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.

More Detail on the Orbital Period Calculation & a Filter Matrix

  The two step least squares process of extracting the Sun's mean motion from the position data can be reduced to the multiplication of the positions by a single matrix, P.

The linear and non-linear functions can be represented by two matrices, f and g.

Repeated multiplication of the modified positions by P moves us closer to the approximate mean motion and the resulting period can be calculated.

The four decimal place accuracy of the previous result for the period may have been a coincidence but the uncertainty appears to be in the 3rd decimal place. The nonlinear portion of the positions no longer appears to be tilted.

The Orbital Period for Gadbury's 1672 Data

  When trying to fit observations of the Sun's position to an elliptical orbit one needs to know the mean motion or the period. But when working with a set of data points subject to error this can be a little difficult but not impossible. We can use Gadbury's 1672 positions for the Sun to illustrate the process. To separate the linear and sinusoidal parts of the motion we first do a least squares linear fit of the ecliptic longitude, θ. The matrix f contains two column vectors with the values of the first being all equal to 1 and the values of the second are equal to the row index, k. The difference between the observed and fit values is δ. As mentioned before the least squares fit can skew the fit to get a lower value for the sum of the squares of the error so there is still some mean motion in δ. We can get around this by fitting a series of sinusoidal curves and subtract this from the original data to get a more linear curve θ'.

Repeating this process slowly extracts the linear mean motion and the slope of the line converges to a fixed value. This slope can be used to compute the period, T, of the orbit since the Sun moves through 360° in this time. The value found is very close to that of the tropical year which is the length of time it takes to go from one Spring Equinox to the next.

You may have noticed that a value for the semi-major axis of the orbit was missing from our set of orbital elements but this can be computed using Kepler's Third Law. One can fit the curve above to a constant term plus an exponential curve if the decay rate is known and search for the rate that produces the least error for the fit.