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Math Help - Modelling a Gravitational Field NOT using an inverse square law

  1. #1
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    Modelling a Gravitational Field NOT using an inverse square law

    Dear Group,

    I am hoping for a math genius to help me with a certain problem. Now, I am not that advanced in mathematics at all and only know certain very basic things, and I hope that the following problem is quite simple, or that the solution can be relayed to me in simple terms if there is one.

    I am trying to model a gravitational field (acceleration values) about a celestial body that does NOT extend away from the centre of the body in accordance with a general inverse square law.

    Now, in astronomy, I am aware that on the issue of deriving the curvature of an elliptical body in orbit, of say the sun, you only need 3 points, and that a unique curve will pass through all 3 points to give the total ellipse. I am hoping for something similar by way of a solution to my problem.


    I have 3 distance measures from the centre of an ‘experimental’ celestial body, with the centripetal gravitational values at those points. What I want to know is, is there a way to determine the unique curvature of the field from just this data? I am hoping for a general method so I could experiment with different values. Any help would be appreciated.


    Distance from centre of body = 238857.528 Miles
    Centripetal (Gravity) Acceleration = 0.000000020633339375 Miles/Second^2

    Distance from centre of body = 1114.118548 Miles
    Centripetal (Gravity) Acceleration = 0.0009483872877 Miles/Second^2

    Distance from centre of body = 1079.943132 Miles
    Centripetal (Gravity) Acceleration = 0.006093296399 Miles/Second^2

    Sincerely

    Keith
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  2. #2
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    If you don't have the inverse square law, then there's no restriction being given on how gravity might behave. Thus just knowing the gravitational field at three points tells you nothing about the gravitational field at other points. You'll need to make some (strong!) assumptions about the gravitational field in order to determine it everywhere from just those 3 data points.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    Dear John,

    The inverse square law offers a unique solution to a (any) gravitational field. I wondered if a unique law or equation would be determined from just the 3 points of data that I have given. Is that not so?
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    What I was trying to say was, that's not possible.
    If you reject the inverse square law, and leave the "law of gravity" totally unrestricted other than demanding that it satisfy those three data points - then there are an infinite number of possible "laws" that are consistent with those three data points.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    The inverse square law ultimately was based on observations, and derived from 3 observations. Now I, have 3 other 'observations', or data sets. And, as you say, I am demanding that there will be a unique curvature that connects up all 3 points; that there will be only one solution. It will be different to the inverse square law in terms of the curvature. Why do you think there will be an infinite number of answers? I am not going to be having 'weird ripples' between points. It will be one smooth curve. I am assuming there is only one answer.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    Quote Originally Posted by lightpotential View Post
    The inverse square law ultimately was based on observations, and derived from 3 observations.
    Why would you think that?

    Now I, have 3 other 'observations', or data sets. And, as you say, I am demanding that there will be a unique curvature that connects up all 3 points; that there will be only one solution. It will be different to the inverse square law in terms of the curvature. Why do you think there will be an infinite number of answers? I am not going to be having 'weird ripples' between points. It will be one smooth curve. I am assuming there is only one answer.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    Dear Halls,

    I hope that perhaps you may know the answer to my problem. I will say this though. I regret the way I have attempted to describe my problem in my initial post. I have no interest in getting involved in a historical discussion about astronomy or gravity with anyone. I wish I had not even mentioned gravity at all. This is a math problem. Please ignore everything above, and just consider the following, as if it were my first post:

    Consider that I have a graph: X and Y axis. Now, I have 3 co-ordinate pairs.

    1) X = 238857.528, Y = 0.000000020633339375
    2) X = 1114.118548, Y = 0.0009483872877
    3) X = 1079.943132, Y = 0.006093296399

    What I want to know is how to derive the unique function that will generate a curve that passes through all three points. You see, if I had an equation to begin with, I could work out a value for Y with any given value of X using the equation. I do not have the equation to begin with though in this case. All I have are 3 co-ordinates. What I want to know is if there is a method for determining the equation from the 3 sets of co-ordinates.

