Shadowless Moments

 

Do you know what Lāhainā Noon is? It is also known as a zero-shadow day.

In Kerala, this year, we have shadowless days from the 10th to the 23rd of April 2024 and the 19th of August to the 1st of September 2024. 

 

10-04-2024 was a zero-shadow day for people living in Thiruvananthapuram, Kerala, India.

In this article, I included a brief explanation of this phenomenon.

I took three objects to observe the variations of shadow between 11.35 am - 12.40 pm on 10th April 2024. I took the side-view and front-view images to observe the shadow. At around 12.25, shadows disappeared, and it was under the object itself. After 12.30, I could see the shadow on the left side of the objects.


Shadow at 11.35 am

It is a semi-annual tropical solar phenomenon, and we can observe it twice a year for locations in the tropics.

                                                       Shadow at 12 noon

The light rays will fall vertically relative to an object on the ground and cast no observable shadow [2] when they reach overhead

Shadowless at 12.25 pm

We can observe a zero-shadow day twice a year.

Front view at 12.25

  • Earth's axial tilt has a link to the zero-shadow day, which causes the planet to be at its zenith, or directly overhead, at specific latitudes. 
  • The Tropic of Cancer experiences it at the summer solstice, while the Tropic of Capricorn witnesses it during the winter solstice.

Shadow at the other side at 12.40 pm

The date varies by location. [3] The Bishop Museum in Hawaiʻi coined the term Lāhainā Noon. [4]

Details

The subsolar point travels through the tropics. Locations between the Tropic of Cancer and the Tropic of Capricorn receive the sun's direct rays overhead before and after the summer solstice.

Lāhainā Noon can occur anywhere from 12:16 to 12:43 noon. At that time, objects that stand straight up cast no outward shadow.

Chosen in a contest sponsored by the Bishop Museum in the 1990s, Lāhainā Noon was the selected appellation because lā hainā (the old name for Lāhainā, Hawaii) means cruel sun in the Hawaiian language. [7] The ancient Hawaiian name for the event was kau ka lā I ka lolo, which translates as the sun resting on the brain.[5][8]

 

 The relationship between the zenith, the nadir, and different types of horizons   (image source: Wikipedia)

In popular culture

 

  • Honolulu possesses a unique sculpture called Sky Gate, created by artist and landscape architect Isamu Noguchi. 
The sculpture is at the Frank Fasi Civic Grounds. It features a bendy, bumpy ring that drastically changes height as it goes around. 

  • Most of the year, except during Lāhainā Noon, it makes a curvy, twisted shadow on the ground. 
  • During Lāhainā Noon, the height-changing ring casts a perfect circular shadow on the ground. [13] 
They held activities on that ground by the City & County of Honolulu around the time of the event.

Sky Gate produces a perfect circular shadow on the ground(image source:  Wikipedia)

 Axial tilt

 

  • Axial tilt is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane. [22] 
  • It differs from orbital inclination. 
  • Two axes point in the same direction, then the axial tilt is zero degrees.  

The angle between the Sun and the local horizontal level is exactly 90° at the subsolar point (image source: Wikipedia)

The rotational axis is the imaginary line that passes through the North and the South Pole. 

  • Earth's orbital axis is the line perpendicular to the imaginary plane through which it revolves around the Sun. Earth's axial tilt is the angle between these two lines.

Throughout an orbital period, the tilt usually does not change considerably, and the orientation of the axis remains the same relative to the background of stars. 

  • Because of this, one pole is pointed more toward the Sun on one side of the orbit and away from the Sun on the other side- cause of the seasons on Earth.

 The two standard methods of specifying a planet's tilt are:

1 Based on the planet's north pole, defined by the direction of Earth's north pole.

2 Based on the planet's positive pole, defined by the right-hand rule.

 

  • The International Astronomical Union (IAU) defines the north pole of a planet as that which lies on Earth's north side of the invariable plane of the Solar System;[2] under this system, Venus is tilted 3° and rotates retrograde, opposite that of most of the other planets. [23][24]
  • The IAU also uses the right-hand rule to define a positive pole[25] [26]to determine orientation. Using this convention, Venus is tilted 177° ("upside down") and rotates prograde.
History

The ancient Greeks had good obliquity measurements since about 350 BCE when Pytheas of Marseilles measured the shadow of a gnomon at the summer solstice. [27] About 830 CE, the Caliph Al-Mamun of Baghdad directed his astronomers to measure the obliquity, and the Arab world used the result for many years. [28] In 1437, Ulugh Beg determined the Earth's axial tilt as 23°30′17″ (23.5047°). [29]

  • During the Middle Ages, it was a belief that both precession and Earth's obliquity oscillated around a mean value, with a period of 672 years, an idea known as trepidation of the equinoxes. 

