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Watsisname
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21 Feb 2018 08:31

The singularity is at the very center of the black hole. Its properties, whatever they might be, don't affect the properties of the outside of the hole, or the event horizon, or even most of the interior. You could replace singularity with a small but not infinetissimal sized spherical mass and all other properties of the hole will be precisely the same.

The reason general relativity predicts a singularity at the center is that, once a horizon forms, nothing can prevent continued collapse. Where this prediction probably breaks down is on a scale where quantum gravitation effects become important for describing the curvature, which will be extremely close to the singularity point anyway.
 
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21 Feb 2018 10:16

Watsisname wrote:
Source of the post It was theorized as a solution to an outstanding problem in the standard model, and it was verified by experiment.  That's as close to necessary and reasonable as anything gets.

My concern is that all they did was predict a particle and observe it, and did not verify its properties. So far what I have read on the Higgs mechanism, there is a lot of talk, a lot of math, but I see nothing saying "look, this is what it does", just "this is what it should do"... What's concerning to me about that is that they could easily get the properties of a particle and the circumstances of its stability mixed up, being able to predict a particle, but not its properties. As in, like correlating a planet's orbit in a resonance with its composition, sure it can be accurate, "there is this resonance, so a planet should be at this distance", and it could align with composition being that different materials exist differently depending on its distance to the star, but the correlation is actually two separate things. Just concerned, thats all.
Watsisname wrote:
Source of the post If you recall from special relativity that simultaneity is relative, this means that different observers have different notions of what constitutes a simultaneous slicing of the space-time.  They will have different definitions for "constant time" slice with which to define the volume enclosed, and for a black hole they will yield different answers.  It may even be zero!

Hmm... Would it be possible to find the volume for every possible observer?
Watsisname wrote:
Source of the post I don't mean for this to sound condescending, but what's more likely is that you don't understand it at a sufficiently technical level to be able to make sense of it.

I do not. But I hope someday I will! Believe it or not physics is my worst subject, partially due to the fact that I only use algebraic notation in my math and cannot comprehend the other notations (which people around me seem to get immediately :/), but this is also why I am so persistent. But something about this is different, often my mistakes would be not realizing something, or looking at things incorrectly, but something about the Higgs mechanism feels off. Aside from the weird natures of the time variable neutrinos and some how defined difference between a Photon and Gluon, I fundamentally do not understand the Higgs field. The boson itself is reasonable, what it does seems to contradict too much for me. Perhaps, by learning more, I will come to understand its nature and comprehend its existence, but out of everything, it is the most strange. The very process which the universe came into existence would make far more sense than the Higgs field, which could be said that since it has happened it is consequential to some process, that, at some level, makes sense. I just am concerned that we might be seeing the right things but saying the wrong perceptions, idk :/


And as always, leaving off with a question...

Is spacetime hyperbolic? I asked the question on stack exchange but got ill reception on it... I think either people misunderstood my question, or I misunderstood their answers... But what I mean by hyperbolic, is not the actual geometry of space, but rather of space over time. If the universe exponentially expands, if observed past to future at once, would it not appear to be hyperbolic? Less space back in time and more space forward in time?


Finally, is there a symmetry in gravity-bound systems? Two planets orbiting for instance, if we traced out their past and future, wouldn't reflecting over the time axis not change anything? It should be the same system if it was right handed or left handed in time, so would this be a real symmetry or can it be violated?
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21 Feb 2018 10:17

But if some property is discovered or theorised that prevents the ultimate collapse, that wouldn't kill general relativity if it doesn't affect the properties outside, would it?
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21 Feb 2018 19:39

midtskogen wrote:
Source of the post But if some property is discovered or theorised that prevents the ultimate collapse, that wouldn't kill general relativity if it doesn't affect the properties outside, would it?

No, but depending on where/how the collapse is halted, it may be more or less surprising and difficult to reconcile with general relativity.  Suppose for example that the collapse is somehow halted just after forming the horizon of a supermassive black hole.  This would be a serious contradiction to principles of GR, because the space-time curvature at that point is still very weak.  Nothing terribly exotic is happening to the space-time there, so GR should still be perfectly valid in predicting continued collapse.

Whereas what we expect is that GR's core assumptions (like space-time being smooth on all scales) must break down near the singularity condition, since they contradict principles of quantum mechanics.  It would not be too surprising at all if this turns out to be how singularities are removed.