    Now I know that with ellipse shapes, you can derive the curvature of an ellipse from just 3 points. I assume my problem is of a similar type and that there will be a unique answer and equation. Do you know how I can derive the equation from the 3 co-ordinates?

    Keith
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    What the previous posters have been explaining is that there are an infinite number of functions passing through your 3 points.

    It can be shown that at most one of these will be a quadratic which is why you can use 3 points to uniquely identify an elipse. So if you are allowed to assume a quadratic relationship there will be a unique answer which you can derive.

    However you are not allowed to assume the realtionship is quadratic, and all you can say is that the formula is one of the infinitely many possibilities....
    Last edited by SpringFan25; October 20th 2012 at 08:15 AM.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    Dear Spring,

    Thank you for your answer. I believe I intuitively grasp what you are saying. Not necessarily in a true technical way.

    You see, I am not a mathematician. I have a very basic understanding of mathematics. I am at a certain level where I do understand simple equations relating to acceleration e.g. acceleration = Velocity Squared/radius. And, I understand the idea of an inverse square law, and get the idea of a 'weakening' of the centripetal acceleration values moving away from a given mass. I am investigating the notion that gravitational fields (and their acceleration values) as extend away from a given mass, may not follow a precise inverse square law.

    From your response, you suggest that there are indeed an infinite number of curves going through my 3 points, because I cannot assume a quadratic relationship. Well, first I must say, that an infinite number of answers is infinitely useless. But I like the idea of a unique answer. I am intrigued to know how this unique answer is attained.

    Now, I must state that the word quadratic means practically nothing to me. But I did mention about ellipses. My mind can grasp a conic section cut e.g. circles, ellipses, parabolas… Are you saying that if a conic section curve of some sort goes through my 3 points, that I have a unique answer? And that ‘quadratic’ curves are conic sections? But that if it is not a conic section but some ‘made-up’ curve generated by me fashioning a mathematical function based on ‘pure numbers’ and NOT geometry, that I have an infinite number of answers? Moreover, can I ask, is a general inverse square law ‘quadratic’. Is it a law that follows a curvature from a conic section or not?

    One further question: Suppose I had given you 4 co-ordinates. Would that have guaranteed a unique answer and eliminated all the infinite possibilities?

    I most appreciate your help. I hope you stay with me.


    Keith
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    One further question: Suppose I had given you 4 co-ordinates. Would that have guaranteed a unique answer and eliminated all the infinite possibilities?
    Basically, no. There would be some other shapes which can be uniquely identified by having 4 points, but i think youd get more confused if we start talking about that. Suffice to say, no matter how many points you pick there will always be an infinite number of functions (or none) which pass through those points.

    But I like the idea of a unique answer. I am intrigued to know how this unique answer is attained.
    Lets start by clearing up some notation. Here are two statements for you:
    [A]In newtonian gravity the strength of attraction between two objects follows an inverse square law, nameley F = \frac{G m_1m_2}{r^2}
    [B]It can be shown that the Path followed by a point mass orbiting another point mass is an elipse, if the only force acting on the point mass is as defined in [A].

    Take a simple system involving one planet and its star. After observing 3 point's in a planet's motion, you can deduce the entire path of the planet by following these steps:
    (1) Assume the planet acts as a point mass [this is actually valid if the planet is a perfect sphere].
    (2) Assume the force acting on the planet is newtonian gravity, as per statement [A]
    (3) then, by statement [B] the path of the planet must be an elipse
    (4) You then have a geometry problem to solve which is "how many elipses pass through these points". It can be shown that the answer is either 0 or 1.


