Of Leap Days, Leap Years & Altered Calendars

Today, February 29th -  the 'extra' day this February - is designated a 'leap day' in this leap year of 2016. Few people, however, understand how leap days and leap years arose and just how they are tied to astronomical timekeeping and the calendars we use.

Time keeping by calendar.

One might say it all began with the development of the Julian calendar, named after Julius Caesar. He had introduced a calendar with 365 days and a leap year every 4 years with 365 days. This meant an average length of year of 365.25 days.  But that extra 0.25 days for the mean Julian year made it in fact a smidgeon longer than what we define as the tropical year of 365.242374 days (or 365 days 5 h 48 min 46 s).

This set the stage for future problems, especially as the  role of the astronomical point known as the vernal equinox (also associated with a date) came to prominence. (The vernal equinox marks the intersection of the ecliptic with the celestial equator. The ecliptic is the projected path of the Sun - as it appears from Earth - onto the celestial sphere, but in reality is the plane of the Earth's own orbital motion).

In the wake of the Edict of Milan, the Council of Nicaea in 325 A.D. defined the dates of Easter and certain other religious holidays by reference to the vernal equinox. . In particular, March 21 was re-specified as the date of the vernal equinox while Easter was defined to occur on the first Sunday after the 14th day of the Moon (e.g. days after the full Moon). It ought to be noted here that the Christians at Nicaea didn't willy- nilly just change the date of the vernal equinox from March 25 back to March 21. No, what happened is that between 45 B.C. and 325 A.D. that date had slipped back from March 25th to March 21st. The reason is based on simple math: Because the Julian year was defined with an average of 365 ¼ days and is 11 mins. 14 secs longer than the tropical year of 365 days 5 h 48 min 46 ec then the slight difference had accumulated to 3 days in those 4 centuries.

Anyway, the ecclesiastical choice of March 21 as the vernal equinox did not always agree with the precise astronomical definition, i.e. when the astronomical point described above transited the meridian. This year, for example, that occurs on March 20th. The proximity is close, but because it isn't exact divergence occurs.

The divergence was such that by 1582 Pope Gregory XIII was tasked with another adjustment. This is because the original tiny deviation between Julian and Tropical year, e.g. of 11 mins and 14 seconds,  had grown to another 10 days so that the first day of spring was now occurring on March 11 and not the prescribed March 21.  What to do? In order to re-align his now (Gregorian) calendar with the actual vernal equinox (or the Church's approximation to it) Pope Gregory removed ten days from the middle of October.  This was undertaken on October 4, 1582 when the next day was proclaimed as October 15, 1582. (Some idiots at the time actually complained that the pope had "taken 10 days out of our lives".)

The effect of the Gregorian change was to realign the seasons but the primary goal was to calculate the  date of Easter based on the Church's definition of the vernal equinox on March 21.  Alas, not everyone in the world was committed so that much confusion reigned with different calendars and dates being kept in different places. Britain, for example, kept the old Julian calendar until 1752.

In the wake of the above the rule for the leap year was changed so that the average length of the year would closely approximate the length of the tropical year. The rule then applied was that only century years divisible by 400 would be leap years. Thus, 1700, 1800 and 1900 - all leap years under the Julian calendar - were not under the Gregorian, while 1600 and 2000 were.

Meanwhile, the average length of the Gregorian calendar, at 365.2425 days, was correct to within 1 day every 3300 years.

Incredibly, a more rational World Calendar was proposed in the 1950s, according to USF visiting astronomy professor Anthony Aveni. The device will replace the Gregorian calendar with its numerous peculiarities. The benefit of this calendar would mainly be that it is identical from one year to the next - so fewer calendars would have to be printed.  The design gives each quarter of three months of 31, 30 and 30 days, respectively, for a total of 91 days or 91 days x 4 = 364 days for each year.   The 365th day, deemed, "Year end day" is simply denoted December W or "Worldsday".  The economy is thereby built into the design so you get just one calendar and can follow it for x years.