In either case (whether something surprising happens that requires revising general relativity, or if the expected turns out and we develop GR into quantum gravitation), I would not say that GR is killed so much as extended to a wider set of conditions.  The extended theory must still reduce to GR and produce the same predictions as before for the geometry outside the hole.  It must be consistent with all observations that so far agree with GR.

Terran wrote:
Source of the post Hmm... Would it be possible to find the volume for every possible observer?

Sure, although I am not sure how useful that would be for understanding a black hole's properties.  In addition to the volume not being invariant, it is generally also not time-independent.  In particular, you could choose to define the maximum possible interior volume of the black hole.  It turns out that this volume grows with time, even if the black hole's mass and horizon area stay the same!  So you definitely cannot think of a black hole as a region that encloses a volume related to its surface area.  The amount of volume it contains is a very tricky concept!


Terran wrote:
Source of the post Is spacetime hyperbolic?

Edited:  I miswrote here earlier and said that it is hyperbolic, but that isn't quite right.  I was mixing up the geometry with how events transform in that geometry.  I've clarified this below.

The geometry of space-time is Minkowskian.  This is not in itself hyperbolic, but it does have important features of hyperbolic geometry.  To see this, it is useful to start in the space-time of special relativity.  Expansion of space over time is just an extension of that, where the spatial component grows with time.

The space-time of special relativity is flat (the spatial component of it is Euclidean) with a time component that does not merge with it in a Euclidean way (so it is not like Euclidean 4D space).  It has the metric ds2 = dx2 + dy2 + dz2 - c2dt2.  That minus sign is crucial, and is what makes it Minkowskian rather than Euclidean.  

The quantity ds2 has a deep physical meaning: it is the (squared) separation between events measured through space-time, or the "space-time interval".  Some weirdness crops up here: because of the minus sign, it is possible for what seem to be widely separated events to actually have zero space-time separation.  These correspond to events that are equally distant in space and time (like one light year apart in space and one year apart in time), and as you might guess, this exactly corresponds to paths that light can take.  Light in vacuum travels a distance of zero in space-time.  So if you set ds2 = 0, you construct the light cone out of the geometry.

Another key property of the space-time interval is that it is invariant.  All observers agree on the space-time interval between two events.  So if you measure space and time coordinates of events and build a space-time map, and then move into a different reference frame, the space and time coordinates of events will change, but they must change in a specific way in order for the space-time interval to remain the same.  That transformation will be along hyperbolas.  To see this, think of the curves generated by y2 - x2 = k, for k a constant.

We might ask of other types of space-times, like for the large scale expanding universe (FLRW metric), or around a spherical mass distribution (Schwarzschild metric), or a rotating black hole (Kerr metric).  The global geometry in these cases may be complicated.  The rotating black hole for example has a cross relationship between time and longitude angle, because of the rotation (frame dragging).  But in all cases, the geometry is at least locally Minkowskian.  If you limit your view enough, curved spaces appear flat.  This is in essence a principle of special relativity.  "Special" relativity means "limited" relativity, in the sense that we are limiting the size of the region being analyzed such that the effects of curvature (the tidal forces) are too small to measure.


Terran wrote:
Source of the post Finally, is there a symmetry in gravity-bound systems?  Two planets orbiting for instance, if we traced out their past and future, wouldn't reflecting over the time axis not change anything?


Absolutely, provided there are no damping mechanisms (gravitational wave emission, for instance).
 
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22 Feb 2018 09:25

Watsisname wrote:
Source of the post provided there are no damping mechanisms

How does that break the symmetry? I'm gonna guess, unless I am wrong, that you mean a changing orbit over time. Wouldn't two spiraling-inward planets still have this symmetry regardless?
Watsisname wrote:
Source of the post The geometry of space-time is Minkowskian.  This is not in itself hyperbolic, but it does have important features of hyperbolic geometry.  To see this, it is useful to start in the space-time of special relativity.  Expansion of space over time is just an extension of that, where the spatial component grows with time.

