging from the oldest galaxies observed, our Galaxy is about 13 billion years old, and that means over 8 billion years before the Solar System formed. Assuming a constant rate, the effective formation time for U-235 is (1 - (fraction left from beginning)) * ((its mean life) = (its half-life) / ln(2)) = 1.015 billion years. Likewise, for U-238, it is 4.6 billion years, and for Th-232, it is 6.6 billion years.
That means that at the Earth's formation, only about 60% of the U-238 that formed before then had remained, and I estimate formation rates relative to U-238: for U-235, 1.4 and for Th-232, 1.1. That seems about right for their having similar nuclear composition and likely for their being produced in the same way, by the "r-process".
Looking at another long-lived radioisotope, thorium-232, there is about 3.6 times as much of it in the Earth's crust as there is of uranium. For the Solar System overall, likely from meteorites, it's 2.5 - 2.7, and I'll use 2.6. Its half-life is 14.05 billion years, and looking back to the formation of the Solar System, I find 2.2 for the Earth's crust and 1.6 for the Solar System.
Judging from the oldest galaxies observed, our Galaxy is about 13 billion years old, and that means over 8 billion years before the Solar System formed. Assuming a constant rate, the effective formation time for U-235 is (1 - (fraction left from beginning)) * ((its mean life) = (its half-life) / ln(2)) = 1.015 billion years. Likewise, for U-238, it is 4.6 billion years, and for Th-232, it is 6.6 billion years.
That means that at the Earth's formation, only about 60% of the U-238 that formed before then had remained, and I estimate formation rates relative to U-238: for U-235, 1.4 and for Th-232, 1.1. That seems about right for their having similar nuclear composition and likely for their being produced in the same way, by the "r-process".
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That tells us exactly nothing, since we don't have a reliable way to estimate its abundance in the Earth's mantle
I seriously doubt that there has been much fractionation of uranium isotopes in the Earth's crust. They are chemically the same and their atomic masses are not very different.
I'll set the Earth's present U-238 to 1. The abundances of U-238, U-235, and Th-232 are thus (1, 0.0072, 2.6). To get the radioactive heating that they make, I'd need their energy per radioactive decay, and I'll hand-wave that by treating them as equal. Thus, relative to the present total, it's (U-238, U-235, Th-232) = (0.53, 0.024, 0.44) adding up to 1.
When the Earth was formed, it was (1.09, 2.16, 0.55) with total 3.79. At 4 billion years ago, it was (0.99, 1.25, 0.54): 2.79, and at 3.5 billion years ago, (0.92, 0.77, 0.52): 2.21. At 2 billion years ago, (0.73, 0.17, 0.49): 1.39. Five billion years from now: (0.25, 0.00018, 0.35): 0.59.
Now for a planet that formed when our Galaxy was only a billion years old. The Earth started out with abundances of (U-238, U-235, Th-232) = (2.03, 0.64, 3.25). This planet would start out with abundances (0.41, 0.40, 0.48). Its heating at formation would be (0.22, 1.35, 0.08):1.65. After a billion years, it's down to (0.19, 0.50, 0.078): 0.77. After 4.5 billion years, (0.11, 0.016, 0.065): 0.19.
So a very early planet won't get heated up nearly as much as the Earth does.
I'll set the Earth's present U-238 to 1. The abundances of U-238, U-235, and Th-232 are thus (1, 0.0072, 2.6). To get the radioactive heating that they make, I'd need their energy per radioactive decay, and I'll hand-wave that by treating them as equal. Thus, relative to the present total, it's (U-238, U-235, Th-232) = (0.53, 0.024, 0.44) adding up to 1.
When the Earth was formed, it was (1.09, 2.16, 0.55) with total 3.79. At 4 billion years ago, it was (0.99, 1.25, 0.54): 2.79, and at 3.5 billion years ago, (0.92, 0.77, 0.52): 2.21. At 2 billion years ago, (0.73, 0.17, 0.49): 1.39. Five billion years from now: (0.25, 0.00018, 0.35): 0.59.
