Compendium* A Mean Time Sundial by Hendrik Hollander

Transcription

1 Volume 3 Number ISSN (printed) ISSN (digital) June 006 The Journal of the North American Sundial Society Compendium* A Mean Time Sundial by Hendrik Hollander Time and reflection change the sight little by little till we come to understand. - Paul Cézanne * Compendium... "giving the sense and substance of the topic within small compass." In dialing, a compendium is a single instrument incorporating a variety of dial types and ancillary tools. 006 North American Sundial Society

2 Bi-Gnomon Sundials Hendrik Hollander (Amsterdam, the Netherlands) The principle of the bi-gnomon sundial is to mount a shadow casting object which casts shadow lines on a sundial plane. Each shadow line can be associated with a separate time system and orientation of the sundial plane. This will enable us to build sundials which can be used for different locations on earth or sundials which function with time systems or sundials which indicate mean time or clock time. [Ed. Note: For an early version of this idea, see F. Sawyer s Compressed Gnomonic Sundials, The Compendium, Mar 005, ():-8 + Addendum.] In general the bi-gnomon sundial s manual is: put the sundial in the right position in the sun, look where the shadow of the relevant gnomon intersects the date line of today and read the time. The restrictions for the gnomons are minor:. any location of the sundial for the separate gnomons is possible (also the same location for both gnomons). any orientation of the sundial plane is possible for both gnomons (also the same orientation for both gnomons) 3. any shadow casting object which generates two straight shadow lines is possible (for example straight nails with random positions, or for example a cone if one looks at the left and right shadow of the cone) 4. any time system can by chosen for the separate gnomons (local, mean, clock time, ) No formulas are presented in this article. The formulas needed to calculate a sundial with a cone gnomon which is parallel to the earth s axis are presented in a separate article. Let us discuss some nice sundials below: - a sundial with perpendicular gnomons which can be used for daylight saving time. The different gnomons simply indicate hour time difference (no equation of time incorporated in this design) - a sundial which indicates the clock time (or daylight saving time). So including the equation of time but without the analemmas - using straight gnomons, not perpendicular to the sundial plane. 3- a sundial with a perpendicular cone gnomon 4- a sundial with cone gnomon which is parallel to the earth axis The required algorithms are discussed for each sundial.. A sundial with perpendicular gnomons that indicate hour difference. The design algorithm is straightforward: - the sundial plane is fixed - time system for gnomon is the local solar time including longitude correction - time system for gnomon is the local solar time including longitude correction + hour - draw the shadow lines for a fixed date for both the gnomons in their own time system - mark the place where the shadow lines cross for that date. Apparently, the shadows of the different gnomons (with their own time system) will cover this point on this date. - repeat this procedure for different dates and hours. - connect the points for datelines and hour lines Below is a design for the Netherlands (lat. 5 long. 5): The Compendium - Volume 3 Number 3 September 006 Page

3 The gnomons are perpendicular to the sundial plane. They are east-west oriented. The hour lines are marked with the time. Depending on which gnomon you choose, this is normal time or daylight saving time (without the equation of time). Check: draw parallel lines over the gnomons and you will see that they differ hour on a specific date line. This design can easily be made for a random vertical wall. When the date lines are drawn for the nightly hours, the ellipse-shaped datelines are completed, see the next drawing. The Compendium - Volume 3 Number 3 September 006 Page