I guess what I mean is... If I looked farther out, all the way to the universe's horizon, would it not appear anti-hyperbolic? The farther away things are, the larger they appear, and the less space there is? And then if I look "inwards", referring to a negative distance in spacetime where the distance away from me still grows but referring now to a future time coordinate, would the future horizon now look exponentially larger? The future horizon, to clarify, being what you would see if you saw further in time the farther you looked (negative distance). For me, a flat spacetime refers to a Euclidean-flat, the angles add up to 180 and follow Euclidean rules, "flat". Hyperbolic being that the angles do not add to 180, less than, and the observed phenomena is more space over distance. When I say anti-hyperbolic, I mean at least, where the phenomena is the reverse, where you get less space over a distance. So for me, more space over distance sounds like hyperbolic, yes its Minkowski space but our universe is not eternal so its size changes over time, and if you include time in the definition of flatness, it does not sound flat.
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23 Feb 2018 00:49

Terran wrote:
Source of the post How does that break the symmetry?

A system where energy is dissipated is not time symmetric.  Almost by definition.  Rewind the clock on a system that dissipates energy, and it will look very different in the past than it does in the future.  Or to see it another way, reverse all the motions but have it obey the same equations.  It won't reverse the system's evolution, because the  dissipation mechanisms are not reversed.  For orbits, reversing the motion does not reverse the inspiral due to gravitational wave emission.  In essence this is the 2nd law of thermodynamics.  The universe is generally not time symmetric.

https://en.wikipedia.org/wiki/T-symmetry

Terran wrote:
Source of the post I guess what I mean is... If I looked farther out, all the way to the universe's horizon, would it not appear anti-hyperbolic? The farther away things are, the larger they appear, and the less space there is?

I think I get what you mean.  Absolutely, if you look far enough out in the universe, objects do start increasing in angular size.  In cosmology we must account for this by defining the angular diameter distance, and you can think of it as a feature of non-Euclidean geometry.  It is like being in a positively curved space, qualitatively similar to what it would look like if you lived on a spherical surface that light rays travel along.  Objects on the opposite hemisphere start looking larger than they would for their distance (measured along the surface) in flat geometry.  An object on the antipode would fill your horizon.

However, this is not actually telling you the space-time geometry (at least not in such a straightforward way). The problem is that you are only seeing a very limited part of the space-time!  You are only viewing the specific slice along which light rays travel.  So to connect with the rest of the space-time you must use some other knowledge.  Since the universe is homogenous and isotropic, you can apply general relativity, thereby obtaining the Friedmann–Lemaître–Robertson–Walker metric (FLRW) metric, and the Friedmann equations.  These are what describe the global geometry, as well as how the universe evolves over time.

What we find by connecting this math with the observations is that the universe is locally Minkowskian, and has globally flat spatial curvature (to within current precision of measurements).  To describe the full space-time geometry, the closest description is that it is like a 4D hyperboloid (again can see from the metric), but this is complicated because the expansion rate is a function of time.  Furthermore, how you describe the space-time geometry in the usual 2D or 3D terms will depend on the slicing.  You can't conclude that the global space-time geometry is simply "hyperbolic", nor is it particularly useful for understanding the universe or doing calculations with it.

To summarize, understanding that the expansion affects how things look across large distances is very important in cosmology.  We cannot apply Euclidean relationships between sizes, distances, angles, and apparent magnitudes of objects. The expansion rate and how it changes with time are connected to the space-time geometry, and the mass-energy contents of the universe, and we use the FLRW metric to describe it. 


Terran wrote:
Source of the post And then if I look "inwards", referring to a negative distance in spacetime where the distance away from me still grows but referring now to a future time coordinate, would the future horizon now look exponentially larger?  The future horizon, to clarify, being what you would see if you saw further in time the farther you looked (negative distance). 

Looking toward the future horizon does not mean looking to negative distances in space-time.  Not only can you not look to negative space-time distances, you also cannot look to positive space-time distances.  What you view is always along null paths, where the space-time distance ds2 = zero, because these paths are defined by the speed of light.  Again we only see a specific slice of the space-time.

With that aside, the future horizon depends on the evolution of expansion rate (which for the future evolution depends critically on the equation of state for dark energy), and it also depends on the event with which you generate the light cone to define it.  If we choose the standard LCDM model with dark energy equation of state w=1 (the cosmological constant), then the future light cone generated from us on Earth today will reach to about 5Gpc in co-moving distance.  This is a consequence of the exponential, dark-energy driven expansion, which brings the cosmic event horizon inward.
 
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23 Feb 2018 09:52

Watsisname wrote:
Source of the post Rewind the clock on a system that dissipates energy, and it will look very different in the past than it does in the future.