  Alas, the U.S.  - after objections  from religious groups (about the dates of Easter, Christmas, etc.), rejected it.  Some religious reactionaries have gone so far as to claim "adoption will mean the end of religious liberty for million" since special holydays will be absorbed into the design and no longer stand out. 

The takeaway? While we often view timekeeping as a precise enterprise, the interjection of politics and religious' dogmas often undermines that assumption. Today, people need to remember that as they do whatever on this special day.
 

Enjoy today as the leap day it is and be thankful you weren't living back in 1650 or 1700.

A Cynical Brit Film Critic (David Thomson) Misfires On The Oscars

David Thomson, who wrote the WSJ piece 'The Oscar Farce',   thinks  the Oscars'  time has passed.

It was kind of amusing, but also infuriating, reading David Thomson's takedown of the Academy Awards in the Weekend WSJ. (Feb. 27-28, p. C3)  Amazingly, after reading the first several paragraphs, I initially thought it was some angry young black guy trying to draw attention and gain notoriety. Imagine my surprise on learning he was a British film critic, older than I am, and a white guy to boot!

On mentioning going to an Oscar party tonight he cited, for example, his friends' AARP membership and they're considering boycotting the show, over the lack of black nominees, writing:

"It's charming that people so old and wise will take Oscar that seriously but I may have to explain how the Academy of Motion Picture Arts & Sciences was always a self-serving sham'>

Hmmm.....written like a true film critic. What was it director Federico Fellini once said about them?

"Eighty percent of film criticism comes from illiterates and deficient mentalities"

Then there's this choice chestnut from Brendan Behan:

"A critic at a performance is like a eunuch at a harem: He sees it performed nightly but can't do it himself".

This guy then questions whether Oscar will still be around in 2036 (he will) and writes:

"Can we agree the Academy is a ridiculous, archaic, and retrogressive club?"

Uh, no,  we can't. It isn't any of those because thankfully most members are actual doers and performers of the movie art, not jaundiced critics. They earned their membership by paying their dues, making or producing actual movies (many of them over decades) as opposed to just panning them, and writing crap op-eds to attempt to render the whole ceremony moot and its supporting organization passe.

Thomson does appear to have a thing for movies like "The Revenant"  which he believes is "the most beautiful and audacious film of the year". Why? Well, because "of its faith in action and heroism and its sense of the cruel beauty of nature, for the natural light...and the extended traveling shots"

He loves it, in other words, because it "harks back to an original cinematic duty- to go out to the real places and the real things."

Which is rank balderdash because not every film can or SHOULD be "Call of the Wild" or "Grizzly Man" (where the star actually ended up being eaten). I mean, that's just plain bollocks. Look at Spotlight, for example, A terrific film - of substantial heft- but not shot in bear country or high country! It's as if the action and adventure theme to Thomson is the only one with  genuine cinematic validity. To accept that premise would be too disavow or reject films like '2001 -  A Space Odyssey', 'Silent Running' or indeed, any of the great science fiction fare of the past 40 years (like the 3D  'Avatar') mostly filmed in specialized sound and special effects studios.

The biggest howler from Thomson is when he whines that:

"There's too much to see now. the avalanche of YouTube and all the other diversions of the Internet are an attention killer".

Thereby implying that in this cornucopia of mostly bullshit (99% of it is), one's appreciation for good films must suffer. Not at all! Most of what's on YouTube is twaddle that in decades past would appeal only to kindergartners. I mean kittens boxing puppies and other such crap?  Farcical performances by wannabe hotshots - magic vanishing tricks,  kid superheroes,  or making gastronomic atrocities? All wastes of time.

For the discriminating mind there is not "too much to see" but rather too little of quality to see. For more on this read Chris Hedges' 'The Empire of Illusion'.

The precise problem today which Thomson touched on but didn't fully explore is that there is simply a surfeit of plain old garbage flooding not only the media airwaves but the web too. Everyone wants to be some kind of somebody - starboy or stargirl - so adds to the total mass of dreck which they misjudge as quality. The end result is an excess of juvenile merde which creates a paradox of choice for too many minds too immature to separate wheat from chaff.