4 . A sundial that indicates the clock time, without analemmas, with straight gnomons The algorithm is straightforward: - the sundial plane is fixed - time system for gnomon is the clock time during the period winter to summer (Period w-s ) - time system for gnomon is the clock time during the period summer to winter (Period s-w ) - choose pares of dates with the same declination of the sun ( date in the Period w-s and for date in the Period s-w ) - draw the shadow lines for these dates for both the gnomons in there own time system - mark the place where the shadow lines intersect for that declination. Apparently, the shadow of the different gnomons (with their own time system) will cover this point on these dates. - Repeat this procedure for different dates and hours. - Connect the points to datelines (actually: declination lines) and hour lines When we place a vertical rod and we draw the horizontal sundial with clock time around it, we will find the well known horizontal sundial with the analemmas. I have incorporated the lines for pairs of dates as indicated above. These date lines are calculated for multiples of 30 degrees of the longitude of the sun. Often these lines are marked with the signs of the zodiac. The lines will mark date in the period of winter to summer (Period w-s ) and date in the summer to winter (Period s-w ). See the next figure. The red (highlighted) lines are the datelines for the days that the equation of time is the same during the Period s-w and Period w-s. We need these lines later on. Period s-w is the period from Summer to Winter Period w-s is the period from Winter to Summer winter line Red lines indicate the dates for which the equation of time is equal during the period s-w and period w-s summer line To read the time, one has to know which side of the analemma to use. We can build a bi-gnomon sundial with gnomon for the Period s-w and one gnomon for the Period w-s. Doing so, the analemmas disappear. I have chosen gnomons which are not perpendicular to the sundial plane. See the next drawing. The gnomons intersect above the sundial plane, above the small circle. The gnomons are mounted to the sundial plane at the small + signs. If the distance between the + signs is, the distance of the intersection point of the gnomons is 0.5 above the plane. The line through the + signs is east to west. The Compendium - Volume 3 Number 3 September 006 Page 3

5 gnomon for the periode s-w gnomon for the periode w-s gnomon for the periode s-w gnomon for the periode w-s The datelines are marked with the zodiac signs. Note that the declination of the sun is the same during Period s-w and Period w-s for each dateline. A photo of a paper model of this sundial is printed below: The shape of the double gnomon is visible in the photograph. Note that this gnomon construction can indeed mark different times at dateline. When the shadow of the intersection point of the gnomons is above a declination line, the shadow of each gnomon will mark a different time. This time can be associated with the dates of the declination line. When the shadow of the intersection point of the gnomon is exactly on a declination line, both gnomons indicate the same time. This has to be the case The Compendium - Volume 3 Number 3 September 006 Page 4

6 when the equation of time is equal for that solar declination during the Period s-w and Period w-s. In the next figure I have marked these datelines red (highlighted). Red lines indicate the declination lines for which the equation of time is equal during the period w-s and the period s-w We can easily check whether these lines are the same for the bi-gnomon sundial and the standard sundial with the analemmas. Both sundials are printed together in the above figure (right). Although a lot of differences between the lines are visible, it is clear that the red lines are the same for both sundials. Cones The sundial with the straight line gnomons as described above has some shortcomings.. the datelines and hour lines appear mostly on side of the gnomons. therefore it is not always possible to show all hours in the sundial To overcome this issue, the straight line gnomons can be replaced by a cone. The shadow of the left side of the cone (with the sun behind you) will indicate the first time system. The shadow of the right side of the cone (with the sun behind you) will indicate the second time system. 3. Sundial with a perpendicular cone gnomon To get a feel for the date lines and hour lines, let s use a cone which is perpendicular to the sundial plane. During the period w-s the shadow of the right side of the cone is used. During the period s-w the left side. In this figure, we see how an hour line is built up. The algorithm: - choose a declination of the sun - find both dates on which the declination is valid - draw the shadow lines, using the valid side of the cone - mark the intersection point of the shadow lines - repeat the steps above several times - connect the points to the date lines and hour lines Creation of an hour line. The shadow lines of the different sides of the cone are combined The cone with the apex above the red cross The Compendium - Volume 3 Number 3 September 006 Page 5

