Kepler - plus?

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If the velocity of a satellite is required to be proportional to its distance from the star (ie Kepler’s law), how would the coming together of smaller satellites, form a satellite that is stable in its orbit?

 

Just to try and simplify this request further - take the asteroid belt - each asteroid travels at a velocity required to maintain it in a ’stable orbit’, (with the absence of Jupiter) if these asteroids were to come together, the resulting increase in mass would require the new satellite to increase its velocity in order to ’stably’ maintain its orbit…

 

…How is this explained?

 

 

By the way...

 

Warm greetings to all and thanks in advance for your efforts

 

[swansont]

Does mass appear in the equation?

[smooth]

So why does the increase in mass not result in the satellite simply falling in toward the star?

 

I thought it was - The greater the mass, the greater the energy required to attract it, and hence objects of differing mass 'fall' at the same velocity.

 

[Cap'n Refsmmat]

That doesn't relate to Kepler's law. That's more of a universal gravitation thing.

 

Essentially, you're right. If the mass of the object increases, the gravitational force it experiences increases, but its inertia increases at the same time.

[smooth]

Thanks for the reply

 

However your description is not detailed enough to have satisfied my curiosity. If I may try to bring a greater clarification to your statement; That doesn't relate to Kepler's law. That's more of a universal gravitation thing.

 

 

If I were to make a statement such as:

 

Objects orbiting upon the two dimensional gravitational plane are not subject to their acceleration been proportional to the force divided by their mass, as is the case for objects within the constraints of three dimensional gravitational effects.

 

Hence within the two dimensional gravitational plane an increase in mass results in a proportional increase in acceleration, suggesting that the (gravitational) force remains constant.

 

Am I wrong? Does this need clarifying? Is this something new?

 

 

[Cap'n Refsmmat]

Newton's law of universal gravitation says that

 

[math]F_g = G \frac{m_1 \times m_2}{r^2}[/math]

 

Or, in other words, the force of gravity an object experiences is equal to the gravitational constant (G) times the masses of the two objects over the distance between the two, squared.

 

[imath]F = ma[/imath] still applies for objects experiencing gravity, but as the mass of the object increases, so does the gravitational force (as in the first equation above), so its acceleration remains constant.

 

Kepler's Law relates the orbital period of something to the radius of the orbit, and ignores mass. It states that

 

[math]\frac{T_1^2}{r_1^3} = \frac{T_2^2}{r_2^3}[/math]

 

which means that the orbital period (T) squared of an object orbiting something over the orbital radius cubed is equal to the same expression for another object orbiting the same planet/whatever.

[swansont]

Hence within the two dimensional gravitational plane an increase in mass results in a proportional increase in acceleration, suggesting that the (gravitational) force remains constant.

 

Am I wrong? Does this need clarifying? Is this something new?

 

 

An increase in mass and an increase in acceleration must result in an increase in force. The force can't remain constant.

 

Mass isn't present in Kepler's laws because the gravitational effect on orbits doesn't depend on mass, as the equations the Cap'n has provided demonstrate.

[J.C.MacSwell]

To get an intuitive sense of this, picture yourself as one of the asteroids, and your twin brother in the same orbit right beside you. If you reach over and grab him nothing much changes. You don't both go plummeting toward the sun.

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Oaky doky, I don’t mind my cobwebs been blown out.

 

 

If a planet moves closer to the point of attraction it orbits around, the greater the gravitational attraction? And the greater the velocity required to prevent it from descending toward a collision with its central point of attraction?

 

The change in a planets velocity that results in the sweeping out of equal areas during equal intervals of time as the planet travels along its elliptical orbit - is related to the varying degree of gravitational attraction.

 

Or in other words, if the planet did not accelerate as it approached its closest point - to the center of attraction - it would descend out of a stable orbit into an eventual collision. So the fact that this acceleration enables the sweeping out of equal areas during equal intervals of time is a by product of the varying degrees of attraction felt by the planet, and is akin to the differing degrees of attraction an object remaining in a spherical orbit, but increasing in mass - and hence velocity, if it is to remain in a stable orbit.