I wasn't really referring to a forwards-backwards time symmetry, rather a left-right symmetry. If I graphed out two inwards spiraling planets, colliding into one object, I would have an inwards curved hyper-coil becoming a kind of hyper-cylinder with spherical sides, that besides the point, I would have a time axis on the graph, and if I make that axis' origin the middle of the event, shouldn't a reflection over it be perfectly symmetric, yielding the same outcome? So I am not changing the "time" of the "function" but rather its orientation in time, remaining completely in the plane at a 90, or now, 180 degree angle, but still traveling normally through time?
Watsisname wrote:
Source of the post What you view is always along null paths, where the space-time distance ds2 = zero, because these paths are defined by the speed of light.  Again we only see a specific slice of the space-time.


Ya... I was kinda describing an impossible perspective, like if the universe was upside-down? Where I saw the future light cone like the past light cone? As if future events started further back in time.
This question comes from me trying to draw the entire universe (cause, why not), from beginning to end, by having t=0 as a changing curve, like a downwards curved spike, which represents the apparent size of the observable universe (where the entire past light cone is a cross section of it). Drawing a straight line from "us" to the end of the spike measures to be 13.8 billion years, the age of where we stand, but a horizontal line measures something like 46.6 billion light years to the t=0 mark, and this shape warps the paths of everything being drawn, where an object falls away from the future light cone at 7.2 billion years where the curve exceeds 45 degrees. All the light cones are not consistent with each-other, angling away from the perspective point, and when the future light cone goes beyond 90 degrees to the horizontal, it can never reach the perspective point unless it goes faster than c. And at the top in some future time is when it goes asymptotic and the time ends.
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24 Feb 2018 03:01

Terran wrote:
Source of the post I wasn't really referring to a forwards-backwards time symmetry, rather a left-right symmetry. If I graphed out two inwards spiraling planets, colliding into one object, I would have an inwards curved hyper-coil becoming a kind of hyper-cylinder with spherical sides, that besides the point, I would have a time axis on the graph, and if I make that axis' origin the middle of the event, shouldn't a reflection over it be perfectly symmetric, yielding the same outcome?

No, this will generally not yield the same outcome.  To make it easier to see, consider a binary orbit and choose cylindrical coordinates with the center of mass of the system fixed to r=0.  In general, points (t, r, φ) on the orbits will not be mirrored with points at (t, r, -φ).  So it is not symmetric about the time axis.  To be symmetric in the way you're describing the two masses would need to always be exactly equidistant from the center of mass.


Terran wrote:
Source of the post This question comes from me trying to draw the entire universe (cause, why not), from beginning to end

That can be done. :)

Image

This graphic was made by Pulsar on StackExchange, in order to explain a common misconception about superluminal recession velocities in the observable universe.  (Yes, we can see galaxies that have recession velocities faster than light, and this does not contradict relativity).  He opted to use co-moving coordinates (such that distance of objects remains constant as the universe expands), and scale the time axis such that light always travels at 45° (the thin yellow lines).  It is possibly the finest and most information-rich example of a space-time diagram of the universe's evolution in Lambda-CDM cosmology that I have seen, though it may certainly take a moment to digest.  His explanation of it is well worth reading.
 
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25 Feb 2018 13:39

Watsisname wrote:
Source of the post In general, points (t, r, φ) on the orbits will not be mirrored with points at (t, r, -φ).  So it is not symmetric about the time axis.

It won't be mirrored in a sense, but wouldn't the outcome be the same?

Watsisname wrote:
Source of the post This graphic

Woah! I was close, but I measured things differently. My graph curved outwards instead, and light did not remain at 45 degrees. I made mine where it was just focused on the observer's perspective, being that the cross section equates to what is seen from the location. The cross section itself is "flat", being perpendicular to the x-axis. On mine, tracing an object compared to the observer. its path exponentially gets pulled away, and its light cone changes angle, passing 90 degrees at the "point of no return" indicating it has no futures in that direction.
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26 Feb 2018 04:29

Terran wrote:
Source of the post It won't be mirrored in a sense, but wouldn't the outcome be the same?

If the signs of all positions and velocities are flipped then the evolution this system will be the same, since this describes the exact same system with a rotated coordinate system.  This isn't a feature of orbits, it's a feature of nature (almost always -- there are some examples of parity violations).
 
