The Supermoon Period

  A supermoon occurs when a full moon and lunar perigee occur at approximately the same time. There was one just before sunrise this morning, 14 Nov 2016.  To find out when the next one will occur we can compute the average time between supermoons using two lunar periods.

So supermoons are spaced about 13½ calendar months apart and the next one should occur at the beginning of January 2018 followed by one about the middle of February 2019. The Lunarian gives the following circumstances for these events.

Lunar perigee 1 Jan 2018 21:48
Full Moon      2 Jan 2018 02:24

Lunar perigee 19 Feb 2019 09:02
Full moon      19 Feb 2019 15:53

The computed interval for the Supermoon Period seems to work quite well. One can expect some variation for this time interval due to the nonuniform motion of the Earth in its orbit.

Magic Squares

  A magic square is an n x n grid of numbers whose rows, columns and diagonals all have the same sum. One can create a magic square by filling a 4 x 4 matrix with 7 numbers in the positions below. I included the numbers of the election date, 11/8/2016, in the bottom left corner. The remaining 3 numbers were chosen at random. If one knows the value of the sums, Σ, one can immediately solve for some of the unknowns.

We can use the unused equations to separate the the known variables from the unknown variables as follows and solve the system of equations for the unknowns k, l, o & p. One of the equations is redundant so we choose 4 for which the determinant of matrix, M, is nonzero. Here the determinant D = |M| = 2.

Filling the calculated values into our magic we get the following matrix which checks out correctly.

Each value of Σ generates a new magic square. One can try to choose the minimum value for Σ for which all the numbers are non-negative.

The Penny's Decline in Value Over Time

  In 1266 King Henry III  introduced an "English Peny, called a Sterling," 12 of which weighed an ounce with 20 ounces in a pound. The relative values of the penny, shilling and pound remained the same until decimalisation in 1971. But over time the penny has declined in value. The Sterling penny was silver with a little alloy mixed in about the fineness of modern sterling silver. The name Sterling may have been an indication of it value being equal to that of a yearling steer.

We can do a rough calculation to determine the rate of decline in the penny's value over time assuming a yearling steer was worth one penny in the year 1300 and $1500 in 2000.

That's less than half the modern annual rate of inflation in the US of about 4%.

The Only Consistent Period for Gadbury's 1672 Sun Positions

  The estimated period for Gadbury's 1672 Sun positions was based on an assumed value for the sinusoidal terms. A plot of the estimated value as a function of the initial assumption (magenta curve) shows that there is only one consistent value for the period.

The cobweb in the plot above show that the approach to the limiting value is quite rapid.

Effect of the Number of Days on the Orbital Period

  The method used in the last blog does a better job of separating the linear and nonlinear motion with data for an incomplete year but the accuracy deteriorates when the number of days is less than half a year.

The subscripts indicate the number of days used for the fit.

A Simpler Orbital Period Estimate

  It's better to include the sinusoidal terms with the linear terms when doing a least squares fit as the following example shows. Again, Gadbury's 1672 Sun positions were used. It's difficult to tell if the period has changed over the elapsed time. The value used for T appears to be stable, yielding the itself back for the estimate of the period. For comparison, the value given in the Explanatory Supplement to the Astronomical Almanac is 365.2421897 days.

The result varies with the number of sinusoidal terms used for the fit. I have some doubts about this fit even though it gave the minimum error.

A Caution on the Extracting Process

  The extraction of the Sun's mean motion will not always work if one does not know the precise form of the sinusoidal motion and the calculation is not done over the approximate period. However, using cosines and sines for the approximate period and half that seems to work fairly well. Below 301 days were used instead of 366. The first day was t=0.