Calculations

Instruments and software

Instruments used for the measurement of stones were purchased in 2008:

  • Theodolite Geo-Fennel FET-500
  • Laser Meter Leica Disto A5. Somewhat dependant on the reflective qualities of the stone, this can accurately measure distance up to 40-50m.

For calculations of the probable view of the horizon in a deforested landscape, GSD- Height Data using grid 2+, produced by Lantmäteriet, was used. This information is produced using data from airborne laser scanning and contains height data in a two meter grid to a very high tolerances. This makes possible exact calculations of the height of the horizon even in cases where today the burial site is completely surrounded by forest.

Two software programmes from Alcyone have been used.

  • Planetary, Lunar and Stella Visibility. This programme provides data about the position of the sun, moon, planets and stars from 3000 BC until 6000 AD, taking into account the refraction of light in the earth’s atmosphere. I have added an extra correction as the program is not capable of calculating accurate refraction for altitudes under 0.5°.
  • Astronomical Tables. This contains information about, among many other things, solstices and equinoxes from 3000 BC until 3000 AD.

Shadow lines – Three criteria for marking calendar days by the shadow from a standing stone falling in the right position on a lower standing stone

I have used three criteria to enable to decide which shadow lines can be considered actual, that is to say formed deliberately to indicate important days in the calendar. Only those shadow lines that fulfil these criteria are classified as actual.

Winter and summer

During summer and winter the sun’s path deviates very little from one day to the next. This means that it is not possible to indicate a particular day using a shadow.  At the summer and winter solstices it is completely impossible, while on the Blot days that occur a half moon cycle later it may be possible, even if with some uncertainty.

As a deviation expressed in days is not suitable I have instead chosen to use the deviation from the sun’s height as the criterion for acceptable variation in summer and winter days of the calendar.

  • The criterion for all calendar days that fall in summer or winter shall be that the deviation in height of the shadow on the lower stone in relation to the top of the stone is not greater than 0.25°.

Spring and autumn

During spring and autumn the height of the sun’s passage deviates markedly from one day to the next. This makes it simple to show the exact date using stone formations. For this reason deviation measured in day(s) may be used as a criterion when approving or rejecting a formation.

  • The criterion for calendar days during spring and autumn shall be that the maximum deviation in height may be 0.5 days. Both the equinox according to today’s definition as well as the day when day and night are of equal length to be considered.

The shape of the stones and the sun’s path

Aside from the shadow falling correctly on the lower stone it is also a requirement that at least one of the two stones should have a gradient on its top that conforms to the sun’s path. Because the stones can look very different in appearance it is hard to define the criterion in the form of a specific number. Instead a subjective assessment has been made.

  • The criterion for an acceptable formation is that at least one of the two stones has the same gradient as the sun’s path.

Horizon lines – Two criteria for standing stones that mark the calendar day by pointing out where on the horizon the sun rises or goes down

Those shadow lines that fulfil the two following criteria are classified as actual.

  • The line formed by the two stones are allowed to point to a point up to 0.25 ° below the horizon. The line may point toward a point above the horizon, where, however, the second criterion that follow must be met.
  • At sunrise the sun is allowed appear first after it has passed the standing stone and at sunset, it shall be hidden after passing the standing stone. This means that the stone’s shape must be such that such leakage from the sun is prevented. A small leakage of the sun’s light is accepted, however: the criterion is therefore that the leakage must not exceed 0.25 ° in height before the stone passed at sunrise and after the passage at sunset.

Leaning stones and re-erected fallen stones

In the case of all leaning but standing stones a calculation has been made of their exact position if straightened up. It is their position after this adjustment that is used to calculate the sun lines. In the event that an adjustment has been made to a leaning stone this will be noted in the presentation of the sun lines.

In the vast majority of cases I do not have any documentation regarding which standing stones may have been previously leaning or fallen, and have since been re-erected. This uncertainty must remain an intangible factor in the overall results from my measurements.

For which year have the sun’s movements been calculated?

All calculations have been done for the year 645 AD. There has been no dating done that shows exactly when the stones were put in place. My guess is that this occurred during the Vendel period, covering the years 550 AD until 800 AD. The exact times of year when the solstices and equinoxes fall vary by less than ± half a day. So that the year chosen is as neutral as possible I have avoided leap years as well as the year prior to a leap year, as both of these are at the extreme end of the scale. 645 AD is one year after a leap year and should therefore give adequately average result when it comes to where the solstices and equinoxes fall. Otherwise the choice of 645 AD may be viewed as nothing more than a random choice.

The choice of year used for calculating the sun’s position is of fairly little importance as the sun’s path has altered very little since the Iron Age. For example, the sun stood approximately 0.2° lower in winter and 0.2° higher in summer during the 7th century than it does today.