Keplers Laws.

[SweetScientist]

I've bought a book called "50 Physics Ideas You Really Need to Know." In the section on Kepler's laws it states this.

 

"A planet in orbit twice as far away from the sun as the earth would take 8 times longer to go around."

 

Is this correct? I did a little calculation using the equation and it didn't work out. I then looked across other websites which stated it was indeed 4 times as far way would induce an 8x increase in time. My calculations also rung true with this. (I used square root semi major axis of the ellipse cubed.)

 

Is the book wrong or am I misinterpreting it?

Edited February 15, 2012 by SweetScientist

[elfmotat]

4 times as far way to yield 8 times the period is correct. This makes me afraid about what else it could say in there.

 

[math]T^2 \propto r^3[/math]

Edited February 15, 2012 by elfmotat

[SweetScientist]

4 times as far way to yield 8 times the period is correct. This makes me afraid about what else it could say in there.

 

[math]T^2 \propto r^3[/math]

 

Makes me wonder as well. It's not even as if it's hard to prove at all. Took me a 20 second calculation to disprove the statement.

[Moontanman]

I applaud you for checking, so many people assume anything that is written down must be true.

[D H]

Apparently everything I know is wrong.

"Our mobile 'phones connect us via invisible electromagnetic threads to satellites orbiting overhead" (So what are those ugly cell towers for?)

"Acceleration is a change of speed over some time" (What about uniform circular motion?)

"A child whirling on a merry-go-round is tugged outwards by the distant stars." (What about Newton's first law? This is Mach's principle done badly.)

[DrRocket]

Apparently everything I know is wrong.

 

Not quite everything.

[PaulS1950]

At least we know what we have wrong...

Now we just need to learn something - that is correct.

 

Paul

[DrRocket]

At least we know what we have wrong...

Now we just need to learn something - that is correct.

 

Paul

 

You might want to take a look at the book Orbital Mechanics by Conway and Prussing. It is quite accessible.

 

Kepler's laws are derivable from ordinary Newtonian mechanics. In fact the motivating factor for Newton in his development of mechanics was to understand the principles behind Kepler's laws.

[PaulS1950]

But the Newtonian laws are incomplete at best and wrong in some arrangements - say in the study of Mercury's orbit.....

 

Paul

[elfmotat]

But the Newtonian laws are incomplete at best and wrong in some arrangements - say in the study of Mercury's orbit.....

 

Paul

 

They are a good enough approximation that it would be well worth your time to give them some thought.

[swansont]

But the Newtonian laws are incomplete at best and wrong in some arrangements - say in the study of Mercury's orbit.....

 

Paul

 

Wrong in a way similar to 1/2mv^2 not giving you the correct kinetic energy. But it's close enough for many applications, and we know the conditions where it fails to be accurate.

[D H]

But the Newtonian laws are incomplete at best and wrong in some arrangements - say in the study of Mercury's orbit.....

 

Paul

The exact same can be said of general relativity. It too is incomplete at best as it has not yet been melded with quantum physics, and it is presumably wrong in some arrangements such as the prediction of a singularity at the very heart of a black hole.

 

Rhetorical question: If Newtonian mechanics is "wrong", why is it that NASA and other space agencies still use Newtonian mechanics to model the trajectories of their spacecraft, and why do engineers still use Newtonian mechanics to build highways, buildings, and robots? The answer is that "wrong" is a bit of an overstatement. The disagreement between the observed and predicted behavior of Mercury per Newtonian mechanics is very, very small. There's a mismatch of 43 seconds of arc per century in the anomalistic precession.

 

43 arcseconds is tiny, tiny angle. Stack a yardstick (meter stick) on top of another. Raise one end by about 5/8 inches (1.75 cm in the case of a meter stick). The angle subtended between the two measuring sticks is about a degree. Now put a postcard on top of the lower stick and lower the upper stick so it rests on the postcard. The angle subtended between the two measuring sticks is now about 40 to 50 arcseconds. That tiny angle is a measure of the wrongness of Newtonian mechanics with regard to Mercury's orbit -- and that is over a course of a century.

 

Rather than saying that Newtonian mechanics is wrong, a better statement is that Newtonian mechanics is not universally true. It is an approximation of reality that is approximately correct in the narrow domain of velocities that are very small compared to the speed of light and distances that are large compared to the Schwarzschild radius (and compared atomic scales; Newtonian mechanics misses the boat on quantum scales as well).

[PaulS1950]

DH and swansont,

I agree with you both. we don't have the complete answers for much of anything we "know". we have a set of working formula that we are able to "get by with" in those places we use them. we can make corrections to allow for errors but unless we open our minds to the errors we will never look for the sollutions.

I use the same math that everyone else does but there are situations where that math is the wrong tool or is being used with errors. It has been said that there is no such thing as a "singularity" because infinitly small and large cannot exist. The "spheroid" shape of a singularity has dimensions and it contains less than infinite mass. We need a new mathmatics approach to realize that what is breaking down is our understanding and not the laws that govern the universe.

 

Paul