Now for a planet that formed when our Galaxy was only a billion years old. The Earth started out with abundances of (U-238, U-235, Th-232) = (2.03, 0.64, 3.25). This planet would start out with abundances (0.41, 0.40, 0.48). Its heating at formation would be (0.22, 1.35, 0.08):1.65. After a billion years, it's down to (0.19, 0.50, 0.078): 0.77. After 4.5 billion years, (0.11, 0.016, 0.065): 0.19.
So a very early planet won't get heated up nearly as much as the Earth does.
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Now to an interesting conundrum about the numerous warm and hot Jovian planets. Why are the Solar System's Jovian planets cold ones? Why haven't they spiraled in? There is a proposed scenario called the Grand Tack where Jupiter and Saturn did that for a while. But then the two planets got into an orbital resonance that led to Saturn pulling Jupiter back out again.
A low mass for Mars from Jupiter/'s early gas-driven migration : Nature : Nature Publishing Group: Full text here
How Did Jupiter Shape Our Solar System?, NASA - Jupiter's Youthful Travels Redefined Solar System
Jupiter got to where Mars is, before Saturn pulled it outward. As Jupiter went inward and then outward, it scattered a lot of watery-rock material inward and a lot of dry-rock material outward. That explains why the inner planets have their water. It also scattered away a lot of material near where Mars would eventually form, and when it did, it has less than 1/9 the Earth's mass.
I also wish to note [1610.03460] Formation and composition of planets around very low mass stars -- some of the simulated planets end up with sizable fractions of water. A big scatter, but with an average around 20% - 30% water by mass. For the Earth's size, an ocean around 1000 km deep.
A low mass for Mars from Jupiter/'s early gas-driven migration : Nature : Nature Publishing Group: Full text here
How Did Jupiter Shape Our Solar System?, NASA - Jupiter's Youthful Travels Redefined Solar System
Jupiter got to where Mars is, before Saturn pulled it outward. As Jupiter went inward and then outward, it scattered a lot of watery-rock material inward and a lot of dry-rock material outward. That explains why the inner planets have their water. It also scattered away a lot of material near where Mars would eventually form, and when it did, it has less than 1/9 the Earth's mass.
I also wish to note [1610.03460] Formation and composition of planets around very low mass stars -- some of the simulated planets end up with sizable fractions of water. A big scatter, but with an average around 20% - 30% water by mass. For the Earth's size, an ocean around 1000 km deep.
That also doesn't mean anything. The distribution of 238 to 235 would depend entirely on what the ratio originally was to begin with. If the early mantle had a much higher proportion of the former than the crust, the proportions will still be higher today. That's just the way half life works: the heavier isotope decays over time, but SINCE it takes a predictable amount of time, the only variable is actually the starting conditions, which we really have no way of knowing.
Why? Again, you have no way of establishing whether or not these really WERE the initial conditions, let alone whether or not alpha or beta decay were their only behaviors at this point. On the one hand, forming closer to the sun than most of the rocks of the belt or even the wandering NEOs means the proportion of heavy metals could easily be much higher, favoring uranium more strongly than in the outer solar system. On the other hand, there are circumstances where even uranium 235 can reach critical mass NATURALLY and can undergo spontaneous fission into lighter elements, in which case the initial conditions could be dramatically different than that found in modern day asteroids. Other radioactives and secondary decay products also produce heat and have their own critical masses that could potentially be reached under the high pressure and density of the lower mantle. We have virtually no reliable data about what's actually happening down there, let alone what was happening 4 billion years ago.
That, again, is not something you have any way of saying with confidence. A planet generated from a third or fourth generation star might actually have little or no uranium at all because criticality events in the cores of ancient planets broke it down into lighter elements early on. Or it might have a stupendously high abundance of it due to the influence of another gravitating body creating a mass sink for the heaviest elements. There are too many variables in this -- and also too many unknowns BY FAR -- to attempt to make the kind of predictions you're making.
Put it another way: you're basically putting yourself in the position of trying to predict a person's life expectancy based on the size of his mother's car. That's useful information and all, but you can only extrapolate so far.
That would require some extreme isotope fractionation.
There is an indirect way of checking. Comparing ages measured with different radionuclides. U-238 and U-235 and Th-232 and K-40 ages ought to show systematic discrepancies. But we don't see any evidence of that. Furthermore, the radiometric-dating ages of the oldest Solar-System material agree with the age of the Sun derived from helioseismology interior probing and stellar-evolution calculations.
