- #36
sjbauer1215
- 20
- 3
Can't avoid getting the problem right. so you can't just answer the question you want to answer.PeroK said:There's always a "but".
Can't avoid getting the problem right. so you can't just answer the question you want to answer.PeroK said:There's always a "but".
...but I thought we agreed that the gravitational field strength increases with depth outside the inner sphere. And you have quoted yourself talking about reductions in field strength which is not the same as the "the gravitational rate of acceleration" you were talking about in the post I quoted.sjbauer1215 said:Yes again, reading comprehension:
"... the reduction of field strength inside the inner sphere will always be greater than the reduction of field strength outside of the innermost sphere [i.e. from planet surface to innermost sphere]." The use of 'than' to indicate comparison.
Only if your density approaches infinity faster than ##1/r## for all radii, which isn't plausible.sjbauer1215 said:However over all, in a nonlinear model of stratified layers of increasing density, gravity increases from the planet surface to innermost sphere towards the center of volume as long as there is any distance whatsoever.
I think the idea is that you take the 4/5 mass, 1/5 radius inner sphere, and divide itself into the outer 1/5 mass shell and 4/5 mass sphere. And so on, ad infinitum, in steps. Which, yeah. If so constructed would result in acceleration increasing without bound as the radius decreases. But it's of course unphysical, as density of real world objects doesn't increase to infinity towards the centre.Ibix said:Talking of reading comprehension, greater than what? I have been reading you as meaning "greater than the gravity at the surface of the inner sphere", in which case this is wrong - it falls linearly to zero. But perhaps you mean to compare to something else.
Ok. I didn't get that from the OP's writing at all, but it at least explains why he thinks his density is increasing without bound.Bandersnatch said:I think the idea is that you take the 4/5 mass, 1/5 radius inner sphere, and divide itself into the outer 1/5 mass shell and 4/5 mass sphere.
Bandersnatch said:I think the idea is that you take the 4/5 mass, 1/5 radius inner sphere, and divide itself into the outer 1/5 mass shell and 4/5 mass sphere. And so on, ad infinitum, in steps. Which, yeah. If so constructed would result in acceleration increasing without bound as the radius decreases. But it's of course unphysical, as density of real world objects doesn't increase to infinity towards the centre.
And in particular, all it takes for the density ##\rho (r)## to increase with decreasing radius at a slower rate than ##\rho (r) \propto 1/r## for the gravity to keep decreasing with radius.
edit: we seem to have posted at the same time
Infinity really? At least lets keep the numbers real, and the 'ad infinitum' argument is unnecessary. I don't believe we will ever reach the extreme density of a black hole.Bandersnatch said:I think the idea is that you take the 4/5 mass, 1/5 radius inner sphere, and divide itself into the outer 1/5 mass shell and 4/5 mass sphere. And so on, ad infinitum, in steps. Which, yeah. If so constructed would result in acceleration increasing without bound as the radius decreases. But it's of course unphysical, as density of real world objects doesn't increase to infinity towards the centre.
And in particular, all it takes for the density ##\rho (r)## to increase with decreasing radius at a slower rate than ##\rho (r) \propto 1/r## for the gravity to keep decreasing with radius.
edit: we seem to have posted at the same time
Yeah, that's in the Wiki page PeroK linked in #2, although left-right reversed.DaveC426913 said:Man, this thread would be only one page long if it were easy to draw and post sketches like these, rather than all this writing out of paragraphs wrought with ambiguity...
View attachment 320230
Yes, I just think we could use one that's tailored more to the OP's scenario.Ibix said:Yeah, that's in the Wiki page PeroK linked in #2, although left-right reversed.
If you want the gravity to increase with depth for all depths, it's a requirement that your density increase without bound.sjbauer1215 said:Infinity really?
Well, if we use what I understand it looks like this, if you'll forgive my lousy finger sketching (that's meant to be a ##\rho## on the vertical axis:DaveC426913 said:Yes, I just think we could use one that's tailored more to the OP's scenario.
I'd sketch it up if but I think we couldn't communicate succinctly enough to not pollute the thread with a sidebar...Ibix said:Well, if we use what I understand it looks like this, if you'll forgive my lousy finger sketching (that's meant to be a ##\rho## on the vertical axis:
Do I understand that you are assuming that the 80/20 ratio holds good at all radii, all the way down to the center?sjbauer1215 said:I always have been talking about an innermost radius of 20% the size of the original volume having a mass density that is 80% of the total volume's density. And I have been stating near the center and not at the center, all this time.
While you can take the extreme where zero distance provides for zero gravity, anything other than zero demonstrates that gravity increases toward the center of volume for this problem.
Why don't you simply write down the formula for density as function of radius that you have in mind?sjbauer1215 said:I don't believe we will ever reach the extreme density of a black hole.
Saturn may be physically large but it is not very dense. Given a sufficiently large bathtub, it would float.sjbauer1215 said:This corresponds to a gravitational acceleration of about 10.7 m/s2 (which is comparable to the gravity at the surface of Saturn!). For comparison, the gravitational acceleration at Earth’s surface is 9.81 m/s2.
The idea that black holes have a density at all is problematic. You could divide a black hole mass by ##\frac{4}{3}\pi r^3## where r is its Schwarzschild radius. But this turns out not to be a constant. The larger the hole, the lower the "density" in this sense.sjbauer1215 said:I don't believe we will ever reach the extreme density of a black hole.