    Now, I must state that the word quadratic means practically nothing to me. But I did mention about ellipses. My mind can grasp a conic section cut e.g. circles, ellipses, parabolas… Are you saying that if a conic section curve of some sort goes through my 3 points, that I have a unique answer? And that ‘quadratic’ curves are conic sections? But that if it is not a conic section but some ‘made-up’ curve generated by me fashioning a mathematical function based on ‘pure numbers’ and NOT geometry, that I have an infinite number of answers? Moreover, can I ask, is a general inverse square law ‘quadratic’. Is it a law that follows a curvature from a conic section or not?
    Im no good at conic sections so i wont respond to that part of your post; but i think part of your confusion is that "inverse square law" and "elipses" are getting mixed up which hopefully has been cleared up above.
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    Re: Modelling a Gravitational Field NOT using an inverse square law

    Dear Spring,

    Let me give you a bit of further information and try to tell you exactly where I am coming from. I am familiar with the gravity equation that you have just stated involving the universal gravitational constant, multiplied by two mass values, and divided by radius squared.

    This is a very intriguing law. And though it appears to be correct, there is a certain assumption that is made when one uses it. It is assumed that the total mass of a given body can be concentrated into a singularity point right at the centre of the body itself. Now I have been investigating gravity fields as surround celestial bodies which are essentially hollow. Basically, imagine such as the moon. We know it's mass value, but what if the total mass were uniformly distributed in a shell (of say) 100 miles thick from the surface. And that the rest of the body was hollow.

    Now I have done a major investigation making use of Newton’s equation, as you have stated. But what I did was as follows. I assumed I had a man stood on the surface of the moon (NB: moon radius = 1080 miles). And instead of assuming all the mass was concentrated at a point like position at the centre of the moon, I split up an inner circle - which was 100 miles less than the physical radius of the moon (980 miles radius for my inner circle) – into a semi-circle. Now, there are 180 degrees to half a circle, and thus 181 points covering such a half circle. I divided the total mass of the moon by 181. I then used basic trigonometry and a Microsoft Excel spreadsheet, to work out the straight-line distance between my man stood on the circumference of the moon, and each of the 181 points on the inner half circle - 100 miles radius less than the moon physical circumference. I then used Newton’s equation to calculate the force of gravity from each point of matter on the 181 points on the inner circle, as felt by the man, and then added the whole lot together.

    In doing this I found that you could get the surface gravity a lot higher than you might expect than if you were to assume that the whole mass was concentrated at the centre of the moon. However, here is the thing. The more you move away from the moon into deep space, the gravitational field appears to closely align itself to a perfect inverse square law. And this is because the more you move away from the moon, the gravitational acceleration values weaken in the curvature of the gravitational field. And, when you reach an infinite distance from the moon, you appear to align with a perfect inverse square law so to speak. I hope you grasp what I am saying.

    Basically, gravitational fields surrounding what are essentially hollow celestial spheres do not follow a perfect inverse square law as is standard to the equation as you posted.

    Now what I have been looking at, in following on from this is the idea that there may well be different mathematical laws which govern the weakening of gravity fields away from a celestial body which may well be hollow, which do not conform to an inverse square law. This is essentially what I am looking at.

    When I gave the three points of data, this was an experimental set of data points that I thought might correctly model the gravitational field surrounding a body which would be primarily hollow and which would not follow an inverse square law. This is essentially where I am coming from.

    Now as I said previously, I am not a mathematician. Not in any technical sense. And this is why sometimes it is difficult for me to grasp what mathematicians are saying because they may be assuming I know things that I don't.

    I have done a lot of work and am even a published author, looking at the gravitational movements of the planets about the Sun, and have gathered evidence to suggest that the planets do not conform to an inverse square law about the Sun, but they each have a different mathematical function which is subtly different from an inverse square law which governs their movement about the central body. I do not wish to go too much into the details of this as it is extremely complicated to explain.

    Sometimes when I want to test theories I am stretched to the limit because I literally do not know how to mathematically test my theories. I know how to describe them in ‘commonsense’ language but I don't know sometimes whether or not there is a mathematical solution to my theories, because of the jargon used I am not certain what mathematicians might describe as being a solution to some of the ideas I have. This is the most difficult aspect of my work, for as I say, I am not a mathematician. And it is difficult sometimes to test my theories because oftentimes I do not have the mathematical knowledge to test them.

    Sincerely

    Keith
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