Perhaps this was incorrect [30], and the first to realize that the obliquity was decreasing at a relatively constant rate was Fracastoro in 1538. [31] The first accurate, modern, western observations of the obliquity were probably those of Tycho Brahe from Denmark in about 1584.[32] The observations by several others, including al-Ma'mun, al-Tusi,[33] Purbach, Regiomontanus, and Walther, could have provided similar information.

 

Seasons

The axis of Earth remains oriented in the same direction concerning the background stars regardless of where it is in its orbit. 

Earth's axis remains tilted in the same direction concerning the background stars throughout the year (regardless of where it is in its orbit) due to the gyroscope effect. 

  • One pole will be directed away from the Sun at one side of the orbit, and half a year later, this pole will be towards the Sun.
  • It is the reason for Earth's seasons. 
  • Summer occurs in the Northern Hemisphere when the north pole is toward the Sun. 
  • Variations in Earth's axial tilt can influence the seasons and are likely a factor in long-term climatic change 

 


How the tilt of 
Earth's axis aligns with incoming sunlight around the winter solstice of the Northern Hemisphere 
(image source: Wikipedia).

Oscillation

 

Short term

We can calculate the exact angular value of the obliquity by observation of the motions of Earth and planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, and from these ephemerides, including the obliquity, are derived.

Annual almanacs are published listing the derived values and methods of use. Until 1983, mean obliquity for any date was calculated based on the work of Newcomb, who analyzed the positions of the planets until about 1895.

ε = 23°27′8.26″ − 46.845″ T − 0.0059″ T2 + 0.00181″ T3

where ε is the obliquity, and T is the tropical centuries from B1900.0 to the date in question. [34]

From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated:

ε = 23°26′21.448″ − 46.8150″ T − 0.00059″ T2 + 0.001813″ T3

where hereafter T is Julian centuries from J2000.0.[35]

They updated JPL's fundamental ephemerides continually. For instance, according to the IAU resolution in 2006 in favor of the P03 astronomical model, the Astronomical Almanac for 2010 specifies:[36]

ε = 23°26′21.406″ − 46.836769″ T − 0.0001831″ T2 + 0.00200340″ T3 − 5.76″ × 10−7 T4 − 4.34″ × 10−8 T5

These expressions for the obliquity are for high precision over a relatively short period, perhaps ± several centuries. [37] J. Laskar computed an expression to order T10 good to 0.02″ over 1000 years and several arcseconds over 10,000 years.

ε = 23°26′21.448″ − 4680.93″ t − 1.55″ t2 + 1999.25″ t3 − 51.38″ t4 − 249.67″ t5 − 39.05″ t6 + 7.12″ t7 + 27.87″ t8 + 5.79″ t9 + 2.45″ t10

where here t is multiples of 10,000 Julian years from J2000.0.[21]

These expressions are for the so-called mean obliquity, free from short-term variations. Periodic motions of the Moon and Earth in its orbit cause much smaller (9.2 arcseconds) short-period (about 18.6 years) oscillations of the axis of Earth, known as nutation, which adds a periodic component to Earth's obliquity. [38][39] The true or instantaneous obliquity includes this nutation. [40]

 

Long term

 

Studies are there for the long-term changes in Earth's orbit and its obliquity using numerical methods to simulate Solar System behavior over several million years. For the past 5 million years, Earth's obliquity has varied between 22°2′33″ and 24°30′16″, with a mean period of 41,040 years. This cycle is a combination of precession and the term in the motion of the ecliptic. For the next 1 million years, the cycle will carry the obliquity between 22°13′44″ and 24°20′50″.[41]

  • The moon has a stabilizing effect on Earth's obliquity. 
In the absence of it, obliquity could change rapidly due to orbital resonances and chaotic behavior of the Solar System, reaching as high as 90° in as little as a few million years. [42][43] 

  • However, more recent numerical simulations [44] made in 2011 indicated that obliquity might not be unstable even in the absence of the moon. 
  • It varies only by about 20–25°. 
To resolve this contradiction, they calculated the diffusion rate of obliquity and found that it takes more than billions of years for Earth's obliquity to reach nearly 90°.[45] The stabilizing effect of the moon will continue for less than two billion years. As the moon continues to recede from Earth due to tidal acceleration, resonances may occur, causing large oscillations in obliquity. [46]

 Position of the Sun

The position of the Sun in the sky is a function of:

  • Time and the geographic location of observation on Earth's surface. 
We know that Earth orbits the Sun over a year. The Sun appears to move concerning the fixed stars on the celestial sphere along a circular path called the ecliptic.

