Editing Talk:2908: Moon Armor Index
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:A=4πr², so A<sub>dif</sub> of A<sub>2</sub>-A<sub>1</sub> would be (4πr<sub>2</sub>²)-(4πr<sub>1</sub>²) or 4π(r<sub>2</sub>²-r<sub>1</sub>²) ((which looks like you could work it out as a pythogorean calculation, i.e. model a new line-length that would go at a tangent out from r<sub>1</sub> until it hits the endpoint of the r<sub>2</sub> radius elsewhere ... but that's probably not useful!)). | :A=4πr², so A<sub>dif</sub> of A<sub>2</sub>-A<sub>1</sub> would be (4πr<sub>2</sub>²)-(4πr<sub>1</sub>²) or 4π(r<sub>2</sub>²-r<sub>1</sub>²) ((which looks like you could work it out as a pythogorean calculation, i.e. model a new line-length that would go at a tangent out from r<sub>1</sub> until it hits the endpoint of the r<sub>2</sub> radius elsewhere ... but that's probably not useful!)). | ||
:Given Earth at a normal 6371km (between equatorial and polar radii, to simplify as a true sphere), Earth+Moon therefore 6371+43 (using figure stated by comic), that gives ...if I've done it right... now an extra 7 million km² on top of the roughly 510 million that it normally has. An increment of 5%, by the time you start spreading your arbitrarily thin final layer (so approximate back to being 2.5% extra by volume, without actually using Eebster's alternate direct shell-volume calculation or doing an integration). | :Given Earth at a normal 6371km (between equatorial and polar radii, to simplify as a true sphere), Earth+Moon therefore 6371+43 (using figure stated by comic), that gives ...if I've done it right... now an extra 7 million km² on top of the roughly 510 million that it normally has. An increment of 5%, by the time you start spreading your arbitrarily thin final layer (so approximate back to being 2.5% extra by volume, without actually using Eebster's alternate direct shell-volume calculation or doing an integration). | ||
| − | :Pluto (saying 44km of layering, as slightly more than Earth's 'pile', on its far smaller radius) isn't that much more 'off'. It would increase the surface by about 8% (so says my mental arithmatic, at least) so maybe 4% more volume than a "flat surface raised up | + | :Pluto (saying 44km of layering, as slightly more than Earth's 'pile', on its far smaller radius) isn't that much more 'off'. It would increase the surface by about 8% (so says my mental arithmatic, at least) so maybe 4% more volume than a "flat surface raised up primsmatically". |
:(Not quite the same as "wrap a string around a tennis ball, add an inch to its length, what is its additional radius? / wrap a string around the Earth, add an inch ..." sort of thing, due to the extra dimensionality involved, but I don't feel like doing the full algebraic differentiations necessary to establish the trend of departure.). | :(Not quite the same as "wrap a string around a tennis ball, add an inch to its length, what is its additional radius? / wrap a string around the Earth, add an inch ..." sort of thing, due to the extra dimensionality involved, but I don't feel like doing the full algebraic differentiations necessary to establish the trend of departure.). | ||
| − | :It certainly initially looks like the '≈'ing of the result holds fairly well under even the two most extreme examples (cases of particularly large moons-by-volume). And, at a certain point, a planet's (single largest) moon cannot be made bigger without drifting into double-planet territory (indeed, Pluto/Charon may be | + | :It certainly initially looks like the '≈'ing of the result holds fairly well under even the two most extreme examples (cases of particularly large moons-by-volume). And, at a certain point, a planet's (single largest) moon cannot be made bigger without drifting into double-planet territory (indeed, Pluto/Charon may be considerd double-dwarfs!), and then, soon after, you're switching their roles around and dismantling the 'planet' (really a moon) to armour the 'moon' (now the planet). So that probably suggests we're at our limit, with twin-binary capping our one-satellite scenarios, until you get into 'busy' N-ary systems with many not-insignificant moons but somehow an identifiable 'main body' planet in the midst of them. |
:I don't think "armour the Sun with all the planets (''and'' their moons), dwarf-planets, minor-planets, random detritus, etc" will strain that relationship. Top of my head estimate is that it'd be nowhere near as high as Earth/Pluto examples, if the Oort cloud isn't oddly massive in total. But someone can correct me if I've goofed or overly hand-waved something. [[Special:Contributions/172.69.195.118|172.69.195.118]] 06:35, 19 March 2024 (UTC) | :I don't think "armour the Sun with all the planets (''and'' their moons), dwarf-planets, minor-planets, random detritus, etc" will strain that relationship. Top of my head estimate is that it'd be nowhere near as high as Earth/Pluto examples, if the Oort cloud isn't oddly massive in total. But someone can correct me if I've goofed or overly hand-waved something. [[Special:Contributions/172.69.195.118|172.69.195.118]] 06:35, 19 March 2024 (UTC) | ||
::If you start with a ball of radius r₀, then its volume is V = 4/3πr₀³, its surface area is 4πr₀², and the derivative of its radius with respect to its volume (and thus its mass, to within a constant, roughly), is dr/dV evaluated at r₀, or 1/(4πr₀²). So a linear approximation is r = r₀ + v/(4πr₀²), where v is the added volume. On the other hand, the exact calculation is v = 4/3π(r³–r₀³), giving r = ³√(r₀³+3v/(4π)). This has the following MacLaurin series: | ::If you start with a ball of radius r₀, then its volume is V = 4/3πr₀³, its surface area is 4πr₀², and the derivative of its radius with respect to its volume (and thus its mass, to within a constant, roughly), is dr/dV evaluated at r₀, or 1/(4πr₀²). So a linear approximation is r = r₀ + v/(4πr₀²), where v is the added volume. On the other hand, the exact calculation is v = 4/3π(r³–r₀³), giving r = ³√(r₀³+3v/(4π)). This has the following MacLaurin series: | ||