If more disciplined minds existed, as opposed to uncritical, undiscerning ones, then people would more clearly see and appreciate what an achievement most of the Academy Awards represent. Then the notion of the Oscars as "old hat rubbish" that Thomson trots out would be seen for the bunkum it is.

I Predict The Best Picture Oscar Goes To.......SPOTLIGHT (Minor Spoilers)

"Spotlight" denotes the name of the investigative reporting unit at The Boston Globe. Newly arrived editor Marty Baron (from The Miami Herald) is briefed by Spotlight editor Walter
Robby' Robinson (Michael Keaton) on its warp and woof, in which a given investigation can take 6 months. The critical thing is it be done properly. Well, it's left to the new guy (played by Lev Schreiber), to use a past Globe column as a basis to push the team to look more deeply into the possible coverup of priest sexual abuse in the Archdiocese of Boston.

The bulk of the film follows the Spotlight team of 4 as they dig into the assorted personae and document trail,  tracing back to a lawyer (Mitchell Garabedian) who claimed in an earlier Globe article that Cardinal Law (the Archbishop of Boston) knew that a priest John Geoghan was sexually abusing children and did nothing to stop him.

The suspense in the film is embodied in virtually every scene where roadblock after roadblock is thrown up -  both by Garabedian and Boston's powerful RC Church whose tentacles reach into almost every nook and cranny of the city. But ultimately, it is the victims themselves who are encouraged to finally come forward and lead to the public release of files that break the case open.

For anyone who recalls that investigation and the ones that followed, exposing priest pederasts in hundreds of locations in the U.S. and overseas, it should be a slam dunk that this is the stuff of 'Best Picture' mettle.  It is a serious film about a serious subject that still has left tectonic shifts in the Church itself and in particular undermined its moral authority, especially on sexual issues.

After all, it was only somewhat later that Marty Baron's objective paid off, i.e.  to show the abuse was part of a derelict,  cynical system that was also morally blind. The mandate to hide and shift pedophile priests  from parish to parish extended all the way to the Vatican and the Office of the supreme "inquisitor" (head of the Congregation for the Doctrine of the Faith - the modern name for the Inquisition) Cardinal Joseph Ratzinger - who in turn was following orders from John Paul II.  Most of us at the time perceived  the changes in rules for canonization were done to cast a more "saintly" light on John Paul II and take attention away from the still metastasizing Church sex scandal.

Anyway, one hopes Oscar attention to the movie will spur many more to see it, as I did last night. This is not to take anything away from the other contenders, including "The Revenant" and "The Big Short" but to me, Spotlight ought to be the clear Best Picture winner.  Still, "The Big Short" must be at least a powerful dark horse contender given its light shed on the financial crisis. (Though a number of reviews in the financial media, e.g. the WSJ, panned it for being "too cartoonish". Noting you shouldn't need or use a woman in a bubble bath to explain credit default swaps.)

Amazingly, the much better picture ("99 Homes" )  dealing with the credit meltdown in depth, never got an Oscar nod and did poorly at the box office, with barely a $1.8 m take.  I have to believe the main reason may have been the choice of title. It is incredible how a few words - say in a header or title - can make a huge difference on whether people train eyeballs on it or ignore it.

Lastly, if DiCaprio doesn't win Best Actor for Revenant than surely Brian Cranston will for his role as commie-blackballed screenwriter Dalton Trumbo in 'Trumbo'.  (I was actually amazed the film wasn't also an Oscar Best Picture nominee given all its homage to courageous Hollywood actors that stood up to the 1940s-50s commie witch hunts.) But maybe after last year's win for 'Birdman' the Academy felt one such winner was enough in a two year span.

It's a shame that the "#Oscarssowhite" meme has crept in to challenge the level of diversity of this year's Academy Awards, but hey - what would you expect for a nominating bunch (Academy of Motion Picture Arts & Sciences) that is more than 60 percent composed of old white men over 65?  At least the Academy has taken steps to improve the membership proportions but until all the old white actors, members croak don't look for much.

In the meantime, the host - Chris Rock - should provide some interesting and even suspenseful moments as he draws attention to any deficits.