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27 Feb 2018 16:57

Watsisname wrote:
Source of the post exact same system with a rotated coordinate system.

Got it.
Well thanks! Sorry if I barraged you too much with questions, as I have so many and know so little people who are willing to answer them. And this honestly was the first time someone actually answered my questions, and kept answering them, and didn't hold it against me. In theory I could ask an infinite amount of questions, and thats why I'll stop for now (not meant to be a foreshadowing or anything, rather I'm saying I don't want to find anyone's limit here). I cannot thank you enough for answering these questions, which I have truly progressed from. It is frustrating though, all paths ending in questions, and many you can't answer for yourself, but perhaps I will be able to, to a better degree, in some not so distant future time.
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17 Mar 2018 11:55

I had a dream this morning which inspired me to ask this question. This dream involved a specific angle that allowed things to orbit within the atmosphere.

Now I ask this, what is the lowest possible orbit for a spacecraft? Do things skip off the atmosphere and go out into space, has anyone done that before?
 
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17 Mar 2018 19:05

Since Earth's atmosphere doesn't have a sudden cut-off, but rather smoothly decreases in density with height, all orbits are technically within the atmosphere, and experience some drag because of it.  Even the ISS orbiting at ~400km experiences some orbital decay due to drag, and must be regularly boosted back up:

Image


The lower the orbit, the greater the density of the atmosphere, and the faster the orbit will decay.  If we were to stop reboosting the ISS, its orbit would continue to decay and finally give out in about a year (though this depends on a lot of complicated factors, like the solar activity which can make the atmosphere expand outward more and increase the decay rate.)

The rate at which something experiences orbital decay, and the lowest height where it can complete one orbit, also depends on the properties of the object.  A smaller, denser object will experience less air drag, and less deceleration due to that drag, so it could reach a lower altitude before its orbit finally gives out.  

So, in short, there is not really a unique lowest possible orbit.  It depends on a lot of things.


Starlight Glimmer wrote:
Source of the post Do things skip off the atmosphere and go out into space, has anyone done that before?


Yes, and yes! :)   

Passing through a planets atmosphere mainly has the effect of slowing the object down.  However, an elliptical orbit that passes through the atmosphere, even though it is slowed down by the passage, may still be left with enough speed to return to space.  It will just have a smaller orbit than before.  This is a bit less like "skipping" -- because its path would have returned to space anyway without the atmosphere -- and more like "not slowing down enough".  

Another effect, which is a lot more like skipping a rock off of a pond, is that the spacecraft surfaces may generate some lift as it passes through the atmosphere.  This can be used to further raise the trajactory, even though the spacecraft will still be left with less energy (and thus smaller orbit) than before.

Together, the use of these effects is called "aerobraking", and it has in fact been used by humans.  The Apollo astronauts returning from the Moon did not simply drop straight down through the atmosphere, but actually did skip a little bit.  The reason for doing this is that the speed when returning from the Moon is a lot faster than when returning from low earth orbit, so there is much greater re-entry heating.  To help mitigate this they basically took the re-entry in two steps: first dropping down to about 200kft (60km), but using the surfaces of the command capsule to generate lift and rising back up into space, and then making the final re-entry to spashdown.  

Image

This effect was also very useful and important because it gives greater control to the re-entry.  If they came in a little too steep, they could re-orient the command module to generate more lift.  Or the opposite, if they came in a little too shallow.  More generally, aerobraking can be used to modify your path and exactly where you land.  You can generate some "sideways" lift to cause a turn, or by varying the upward lift can spend more time and travel more or less distance down range.  It can be fun to experiment with this in Kerbal Space Program, or Orbiter. :)

Another nice portrayal of the use of aerobraking in science fiction can be seen in the film 2010:

 
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17 Mar 2018 20:03

Watsisname wrote:
Since Earth's atmosphere doesn't have a sudden cut-off, but rather smoothly decreases in density with height, all orbits are technically within the atmosphere, and experience some drag because of it.  Even the ISS orbiting at ~400km experiences some orbital decay due to drag, and must be regularly boosted back up:

Image



The lower the orbit, the greater the density of the atmosphere, and the faster the orbit will decay.  If we were to stop reboosting the ISS, its orbit would continue to decay and finally give out in about a year (though this depends on a lot of complicated factors, like the solar activity which can make the atmosphere expand outward more and increase the decay rate.)