7 A full sundial design is shown below. The intersection of the cone and the sundial plane is drawn as a (blue) circle. The apex of the cone is above the (red) + sign. When the radius of the circle is, the apex of the cone is 3 above the sundial plane. So: look at the shadow of the proper side of the cone (the Period s-w or Period w-s.), extrapolate the shadow line to the date line if needed and read the clock time. Winter line clock time The radius of the cone is. The height is 3 Summer line daylight saving time Indeed all hours can be shown on the sundial. To undertand how the sundial marks two different times on a specific date line, the next figure is drawn. Oct 9 Feb line The shadow of the different sides of the cone do indicate different times. For example, the date line of October and 9 February is shown. The shadow of the right side of the cone indicates 0:00 wintertijd on February 9th. The shadow of the left side of the cone indicates 0:5 wintertijd on October nd. The Compendium - Volume 3 Number 3 September 006 Page 6

8 4. sundial with a cone gnomon which is parallel to the earth axis As will be noticed, the hour lines are curved. To overcome this, the cone can be tilted in a way that the central axis of the cone is parallel to the earth axis. All hour lines will become straight (well, a very small deviation as we will see later on) Cone gnonom with axis parallel to the earth axis indicates mean time with straight hour lines 4:00 :00 Apex of the cone is above the red + sign, intersection of the cone with the sundial plane is the blue ellipse. The intersection of the cone with the sundial plane is an ellipse. The top of the cone is above the + sign. This design is for lat. 5, long. 0, including the equation of time. The top of the cone is 0.79 above the plane (relative to the scale in the edge of the drawing). To the left is a picture of a similar sundial with correction for the equation of time and longitude correction, so it indicates clock time. 5. The equatorial sundial with a cone gnomon The sundial with the cone gnomon can be designed as an equatorial sundial. First we take a look at the northern part (summer part for northern lat.). See the picture below, left. The Compendium - Volume 3 Number 3 September 006 Page 7

9 Northern side of the equatorial sundial with cone gnomon, height of the cone is radius of the cone is indication: clock time Southern side of the equatorial sundial with cone height of the apex is 3.6 radius is indication: clock time 0 febr febr. 006 As can be expected, the hourlines are the same as for a sundial which indicates the local solar time. However, this sundial indicates the clock time. The inner red circle is the zodiac line for the start of the summer. The middle red circle is the next zodiac line. Unfortunately, the equinox line is not available, it is for out of this picture (although, not at infinity). So this side of the sundial will indicate clock time from approx. April to August. The southern part of the sundial will tell the time from approx. october to February, for the same reason: the equinox date line is at infinity. The design for the northern part is generated for the radius, the top is above the red + sign. The southern part of the sundial is also depicted below, right. Here we have some unexpected features. Since the equation of time changes rapidly, the date lines move rapidly towards the center or outside the center. Carefully designing the size of the cone will put the datelines (zodiac lines) together. So, moving from infinity towards the center we have: - infinity: September *) - outer circle: October - inner circle: November - outer circle again: December - inner circle again: January - outer circle again: February - infinity: March *) (actually the datelines are the zodiac lines, so the st is for reference only, actually the red circles are the lines for multiples 30 degrees of the longitude of the sun, which is what I have implemented). When the radius is, the top of the cone is 3.6 above the plane. To design the sundials with an oblique cone gnomon as user friendly as possible, Fred Sawyer has suggested a way to associate the left and right shadow of the cone with the dates of the year in *)in detail: the lines for spring and autumn (approx. at March and September) are not at infinity. Since the equation of time is not equal on these dates, the shadow of the cone has to intersect this dateline. The Compendium - Volume 3 Number 3 September 006 Page 8