 

Hit me!

 

 

[Janus]

Oaky doky, I don’t mind my cobwebs been blown out.

 

 

If a planet moves closer to the point of attraction it orbits around, the greater the gravitational attraction? And the greater the velocity required to prevent it from descending toward a collision with its central point of attraction?

 

The change in a planets velocity that results in the sweeping out of equal areas during equal intervals of time as the planet travels along its elliptical orbit - is related to the varying degree of gravitational attraction.

 

Or in other words, if the planet did not accelerate as it approached its closest point - to the center of attraction - it would descend out of a stable orbit into an eventual collision. So the fact that this acceleration enables the sweeping out of equal areas during equal intervals of time is a by product of the varying degrees of attraction felt by the planet, and is akin to the differing degrees of attraction an object remaining in a spherical orbit, but increasing in mass - and hence velocity, if it is to remain in a stable orbit.

 

Hit me!

 

The difference in the second situation is that the second case the inertial mass increases by the same factor as the force.

 

Try looking at it this way:

 

Without the Sun's gravity, the planet would fly off in a straight line. It is the centripetal force supplied by the Sun's gravity that causes the planet to follow the curve. So what happens if you double the mass of the planet? For one, you double the gravitational force. At the same time you double the inertia of the planet, and double the amount of force it takes to hold the planet to the same curve.

 

Mathematically it looks like this.

 

The centripetal force needed to hold an object in a circular path is

 

[math]F_c = \frac {mv^2}{r} [/math]

The gravitational force is found by

 

[math]F_g =\frac{GMm}{d^2} [/math]

 

If m is the mass of the planet

M is the mass of the Sun

d is the distance between Sun and planet and is equal to r (the radius of the orbit.

 

Then for a circular orbit:

[math]F_c=F_g[/math]

 

[math]\frac {mv^2}{r}=\frac{GMm}{d^2}[/math]

 

Substitute r for d:

[math]\frac {mv^2}{r}=\frac{GMm}{r^2}[/math]

 

Reducing

[math] v^2=\frac{GM}{d}[/math]

 

Notice that the mass of the planet (m) cancels out of the equation, so that the orbital velocity for a given orbital distance "r" remains the same no matter what the mass. (within limits)

[smooth]

If I may respond briefly...

 

I do not communicate in equations, so most of what you have said means nothing to me. I do accept that if the mass increases so does the gravitational attraction, in proportion to that increase in mass, and hence the velocity of the planet.

 

Which all translates to me at the moment as - the force known as gravity, accelerates objects in orbit when they increase their mass.

 

I do not know how this fits in with:

The heavy and light objects travel at the same rate because

there are two competing factors that cancel each other out.

 

The force of gravity is greater on the heavier object than

on the lighter object, proportional to the object's mass.

This means that an object with twice the mass will be pulled toward

the earth with twice the force.

 

On the other hand, the acceleration is proportional to the force

divided by the mass. This means that an object that is

twice the mass of another object will be accelerated

twice as slowly as the lighter object given the same force.

 

So in order for an object with twice the mass to move

at the same rate as the lighter object, the heavier object

must be submitted to twice the force. And this is exactly

what the force of gravity does.

 

http://www.newton.dep.anl.gov/askasci/gen99/gen99110.htm

 

 

[swansont]

Which all translates to me at the moment as - the force known as gravity, accelerates objects in orbit when they increase their mass.

 

It does, but since the increase in force and increase in mass are in exactly the same proportion, the value of the acceleration does not change, so the orbit does not change.

[smooth]

Thank you swansont, what you have said makes perfect sense

 

In the context of the proportional increases there is no acceleration.

 

 

However in the relative context of two objects where one object increases in mass (and the other remains constant), there is an acceleration, and the heavier object hits the ground before the lighter object - in the event that they have been dropped from the same height at the same time!?