  • Earth's rotation about its axis causes diurnal motion so that the Sun appears to move across the sky in a Sun path that depends on the observer's geographic latitude. 
  • The time when the Sun transits the observer's meridian depends on the geographic longitude.

 

To find the Sun's position for a given location at a given time, one may proceed in three steps as follows:[47][48]

1.     Calculate the Sun's position in the Wecliptic coordinate system,

2.     Convert to the equatorial coordinate system.

3.     Convert to the horizontal coordinate system for the observer's local time and location.

It is the coordinate system used to calculate the position of the Sun in terms of solar zenith angle and solar azimuth angle. 

To depict the Sun's path, we can use these two parameters. [49]

This calculation is significant in astronomy, navigation, surveying, meteorology, climatology, solar energy, and sundial design.

Earth

The north and south celestial poles and their relation to axis of rotationplane of orbit, and axial tilt (image source: Wikipedia)

Earth's orbital plane is known as the ecliptic plane, and Earth's tilt is known to astronomers as the obliquity of the ecliptic, being the angle between the ecliptic and the celestial equator on the celestial sphere. [6] Its notation is by the Greek letter ε.

  • Earth currently has an axial tilt of about 23.44°.[7] 
  • This value remains about the same relative to a stationary orbital plane throughout the cycles of axial precession. [8]
  •  But the ecliptic (i.e., Earth's orbit) moves due to planetary perturbations, and the obliquity of the ecliptic is not a fixed quantity. 
  • It is decreasing at about 46.8″ [9] per century.

What is the subsolar point?   

  • The subsolar point on a planet is the point where the Sun's rays strike the planet perpendicular to its surface (position is directly overhead). [1] 

Imagine you are on a planet with an orientation and rotation same as that of the Earth.

Then, the subsolar point will appear to move westward at 1600 km/h speed. However, it will also move north and south between the tropics over a year.

  • The subsolar point meets the Tropic of Cancer on the June solstice and the Tropic of Capricorn on the December solstice.
  •  The subsolar point crosses the equator twice during equinoxes in March and September.
  •  The summer solstice occurs when one of the earth's poles reaches its maximum tilt to the sun.

  •  It happens twice yearly, once in each hemisphere (Northern and Southern). [1]

The summer solstice occurs during the hemisphere's summer. In the Northern Hemisphere, this is the June solstice (20 or 21 June). In the Southern Hemisphere, this is the December solstice (21 or 22 December). Since prehistory, the summer solstice has been a significant time of year in many cultures, and it is the season of festivals and rituals. [1]

  • Earth's maximum axial tilt toward the Sun is 23.44° On the summer solstice.[7] 
  • Likewise, the Sun's declination from the celestial equator is 23.44°.

 Zenith

  • The zenith [11] is an imaginary point directly above a particular location on the celestial sphere. 
Above means in the vertical direction opposite to the gravity direction at that location. 

  • The zenith is the highest point on the celestial sphere.

                            

                  Angles and planes of a celestial sphere (image source: Wikipedia)

Origin

The word zenith derives from an inaccurate reading of the Arabic expression samt al-ras, meaning direction of the head or path above the head. [12] 

 Relevance and use

 The shadows of trees are the shortest on Earth when the Sun is directly overhead (at the zenith). This happens where the tree's latitude and the Sun's declination are equal at solar noon on certain days in the tropics.

The shadows of trees are the shortest on Earth when the Sun is directly overhead (at the zenith) (image source: Wikipedia)

  • The term zenith sometimes means the highest point, reached by a celestial body on its daily apparent path around a point. [14] 
In this sense, we can use the word to describe the position of the Sun, but to an astronomer, the Sun does not have its zenith.

In a scientific context, the zenith is the direction of reference for measuring the zenith angle, the angle between a direction of interest (e.g. a star) and the local zenith - that is, the complement of the altitude angle.