10 Left shadow side of the cone Approx. April th to June 0th Approx. August 30th to December 0th Right shadow side of the cone Approx. June 0th to August 30th Intersection about August 30th and April th Approx. April th to December 0th such a way that the shadow will always intersect the current date line (See figure on left). This definitely improves the concept and is incorporated in, for example, the paper cut out dial pictured on the previous page. Some last remarks about the straightness of the hour lines. The hour lines (indicating clocktime with the cone gnomon) are straight because the equation of time is almost symmetrical with respect to the sun s declination. The very small deviations are shown in the picture below. Local solar time with standard gnomon Clock time with cone gnomon The actual shape of the equation of time is determined by the elliptic orbit of the earth and the obliquity of the ecliptic. Currently, the perihelion (the point where the earth is the closest to the sun) is at January 3rd. When the perihelion coincides with the start of a season, the equation of time is symmetric. During the year 46 the perihelion coincided with the start of the winter at December 0th and therefore the equation of time was 00% symmetrical. Building the same picture as above for the year 46 returns the picture below right. As can be noticed, the hour lines are 00% straight. The analemmas with standard gnomon I want to thank Fred Sawyer (President of the North American Sundial Society) for his support and good suggestions on the algorithm (see above) and also Fer de Vries (Secretary of the Dutch Sundial Society De Zonnewijzerkring) for his support and checks on the algorithms The well known analemmas Local solar time The clock time with cone gnomon Hendrik Hollander, De Breekstraat LJ Amsterdam, the Netherlands info@analemma.biz The Compendium - Volume 3 Number 3 September 006 Page 9

11 Basic principles of the mean time sundial with a cone gnomon On the sundial plane the hour lines and date lines are present. These lines look very similar to a regular sundial. Also a shadow casting cone is present. The central axis of the cone is parallel to the earth s axis. One can see the shadow of the left side of the cone and the right side of the cone separately. For which period of the year you have to use the shadow of which side of the cone is marked on the sundial. Look at the date line of today and look at the shadow of the correct side of the cone. Read the mean time on the date line. Mean Time Sundial With A Cone Gnomon Hendrik Hollander (Amsterdam, the Netherlands) Below you will find the mathematical background of the mean time (or clock time) sundial with a cone gnomon. The connection between the longitude of the sun, the declination of the sun and date lines on a sundial As regularly done by dialists we use 30 degree multiples (or even better, 0 degree multiples) of the longitude (λ) of the sun to mark the date. The seasons are marked by λ=0 for spring, λ=90 for summer, λ=80 for autumn and λ=70 for winter. Often the lines of the multiples of λ=30 are marked with the zodiac signs. In general each declination of the sun has two λ s associated. One for the period of the winter solstice to the summer solstice (λ w-s ) and one for the period of the summer solstice to the winter solstice (λ s-w ). For instance the line for spring and autumn is the same line (declination of the sun is 0 ) and has λ w-s = 0 and λ s-w = 80 associated. In general λ w-s and λ s-w are related by: o λ w s = 80 λs w with λs w [90 notice that λ w-s and λ s-w represent the same declination of the sun and therefore represent one date line on a standard sundial with a gnomon. We are now to build a mean time (or clock time) sundial with the same date lines. The algorithm of the mean time/clock time sundial Follow these steps:. Choose a valid combination of λ w-s and λ s-w o,70 o ] The Compendium - Volume 3 Number 3 September 006 Page 0

12 . Associate λ w-s with the shadow of one side of the cone and λ s-w with the other side (see addendum for a good suggestion on this choice) 3. Choose a moment in mean time or clock time 4. Calculate the shadow lines for this moment and the λ s of both sides of the cone (formula s are presented below) 5. Calculate the intersection of the shadow lines 6. Mark this point on the sundial plane, this point is part of the hour line of the time chosen in step 3 and is also part of the date line associated with λ w-s and λ s-w 7. repeat step to 6 many times for different λ w-s and λ s-w and different moments and connect these points to hour lines and date lines Addendum Step. I developed these algorithms during February 006. During the very nice discussions I had with Fred Sawyer about this subject, Fred suggested a way to associate the λ s so that the shadow of the cone will always intersect the date lines x. This makes the sundial very user friendly and this concept is incorporated in the algorithm ever since. [Editor s Note: See the following article.] To force this, one has to associate the period that the sun is fast in respect to mean time (or clock time) to the left side of the cone with the sun behind you. Calculation of the cone The central axis of the cone should be parallel to the earth axis and therefore, in general, it is oblique to the plane of the sundial. The cone can be drawn on a flat paper, cut out and put together to the 3 dimensional shape. Defining: h : the length of central axis of the cone from the sundial plane to the apex γ : the apex half angle φ : the angle between the cone axis and the sundial plane (φ = 90 means the cone axis is perpendicular to the sundial plane) (for horizontal sundials φ can be interpreted as the latitude) In polar coordinates (r,θ) the cone can be plotted with: h r = using o θ cosγ + sin γ tan(90 ϕ) cos sin γ θ 0, π sin γ. [ ] An example of an unfolded cone is shown here. To be able to draw the ellipse in the sundial plane where the cone will be mounted we define further: q : the distance between the heart of the sundial (where the cone axis intersects the sundial plane) and the center point of the ellipse x y the ellipse itself: + =, we notice that the apex of the cone is located at a b The Compendium - Volume 3 Number 3 September 006 Page

