Mass distribution and gravitational field

In summary, the field outside a body with spherical mass distribution is well known, but for a non-spherical body (or a almost simmetrical body like the earth) which is the approach?. Do you know any reference?
  • #1
Frank66
11
0
The field outside a body with spherical mass distribution is well known but for a non simmetrical body (or a almost simmetrical body like the earth) which is the approach?. Do you know any reference?
thank you
 
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  • #2
I think you can apply Gauss law. You can draw a spherical surface with center of center of mass of the object. When the sphere includes all the masses, according to gauss law, it is the same. Since Gauss theorem ▽·g=4πGρ, and apply the law ∫▽·g dV=∫g·dσ=4πM, where M is the mass inside the sphere. Therefore on this occasion where it is sufficiently far from the center of the object, you can use Newton.
 
  • #3
Newton's law of gravity in its familiar form gives the gravitational force between two point mass objects (or two uniform spheres which act as point objects). However, it can be cast in a form that handles non-spherical masses. Just treat a non-spherical mass as a collection of integration volumes, each behaving as a point mass, and integrate over the volume.

Newton's Law in full fixed-coordinate vector form:

F= - G m m' (x - x')/|x - x'|3

where F is the gravitational force of mass m' on m located at x' and x respectively. If another massive body m'' is introduced, it's force on m is just additive:

F= - G m m' (x - x')/|x - x'|3 - G m m'' (x - x'')/|x - x''|3

If we bring many point masses into the picture, we just add up all the forces:

F= - G m Ʃ mi(x - xi)/|x - xi|3

In the limit that the point masses get so packed together that they can be treated as constituting a spatial continuum of mass, the sum becomes a volume integral:

F= - G m ρ(x')(x - x')/|x - x'|3 d(3) x'

where ρ is the mass density. This is what you would use for extended non-spherical mass objects.
 
  • #4
You can also use Poisson's equation for the gravitational potential.
 
  • #5
Thank you,
questions:
what effects on the general motion of the Earth are due to its non-spherical symmetry?
astrophysicists will take account of this...
Even artificial satellites are definitely not spherically symmetric objects (although relatively small): as we can account for this?
 
  • #6
DrStupid said:
You can also use Poisson's equation for the gravitational potentialhttp://www.flughafenhotel.net/hotel-flughafen-national/hotel-hamburg-flughafen/" [Broken]

In my mind the easiest way ...
 
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  • #7
I return to the topic after a long time ...
A tangible example, the moon will not be a celestial body in perfect spherical symmetry. How this affect on its motion around the earth?

thanks
 

1. What is mass distribution?

Mass distribution refers to the way mass is distributed or spread out within a given space or object. It can be described in terms of the density and arrangement of the mass within the space.

2. How does mass distribution affect the gravitational field?

Mass distribution directly affects the strength and shape of the gravitational field. Objects with a higher mass concentration will have a stronger gravitational pull, while objects with a more spread out mass distribution will have a weaker gravitational pull.

3. Is mass distribution the same as weight distribution?

No, mass distribution and weight distribution are not the same. Mass refers to the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. Mass distribution affects the gravitational field, which in turn affects the weight distribution of an object.

4. How is mass distribution measured?

Mass distribution can be measured using various techniques, such as gravitational lensing, spectroscopy, and imaging. These methods allow scientists to map the distribution of mass within an object or space.

5. Can mass distribution change over time?

Yes, mass distribution can change over time due to various factors such as the movement of particles, the formation of new objects, or the influence of external forces. These changes can affect the gravitational field and have an impact on the dynamics of the system.

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