  • The Sun reaches the observer's zenith when it is 90° above the horizon, and this only happens between the Tropic of Cancer and the Tropic of Capricorn.
 In Islamic astronomy, the passing of the Sun over the zenith of Mecca becomes the basis of the qibla observation by shadows twice a year on 27/28 May and 15/16 July. [15][16]

At a given location during the day, the Sun reaches its nadir also, at the antipode of that location, twelve hours from solar noon.

  • In astronomy, with the horizon perpendicular to the zenith, the altitude in the horizontal coordinate system and the zenith angle are complementary angles.
  • We can determine the astronomical meridian by the zenith. [18]

 

Meridian

 

  • In astronomy, the meridian is the great circle passing through the celestial poles, the zenith and nadir of an observer's location.

A zenith telescope is designed to point straight up at or near the zenith and used for precision measurement of star positions, to simplify telescope construction, or both. The NASA Orbital Debris Observatory and the Large Zenith Telescope are zenith telescopes that use liquid mirrors. These telescopes could only point straight up. [20]

  • Meridian contains the north and south points on the horizon and is perpendicular to the celestial equator and horizon. 
  • We can determine the celestial and geographical meridians by the pencil of planes passing through the Earth's rotation axis. 
  • For a location not at a geographical pole, there is a unique meridian plane in this axial pencil through that location. 
  • The intersection of this plane with Earth's surface is the geographical meridian, and the intersection of the plane with the celestial sphere is the celestial meridian for that location and time.[19][21]

Analemma

  •  An analemma is a diagram that shows the annual variation of the Sun's position on the celestial sphere relative to its mean position, as seen from a fixed location on Earth. 
  • It shows an image of the Sun's apparent motion during a year. 
An analemma can be pictured by superimposing photographs taken at the same time of day, a few days apart for a year.

An analemma (image source: Wikipedia)

  • An analemma is a graph of the Sun's declination, usually plotted vertically against the equation of time, plotted horizontally. 
  • Usually, the scales are equal distances on the diagram, representing equal angles in both directions on the celestial sphere. 
Thus, four minutes in the equation of time are represented by the same distance as 1° in the declination since Earth rotates at a mean speed of 1° every 4 minutes relative to the Sun. [1]

References:

1.     Wikipedia

2.      "Clock and sun rarely match at noon"Honolulu Star-Bulletin. December 31, 2008. Archived from the original on July 17, 2023, and retrieved July 16, 2023.

3.     ^ Newsd (2019-04-24). "Zero Shadow Day 2019: Date, time & know why you cannot see your shadow". News and Analysis from India. A Refreshing approach to news. You are retrieved 2019-08-22.

4.     ^ "Zero Shadow Day". ASI POEC. 2017-04-07. Retrieved 2019-08-22.

5.     Jump up to a b "Lāhainā Noon". Bishop Museum. Retrieved August 3, 2022.

6.     Jump up to:a b c Nancy Alima Ali (May 11, 2010). "Noon sun not directly overhead everywhere"Honolulu Star-Bulletin. Retrieved November 12, 2010.

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15.  Jones, Daniel (2011). Roach, PeterSetter, JaneEsling, John (eds.). Cambridge English Pronouncing Dictionary (18th ed.). Cambridge University Press. ISBN 978-0-521-15255-6.

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19. ^ van Gent, Robert Harry (2017). "Determining the Sacred Direction of Islam". Webpages on the History of Astronomy.

20. ^ Khalid, Tuqa (2016). "Sun will align directly over Kaaba, Islam's holiest shrine, on Friday". CNN.

21. Jump up to a b Lewis, David (1972). "We, the navigators: the ancient art of land finding in the Pacific". Australian National University Press. Retrieved 2023-06-01.

22. U.S. Naval Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.). Explanatory Supplement to the Astronomical Almanac. University Science Books. p. 733. ISBN 978-0-935702-68-2.

23. Explanatory Supplement 1992, p. 384

24. Jump up to:a b Correia, Alexandre C. M.; Laskar, Jacques; de Surgy, Olivier Néron (May 2003). "Long-term evolution of the spin of Venus I. theory" (PDF). Icarus. 163 (1): 1–23. Bibcode:2003Icar..163....1Cdoi:10.1016/S0019-1035(03)00042-3Archived (PDF) from the original on 9 October 2022.

25. ^ Correia, A. C. M.; Laskar, J. (2003). "Long-term evolution of the spin of Venus: II. numerical simulations" (PDF). Icarus. 163 (1): 24–45. Bibcode:2003Icar..163...24Cdoi:10.1016/S0019-1035(03)00043-5Archived (PDF) from the original on 9 October 2022.

