relativity - Circular motion and time dilation

Einsteins' time paradox was shown when two atomic clocks were synchronized and one was sent in a super sonic plane and the other kept at rest. When the two where then checked it was shown that yes indeed the clock that took the plain trip had lost time when compared to the clock that was kept at rest.

My question is what if instead of one of the clocks going on a plain it was put into a centrifuge or some other device that spun it around and very high speeds would it feel the effects of time dilation in the same manner?

solar system - Why do the planets' orbital distances fall on an exponential curve?

Background: I was recently reading a book on the planets to my son and I noticed a pattern in the distributions of the planets. The planets' distances roughly follow an exponential distribution.

Below is a plot of scaled, log orbital distances $$ \tilde{d_n} = \frac{\log(d_n/d_1)}{\log(d_2/d_1)} $$ with the line $an+b$:

Where $d_1$ corresponds to Mercury and so on. Ceres is included, Pluto is excluded. By linear regression, $a = 0.90, b = -1.06$.

For the statistically minded, the data has a Pearson's correlation of 0.996. Note that this is a well known phenomenon, see Pletser and references. The code used to generate the plot may be provided on request.

Question: What is the mechanism that leads to this distribution?

Aside: Is there a good introductory text on solar system formation for the mathematically inclined?

Update: This is also known as Titius–Bode law.

Answer

This correlation is known as Titius-Bode's law, which is often stated as

\begin{equation} d=0.4 + 0.3 \cdot 2^n \end{equation}

where d represents planet's mean distance from the Sun in Astronomical Units and n = -∞, 0, 1, 2... for Mercury, Venus, Earth, Mars, asteroid belt, Jupiter and so on.

The rule is not satisfied exactly with Neptune's orbit (n=7) constituting a significant departure from it: according to the law Neptune's mean distance ought to be 38.8 AUs, but is in reality just 30 AUs (disagreement of close to 30% with all other planets agreeing to less than 6%). In fact, this departure is what has historically led to diminishing importance of the law. See also the table and chart in wikipedia.

It is currently thought that if the law is not a pure coincidence then it is a consequence of orbital instabilities and the mechanism through which Solar system was formed. It's been shown that rotational and scale invariance of a protoplanetary disk leads to density maxima in the disk appearing periodically in variable

\begin{equation} x = \ln \frac{r_n}{r_0} \end{equation}

which leads to geometric series for planetary distances similar to that expressed in Titius-Bode's law. See this and this paper for details.

Note that the requirements of rotational and scale invariance are very general. As the nebula from which protoplanetary disk is formed collapses under its own gravity, its rotation increases due to the law of the conservation of angular momentum. This eventually leads to the protoplanetary disk's rotational symmetry. Also, gravity does not have intrinsic length scale, so the nebula is highly likely to possess scale invariance. These two requirements are so general that even if the Titius-Bode's law is real it isn't at all useful to select between Solar system's formation models.

I don't know of an advanced book specifically on Solar system formation, but there is a very good book by A.E. Roy on orbital mechanics which certainly would qualify as a book for the mathematically inclined which in addition to chapters on orbital mechanics, rocket dynamics, interplanetary trajectory design includes few solar system formation and many-body stellar systems. So depending on how broad your interests are you may enjoy it.

special relativity - How to derive addition of velocities without the Lorentz transformation?

Lorentz contraction and time dilatation can be deduced without Lorentz transformation. Can you deduce also the theorem of addition of velocities

$$w~=~\dfrac{u+v}{1+uv/c^2}$$

without Lorentz transformation? Using just the constancy of light speed.

spacetime - See behind the black hole

Why in this video does the 2nd black hole appears to change size and appear larger the farther away it gets? How can you see behind it? http://www.youtube.com/watch?v=ENd8Sz0AFOk

Answer

Black holes bend light passing them and this means they act as a lens. The phenomenon is called gravitational lensing. The way black holes bend light is different to the way a conventional lens, for example in a magnifying glass, bends light and as a result there can be some very odd visual effects. In this case the lensing by the black hole at the front is magnifying the black hole at the back and distorting it into an Einstein ring.