 

 

[swansont]

Thank you swansont, what you have said makes perfect sense

 

In the context of the proportional increases there is no acceleration.

 

 

However in the relative context of two objects where one object increases in mass (and the other remains constant), there is an acceleration, and the heavier object hits the ground before the lighter object - in the event that they have been dropped from the same height at the same time!?

 

 

Actually, it's no change in the acceleration.

 

The same concept applies to the objects falling. The acceleration is the same, and they hit the ground at the same time (assuming nongravitational effects aren't included)

[smooth]

99, 98, 97... 3, 2, 1 - Coming ready or not

 

The problem was that when I was babbling about a 2D gravitational plane & 3D gravity, what I was really trying to communicate was what you have been trying to explain to me (and when you said that wasn‘t right, I went off somewhere else)…

 

When an object is in the earths atmosphere and is failing toward the centre of attraction, inertia becomes irrelevant, as the objects momentum does not oppose the force of attraction (meaning that it does not accelerate if increased in mass). However when in orbit an objects momentum must oppose the force of attraction, and it is this inertia that causes the object to accelerate even though it is subject to the very same force (as an object whose momentum is taking it toward the centre of attraction).

 

 

 

[Janus]

Thank you swansont, what you have said makes perfect sense

 

In the context of the proportional increases there is no acceleration.

 

I have feeling here that part of the problem revolves around the use of the word "acceleration" . Acceleration is a change in velocity, where velocity is measured by both speed and direction( IOW a car driving 30 kph heading East has a different velocity than a car driving at 30 kph heading West. They have the same speed, but have different headings)

A planet in a circular orbit is constantly accelerating even though it never changes its speed, by virtue of the fact that it is constantly changing its heading.

 

So what Swansont is saying is that increasing the mass of the object does not change its acceleration (either its speed or heading), and it continues to travel in the same orbit it did before.

 

 

 

However in the relative context of two objects where one object increases in mass (and the other remains constant), there is an acceleration, and the heavier object hits the ground before the lighter object - in the event that they have been dropped from the same height at the same time!?

 

 

 

Just as above, both objects have equal acceleration (this time as a change of speed), fall at the same rate and hit the ground at the same time.

[smooth]

If I took a snap shot of a planet orbiting a star, then drew a line representing the straight path the planets inertia dictates it should take (taking into account its prior momentum), also drawing a line from the centre of the suns attraction to the centre of the planets attraction, the planets orbit would dissect these two lines?

 

If a planet (or object falling to earth) were to increase in mass, its inertia would increase proportionally, and so the proportional increase in gravitational attraction cancels out any change in speed?

 

The gravitational attraction increasing (ie in the case of an elliptical orbit), is as if the mass is proportionally increasing, and so the inertia increases proportionally?

 

Ehhh!?

 

In an elliptical orbit both speed and direction change?

 

[Janus]

If I took a snap shot of a planet orbiting a star, then drew a line representing the straight path the planets inertia dictates it should take (taking into account its prior momentum), also drawing a line from the centre of the suns attraction to the centre of the planets attraction, the planets orbit would dissect these two lines?

 

If a planet (or object falling to earth) were to increase in mass, its inertia would increase proportionally, and so the proportional increase in gravitational attraction cancels out any change in speed?

 

Not any change in speed, any change in acceleration (see above)

 

The gravitational attraction increasing (ie in the case of an elliptical orbit), is as if the mass is proportionally increasing, and so the inertia increases proportionally?

 

Ehhh!?

In an elliptical orbit it is the decreasing distance that results in the increase in pull. Since the mass of the planet does not change neither does its inertia. Inertia only changes if you add more mass to the planet. just increasing the pull does not add more mass.

In an elliptical orbit both speed and direction change?

 

 

Yes.

[smooth]

Sorry to be a pain and all that, but better to fully understand this now so I can move on.

 

In the scenario of an elliptical orbit, when the object descends toward the centre of attraction, it is the lack of an increase in inertia that results in the increase in both speed and direction (acceleration).