13 x apex = h cos ϕ + q, z apex = h sinϕ and z h Apex h sin γ h sin γ a = + sin( ϕ γ ) sin( ϕ + γ ) h sin γ q = a sin( ϕ + γ ) φ γ Cone Sundial plane (x;y) b = ha tan γ a q y The apex of the cone will cast a shadow by a sunbeam at a certain time and declination of the sun. This shadow point of the apex can be calculated with standard methods for sundials with a gnomon. Defining this shadow point as (P, P ), we notice that this point will be part of the shadow line of the cone. Also the shadow line will be tangent to the ellipse. Let this shadow line be y P ) = ( x P ) r. ( P P ± it can be shown that r = ( a P + b P ( a P ) b x y It must be checked that (P, P ) is outside the ellipse which can be done with: + a b q a One has to distinguish between the shadow of the left and right side of the cone with the sun behind you. This can be done with: b ) a x The basis of the cone is an ellipse if if sign( P ) = sign( P ) left sign( P ) = sign( P ) left shadow line is the shadow line is the smaller r larger r I want to thank Fred Sawyer (President of the North American Sundial Society) for his support and good suggestions on the algorithm (see above) and also Fer de Vries (Secretary of the Dutch Sundial Society De Zonnewijzerkring) for his support and checks on the algorithms. Hendrik Hollander, De Breekstraat LJ Amsterdam, the Netherlands info@analemma.biz The 006 Sawyer Dialing Prize has been awarded to Hendrik Hollander for his innovative design of a mean-time planar sundial with oblique conical gnomon and modified hour lines and day curves resulting in a sundial adapted to modern timekeeping while retaining the aesthetic appeal of the familiar dial face. The Compendium - Volume 3 Number 3 September 006 Page

14 Equations For Hollander s Mean Time Sundial Fred Sawyer (Glastonbury CT) At northern latitude ϕ, begin with a rectangular coordinate system with x increasing to the east and y to the north. Consider a right circular cone with apex half-angle α < Slice the bottom portion of the cone off at such an angle ( ψ α ) that the remaining portion of the central axis will be inclined ψ > α degrees above the horizontal. Let the length of this central axis be g. In the case where ψ = ϕ, the resulting hour curves will be very close to the hour lines of a traditional horizontal sundial. Align this cone so that its central axis lies in the meridian plane, inclined to the pole. The base of the oblique cone will now be an ellipse. Set the center of this ellipse at the origin of the coordinate system. The ellipse: ( a) ( y ) x + b = where = g cot α cot ψ Let q = g sinα ( csc ( ψ α ) csc ( ψ + α )) ( ) a, b = g sinα csc ( ψ α ) + csc ( ψ + α ). The base of the axis will be at point (0, q). Its apex will be at (0, q + g cosψ, g sinψ ). This inclined cone will serve as the gnomon. (See development below.) Now consider a declination δ 3.44 and a mean time -80 < τ 80. Except for the two solstice values, the declination δ corresponds to two dates day and day, each of which has an associated equation of time: ε and ε, respectively. View the equation of time as a signed angle, positive when the sun is slow with respect to mean time, and negative when fast. In particular, assign subscripts to these dates such that ε > ε, i.e. day is the date of the two for which the sun is slower with respect to mean time. For each of these days, there are distinct solar times ti t <. = τ ε i. Note that t Consider the points ( x i, y i ) which are the shadows of the apex of the cone on dates day i at mean time τ. sin ti x i = g sinψ, cost cosϕ + sinϕ tan δ i cos ti sinϕ cosϕ tanδ y i = q + g cosψ + g sinψ. cost cosϕ + sinϕ tan δ i Note that, assuming we are in the northern hemisphere, point ( x, y ) is slightly clockwise from point ( x, y ) as viewed from the origin (since t < t ). Given the ellipse ( a) + ( y b) = x (or equivalently, x = a cosω, y = b sin ω ), the equations of all lines with slope m and tangent to the ellipse are: y = mx ± a m + b. To find the tangents to the ellipse that pass through point ( x i, y i ), we substitute these values for x, y and solve the resulting quadratic equation for m i. These tangents are the leading and trailing edges of the cone s shadow on the given date and time. We have here two possible values for m i, corresponding to the slopes of the two tangents to the ellipse from the given point. For our purposes, we want the value of the slope of the lead shadow for m and of the trailing shadow for m. Slope of Leading Edge Slope of Trailing Edge m x = y + a y x + b a x a b x y m = a y x + b a x a b The Compendium - Volume 3 Number 3 September 006 Page 3