26. ^ Seidelmann, P. Kenneth; Archinal, B. A.; a'Hearn, M. F.; Conrad, A.; Consolmagno, G. J.; Hestroffer, D.; Hilton, J. L.; Krasinsky, G. A.; Neumann, G.; Oberst, J.; Stooke, P.; Tedesco, E. F.; Tholen, D. J.; Thomas, P. C.; Williams, I. P. (2007). "Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006". Celestial Mechanics and Dynamical Astronomy. 98 (3): 155–180. Bibcode:2007CeMDA..98..155Sdoi:10.1007/s10569-007-9072-y

27. Gore, J. E. (1907). Astronomical Essays Historical and Descriptive. Chatto & Windus. p. 61.

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29. ^ Sédillot, L.P.E.A. (1853). Prolégomènes des tables astronomiques d'OlougBeg: Traduction et commentaire. Paris: Firmin Didot Frères. pp. 87 & 253.

30. ^ Saliba, George (1994). A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam. p. 235.

31. ^ Dreyer, J. L. E. (1890). Tycho Brahe. A. & C. Black. p. 355.

32  ^ Sayili, Aydin (1981). The Observatory in Islam. p. 78.

33  U.S. Naval Observatory Nautical Almanac Office; H.M. Nautical Almanac Office (1961). Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. H.M. Stationery Office. Section 2B.

34. ^ U.S. Naval Observatory; H.M. Nautical Almanac Office (1989). The Astronomical Almanac for the Year 1990. US Government Printing Office. p. B18. ISBN 978-0-11-886934-8.

35. ^ Newcomb, Simon (1906). A Compendium of Spherical AstronomyMacMillan. pp. 226–227.

36. ^ See table 8 and eq. 35 in Laskar, J. (1986). "Secular terms of classical planetary theories using the results of general theory". Astronomy and Astrophysics. 157 (1): 59–70. Bibcode:1986A&A...157...59L. and erratum to article Laskar, J. (1986). "Erratum: Secular terms of classical planetary theories using the results of general theory". Astronomy and Astrophysics. 164: 437. Bibcode:1986A&A...164..437L. Units in the article are arcseconds, which may be more convenient.

37. ^ Explanatory Supplement (1961), sec. 2C

38. ^ "Basics of Space Flight, Chapter 2". Jet Propulsion Laboratory/NASA. 29 October 2013. Retrieved 26 March 2015.

40 Berger, A.L. (1976). "Obliquity and Precession for the Last 5000000 Years". Astronomy and Astrophysics51 (1): 127–135. Bibcode:1976A&A....51..127B.

41^ Jump up to:a b Laskar, J.; Robutel, P. (1993). "The Chaotic Obliquity of the Planets" (PDF). Nature361 (6413): 608–612. Bibcode:1993Natur.361..608Ldoi:10.1038/361608a0S2CID 4372237. Archived from the original (PDF) on 23 November 2012.

42^ Laskar, J.; Joutel, F.; Robutel, P. (1993). "Stabilization of the Earth's Obliquity by the Moon" (PDF). Nature. 361 (6413): 615–617. Bibcode:1993Natur.361..615Ldoi:10.1038/361615a0S2CID 4233758Archived (PDF) from the original on 9 October 2022.

43^ Lissauer, J.J.; Barnes, J.W.; Chambers, J.E. (2011). "Obliquity variations of a moonless Earth" (PDF). Icarus217 (1): 77–87. Bibcode:2012Icar..217...77Ldoi:10.1016/j.icarus.2011.10.013We have archived (PDF) from the original on 8 June 2013.

44^ Ward, W.R. (1982). "Comments on the Long-Term Stability of the Earth's Obliquity". Icarus50 (2–3): 444–448. Bibcode:1982Icar...50..444Wdoi:10.1016/0019-1035(82)90134-8

45 Meeus, Jean (1991). "Chapter 12: Transformation of Coordinates". Astronomical Algorithms. Richmond, VA: Willmann Bell, Inc. ISBN 0-943396-35-2.

46^ Jump up to a b c d e Jenkins, Alejandro (2013). "The Sun's position in the sky". European Journal of Physics. 34 (3): 633–652. arXiv:1208.1043Bibcode:2013EJPh...34..633Jdoi:10.1088/0143-0807/34/3/633S2CID 119282288.


 

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