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A Problem of REOCCURENCE

  1. Mar 8, 2008 #1
    A Problem of "REOCCURENCE"

    1. The problem statement, all variables and given/known data
    Suppose there are 3 heavenly bodies in space - say sun and 2 other planets A and B which revolve around the fixed sun with some fixed time periods. Also consider that at a particular instant of time they are present EXACTLY along a straight line in space.
    we have various instruments which can measure the time periods with different least counts (L.C's) - say 10 yrs., 1yr. 0.1 yr. etc. The problem is to find when this condition again occurs. The difficulty is that as the measure ment gets more and more accurate this "reoccuring" time becomes larger and larger....
    (N.B : 1.The 2 planets must be aligned along the SAME line once again)
    2.We can consider the planets to be point sized, for the sake of simplicity and
    clarity)
    2. Relevant equations
    The reoccuring time will be the LCM of time periods. in case LCM is not an integer it must be multiplied by a suitable integer so that the result is close to an integer (appx.ly)


    3. The attempt at a solution
    Illustration of my difficulty: let us consider the first case of measurement of time period with an instrument of L.C 10 yrs.
    let time period of planet A be measured as 40 yrs. and that of B be measured
    as 120 yrs. then we can calculate the answer as 120 yrs. (LCM of 40 and 120).
    let us use an instrument of greater accuracy now - say 1 yr LC. then suppose it reads 43 yrs and 122 yrs.then the required answer becomes 5246 yrs.
    suppose we use instrument of LC 0.1 yr and it reads time periods as 43.2 yrs and 122.3 yrs. their LCM is 5283.36 yrs. (their product). by this time A would have completed 122.3 revs B would have completed 43.2 revs they would be along a line with the sun but not along the same initial line. so we multiply this answer with a suitable whole number to get (almost) a whole number as the answer. in this case 3 is suitable as 5283.36 multiplied with 3 yields 15850.08 (appxly a whole number). hence we see that there is again a huge difference.
    if we manufacture instruments having high accuracies then this "answer" will scintillate further to absurdly higher values.
    does it mean that this situation is seen only after infinitely large time ? if not, is there any means to find when exactly this will happens ? how accurate an instrument will yield that answer ?
    Please help me out out of this tight spot.... i am unable to think it out.....
     
  2. jcsd
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