Diurnal plane and apps

Motivation

There was a long discussion about the following brAhmaNa of the taittirIyakas- तैत्तिरीयब्राह्मणे ऽत्र।.

The classical commentators (as well as modern shrauta and smArta ritualists familiar with ancient and modern astronomy I checked with) clearly require measurement of the time required by the sun to “reach” a certain point “behind” the position of a desired naxatra at sunrise. Of course, this means reaching a certain point on the sun’s daily path, [let’s say that it lies in something called the Diurnal circle (defined below)], which is comparable (say in altitude or proximity) to the point where the nakShatra was last seen.

Certain objections were raised, among which were (I paraphrase and correct misstatements):

  • “You cant see the “dirunal plane” with the naked eye! It’s hard to get! They did not have compass and divider.”
    • Response: This objection is addressed with this note. We will show that without needing to stare at the sun, and no fancy instruments, we get the Diurnal plane accurately.
  • “The ancestors at that point in time were not known to calculate.”
    • Response: Since a device for measuring time and ability to multiply and divide is all that is required, this we dismiss.
    • Further, as per BG Tilak’s (very plausible) interpretation (in contrast with sAyaNa) - " स्व॑र्भानो॒रध॒ यदि॑न्द्र मा॒या अ॒वो दि॒वो वर्त॑माना अ॒वाह॑न् । गू॒ऴं सूर्यं॒ तम॒साप॑व्रतेन तु॒रीये॑ण॒ ब्रह्म॑णा (=device or axial division) ऽविन्द॒दत्रिः॑ ।।" refers to a quadrant (तुरीयम्) like the one pictured below (from 19th century bulhomallavarman), or some such as that.
  • “It’s not as easy as you say. Try it.”
    • Response: Challenge accepted. See below.

Initial definitions

Celestial equatorial plane: Celestial equator and the plane corresponding to it is pictured below. celestial equator: 1. the great circle on the celestial sphere midway between the north and south celestial poles. 2. the great circle on the celestial sphere determined by extending the Earth’s equator to the celestial sphere. Daily motions of celestial objects occur in planes parallel to the celestial equator.

Diurnal circle: The apparent path of a star in the sky during one day due to the the rotation of the Earth. Diurnal circles are parallel to the celestial equator. Note that the diurnal circle for a day is NOT the ecliptic (a blunder in an earlier version of this post). Let the plane defined by the diurnal circle be called Diurnal plane.

Ecliptic: From the point of view of a person on Earth, the sun appears to travel through a path of constellations called the ecliptic. The plane corresponding to this is called the ecliptic plane. This is pictured below, in reference to the equatorial plane (specified by “cardinal directions” NSEW).

Getting the celestial equator plane

EW line lies within the equatorial plane. You of course need a third point to completely define the equatorial plane, and this is easy to be had using the apparent altitude of the Celestial north pole. A picture speaks a thousand words, so:

So just line the base of a plank along the EW line, then tilt it by the angle of the pole star. Now the plank is parallel to the celestial equatorial plane and the diurnal plane.

Getting the arc of the sun given the diurnal plane.

This is simple, but I say it explicitly just in case folks don’t get it at once.

  • Just use a stick lying flat on the Diurnal plane (a plank or a screen as described above), with its lower end fixed at a point. Draw a semicircle on the face of the Diurnal plane. There – you have something corresponding to the diurnal arc.
  • At any point in time, you can note the position of the sun by moving the top of this stick (lying on the Diurnal plane as described above) so that it casts no shadow.
  • You can do fancy things like dividing the arc into 5*3 sections corresponding to the 5*3 muhUrta-s in a day, as described in the tattirIya brAhmaNa (तैत्तिरीयब्राह्मणे सायणभाष्ये ऽत्र, भट्टभास्करभाष्ये ऽत्र। ).

Projecting the position of a star on to the Diurnal plane (or plank)

Hold up a rectangular plank with the same orientation as the Diurnal plane – up to your eyes with one hand. Place another rectangular plank on it so that it’s base fully lies on the “Diurnal plane” plank, and so one of the points on its base overlies the center of the semicircle described in the “Getting the arc of the sun given the Diurnal plane” section. Now, you can move this “perpendicular” plank like you were moving a stick in the “Getting the arc of the sun given the Diurnal plane” section. Move it so that the naxatra you’re interested in is collinear with the “perpendicular” plank (you should hold the planks so that only the edge of the perpendicular plank is visible to you). See example where I sight and project a tree leaf below.

There – mark the position of the perpendicular plank. This radial spoke is where the sun must come to be in the position dictated by the brAhmaNa.

Predicting starset time

1. Measure the angular velocity of the star after it rises by measuring two positions an hour apart: v= (s2-s1)/1.
2. Find the angular position p of the star at a given time.
3. Get angular span s of horizon in the equatorial plane the star is in. (Simple by holding a plank parallel to the equatorial plane and sighting the star and points on both horizons.)
4. Use the above to determine when it sets. (s-p)/v .

Acknowledgement and corrections

An earlier version of this post made some mistakes including:

  • the huge blunder of confusing the plane defined by the solar Diurnal plane (defined below) with the ecliptic plane. Corrected thanks to shrI trasadasyu.
  • an untested idea for identifying the plane defined by the Diurnal plane (defined below) by drawing a 23.5 degree line over the EW line – it was solidly dismissed thanks to shrI trasadasyu.
  • wrong methods about tracking something close to the solar Diurnal plane using
    • a plank with a linear shadow
    • two sticks
    • sun rise and set poitions and a middle solar position.