The rate at which something experiences orbital decay, and the lowest height where it can complete one orbit, also depends on the properties of the object.  A smaller, denser object will experience less air drag, and less deceleration due to that drag, so it could reach a lower altitude before its orbit finally gives out.  

So, in short, there is not really a unique lowest possible orbit.  It depends on a lot of things.


Starlight Glimmer wrote:
Source of the post Do things skip off the atmosphere and go out into space, has anyone done that before?


Yes, and yes! :)   

Passing through a planets atmosphere mainly has the effect of slowing the object down.  However, an elliptical orbit that passes through the atmosphere, even though it is slowed down by the passage, may still be left with enough speed to return to space.  It will just have a smaller orbit than before.  This is a bit less like "skipping" -- because its path would have returned to space anyway without the atmosphere -- and more like "not slowing down enough".  

Another effect, which is a lot more like skipping a rock off of a pond, is that the spacecraft surfaces may generate some lift as it passes through the atmosphere.  This can be used to further raise the trajactory, even though the spacecraft will still be left with less energy (and thus smaller orbit) than before.

Together, the use of these effects is called "aerobraking", and it has in fact been used by humans.  The Apollo astronauts returning from the Moon did not simply drop straight down through the atmosphere, but actually did skip a little bit.  The reason for doing this is that the speed when returning from the Moon is a lot faster than simply returning from low earth orbit, so there is much greater re-entry heating.  To help mitigate this they basically took the re-entry in two steps: first dropping down to about 200kft (60km), but using the surfaces of the command capsule to generate lift and rising back up into space, and then making the final re-entry to spashdown.  

Image

This effect was also very useful and important because it gives greater control to the re-entry.  If they came in a little too steep, they could re-orient the command module to generate more lift.  Or the opposite, if they came in a little too shallow.  More generally, aerobraking can be used to modify your path and exactly where you land.  You can generate some "sideways" lift to cause a turn, or by varying the upward lift can spend more time and travel more or less distance down range.  It can be fun to experiment with this in Kerbal Space Program, or Orbiter. :)

Another nice portrayal of the use of aerobraking in science fiction can be seen in the film 2010:


Thats quite an interesting chart there. I knew it slowly decreases in effect as you go up, interesting. Now I remember there was a small asteroid that skipped off of earth's atmosphere in the 90's? 80's? I don't remember the exact name though.
Now for the whole aerobreaking thing. Thats very practical but it doesn't work very well on mars. Going to mars is hard since you need to have rockets and a heatshield. On earth you don't need rockets but on the moon you do. On mars you need both. The first humans could land on mars in a giant donut or just...fly over mountains for a really long time sideways.
Another question. What types of stars are Polaris C and D? Are they bound to the Polaris system?
 
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17 Mar 2018 20:45

Starlight Glimmer wrote:
Source of the post Now I remember there was a small asteroid that skipped off of earth's atmosphere in the 90's? 80's? I don't remember the exact name though.

Midtskogen might know, and I recall he also has observed a meteor a more recently that did the same thing.  Meteors come in pretty fast, so if they aren't slowed down by the atmosphere enough then they'll keep right on going.  They may even not be captured by Earth's gravity at all, and continue on orbiting the Sun (but with a slightly modified orbit.)

Starlight Glimmer wrote:
Source of the post Now for the whole aerobreaking thing. Thats very practical but it doesn't work very well on mars.

Yeah, Mars is very challenging to land on, since the atmosphere is too thin to only use parachutes, and also too thick to ignore heat shields.  However, aerobraking in the Martian atmosphere can in theory still be useful.  For example, you could send a ship from Earth to Mars and use the atmosphere to slow down and be captured into orbit, thereby saving fuel.  As a general rule of thumb, if the object you're going to has an atmosphere, then you can always save fuel by aerobraking instead.

The catch is that aerobraking for any purpose is inherently risky, difficult, and adds to the cost of a mission since it requires appropriate shielding.  The correct aerobraking altitude must also be very precise.  If you are off by even a few kilometers altitude, it could have disastrous consequences, like burning up, or even accidentally lithobraking. ;)


Another question. What types of stars are Polaris C and D? Are they bound to the Polaris system?


I'll have to come back to this question, unless someone else would like to answer first. :)

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