15 We now have two lines: a leading shadow edge through one point ( x, y ), and a trailing shadow edge through a point ( x, y ) slightly clockwise from the first point. The lines are identified by the following equations: y = mi x + ki, where ki = yi mi xi. Note that this selection of pairs of lines will always result in their point of intersection being part of the shadow; i.e. the intersection will be on the line segments between the shadow apex and the point of tangency with the cone s elliptical base. Had we selected the other two possible lines, their intersection would have occurred on the extension of the lines beyond the apex shadow point and would thus have been in sunlight. So we can now identify the point ( x δ, τ, y δ, τ ) of intersection of the two lines as follows: x δ, τ k = m k m y δ, τ k = m m k m m By this construction we know that on each of day and day associated with the solar declination δ one edge of the gnomon s shadow passes exactly through the point ( x, y τ ) at mean time τ - the leading edge for day (dates Dec - Apr and Jun - Aug 3) and the trailing edge for day (dates Apr 3 Jun 0 and Sep Dec 3). We can now draw a single curve for the two days associated with declination δ by calculating the positions of the intersection points for the range of mean times when the sun is above the horizon. Similarly, hour curves (which, as it turns out, are very close to being the straight hour lines of a traditional sundial) may be drawn by locating the intersection points for fixed values of τ and varying declination values. The resulting grid of day and hour curves will produce the dial face for a mean time sundial. Mean time is read from the intersection of the proper shadow edge with the current date line. Cone Development - A cone is a developable shape; it can be produced from a flat surface. To obtain the required oblique cone in this manner, draw and cut out a figure such as is shown here, with the curve s polar coordinates as follows: tanψ ρ = g, for θ 80 o sinα. cosα tanψ sinα cos θ sinα The Compendium - Volume 3 Number 3 September 006 Page 4 δ, τ δ, ( ) Bring the two straight edges together and fasten; adding tabs may facilitate this fastening. Hendrik Hollander, Bi-gnomon zonnewijzers, Bulletin van De Zonnewijzerkring, Jan 006, 06-:6-3. Hendrik Hollander, Bi-Gnomon Sundials, The Compendium, Sep 006, 3(3):???. Hendrik Hollander, Mean Time Sundial With A Cone Gnomon, The Compendium, Sep 006, 3(3):???. Fred Sawyer, Compressed Gnomonic Sundials, The Compendium, Mar 005, ():-8 + Addendum. Fer de Vries, Samengedrukte gnomonische zonnewijzers, Bulletin van De Zonnewijzerkring, 05-3, Sep 005. Fred Sawyer, 8 Sachem Drive, Glastonbury CT fwsawyer@aya.yale.edu