Working out exactly what the gravitational lensing does is exceedingly complicated. If anyone is interested some details of the calculation shown in this image are described in this paper.

Distinguishing real forces and fictitious/pseudo forces in Newtonian mechanics

In understanding the law of inertia I had to consider the motion of bodies screened from the so called "real forces".

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  What characterises these real forces?

  
  


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  What makes us call them real?

  
  


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  Or what is separating the forces called true or real from another group of forces called fictitious/pseudo forces?

  
  
  

Many forces called true are also invisible just like the centripetal force or other pseudo forces.

• Then what is making them special and different?

fluid dynamics - Pressure in giant ball of water floating in space

In this question:

Swimming in a ball of water in space

They seem to have reached the conclusion that the pressure inside a giant ball of water in space is zero.

Quoting the second answer:

As a conclusion, you would be swimming in a bubbling sphere of water, feeling no pressure at all, having a bit less difficulty in moving your arms and legs since the water would be full of bubbles, however I am assuming it would be harder to move around for the very same reason.

Quoting the first answer:

So, the bottom line is swimming in a big ball of water pretty much feels like swimming very slowly in space - until the ball of water gets big enough (2.68 km). Then it just feels like swimming in a giant pool on a distant planet. For practicality, the ball of water doesn't work, but the lunar swimming pool is awesome.

However, I'm still confused about one thing. So far as I know, although the cause of pressure here at the surface of the earth may be the gravitational weight of all the air above us (and the cause of more pressure deeper in the ocean is the weight of all the water above it) at the local level, pressure is really a cause of the random motion of the particles, smashing into the particles around them, and delivering their momentum.

After all, that's why rising temperatures cause the pressure to rise.

Assuming this giant ball of water can hold itself together due to cohesion, wouldn't you still feel the pressure from...well, simply the water molecules themselves, moving randomly in all directions?

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Edit:

To help readers, and as a response to the criticism from the person who answered (criticism taken kindly, you're right, I should clarify. Although you could've said it in a nicer way...), this is why I'm asking this question:

I'm trying to understand pressure.

I'm trying to understand whether it originates fundamentally from having to "sustain" a fluid in place due to gravitational forces, or from something intrinsic to the fluid itself (its random motion.)

Although I know that at a local level, the random motion of particles is the cause of pressure, what I'm trying to understand is whether the random motion is caused by something else, or is intrinsic to the fluid itself.

In short, I want to know if there would be pressure without gravity, and this is the best thought experiment I could come up with to explain what I mean by that.

Thanks.

Answer

The misconception that is probably causing your confusion is that

at a local level, the random motion of particles is the cause of pressure,

Random motion of particles is measured by temperature; the higher the temperature, the more intense the random motion.

If we are to talk about causes, the cause of pressure on some wall is first and foremost mutual interaction of the particles and the wall. The fact that the particles move randomly is secondary. True, in gases increase of pressure often goes with increase in this random motion, because the increase of gas pressure can be done only by putting in substantial energy. But in liquids, it is possible to increase the pressure substantially with negligible amount of work and so with negligible change in intensity of this random motion.

Pressure of such liquid is due to force interaction of the particles with walls and each other, not necessarily due to their random motion. It suffices that particles push or pull each other. They do not have to move rapidly. You can have high pressure in very cold water or in ice cold at 1 K.

When pressure of a liquid water is increased, say, by moving a piston in a blocked syringe filled with water, water temperature increase is very small and is usually neglected.

Now to your question - gravity isn't necessary for pressure either. What is necessary to increase pressure is some other body that will squeeze the gas or liquid into smaller volume. On Earth, this body is the Earth with its gravity, but the same pressure is achieved in a closed vessel, such as the International Space Station, simply by making it robust enough to withstand the pressure and pushing in enough amount of gas. There is no effective gravity there, but there is pressure close to 100kPa, due to walls not allowing the gas to escape.

experimental physics - Why is there a hiss sound when water falls on a hot surface?

Why is there a hiss sound when water falls on a hot surface? I have searched a lot, asked my teachers but none of them seem to give me the logical answer to it.