A Look At Solar Oscillations

A computer-generated image of some of the 10 million modes of acoustic waves on the Sun (red indicating receding wave fronts, and blue approaching).

It is incomprehensible to many people that the Sun’s surface  could be undergoing vibrations in frequency that could be used as a diagnostic tool. This may be due to the phenomenon of solar oscillations being a relatively new field, mainly developed in the last forty years. It allows hitherto unknown tools to be applied to solar observations and predictions. In this chapter we will survey the basic physics associated with these observations.

The first reckoning of non-radial oscillations arrived ca. 1968 with the work of Frazier who made two-dimensional plots of wavenumber vs. frequency or (k -w) diagrams. Several peaks of amplitude were found and it was suggested that these corresponded to the fundamental and first overtones for the solar envelope. Interestingly the patterns of solar oscillations - namely the acoustic or "p-modes" resemble those detected on drum heads by computer holography.

The Sun is clearly not a drum head, but it seems to behave like one in terms of its oscillations.  Solar physicists are particularly interested in what are called p, g and f modes given they are resonant modes of oscillation.  The p-modes are basically for acoustic or sound waves, the g modes are for internal gravity waves and the f modes are for surface gravity waves.   The spherical harmonic function, also peculiar to atomic physics, but here most applicable to the p-modes in the solar oscillations context, is given by:

y _nℓm  =  R _n (r)  Y _ℓm (q,j) exp (iw t)

And for normalized spherical harmonics:

Y _ℓm (q,j) = 

(1)   ^m [2 ℓ +1 (ℓ - m)!/ 4n (ℓ +m)!] ^½ P ^ℓ_m (z ) exp (i m j),

Where z =  cos q and  w = 2 pn  where n = n _nℓm is the frequency of oscillation of mode n, ℓ, m. (Note: j is measured from the meridian of the ascending node of the Sun’s equator.)

Thus, R _n (r)   is applicable to radial patterns (with n the radial quantum number) whereby for a given value of n we elicit a pattern of radial nodes, for which the position is determined by the exact pattern of the function  R _n (r)   The rest of y  gives the surface pattern as a spherical harmonic of the oscillation. The spherical harmonic, e.g.

 Y _ℓm (q,j), determines the angular dependence of the eigenfunctions and hence the surface distribution of the oscillation amplitudes, i.e. as seen by an observer.

The letters n, m and ℓ denote numbers whose meanings should be further clarified. The first is the radial order or the number of nodes in the radial direction. The second is the harmonic degree or azimuthal order which indicates the number of nodes around the equator on the three dimensional spherical surface. Finally we have the angular degree or the number of nodes from pole to pole, e.g. along longitude or meridian lines. The difference  (ℓ  - m) is also of interest as it yields the lines corresponding to parallels of latitude. Any given combination of the numbers n, m and ℓ  allows a unique frequency n to be computed. For example, if we have n= 14, m = 16 and ℓ = 20 one gets a period of 340.61 s or:

n =  2 p/ T  =  2 p/ (340.61 s) =  2.935 x 10 ^-3 /s

Radial oscillations alone have ℓ  =  0 and we see in this case the associated Legendre function            (P _ℓm (q )) has:

P _ℓm (q )  =  (1 – z²) ^m/2 / ℓ! 2^ℓ   d ^(ℓ+m) / dz^(ℓ+m)  (z²   -1) ^ℓ

Recall m= 2 ℓ + 1 = 2(0) +1 = 1

So:

P _ℓm (q )  =  (1 – z²) ^1/2 / 0! 2⁰   d / dz  (z²   -1) ⁰

=  (1 – z²) ^½  =   (1 – cos ² q)^½   =  (sin ² q)^½     = sin q

For q = p/2  , P _ℓm (q )  =  1

And: P _ℓm (q ) exp (i m j)  =  (1) exp (i (1) 0)  = 1

If  n _nℓm =   1 c/s  then:  Y _ℓm (q,j) = 1 and y _nℓm  =  R _n (r) 

The degree ℓ of the spherical harmonic can assume any integer value, i.e.:

ℓ  =  0, 1, 2, …….

At each such ℓ the azimuthal number m assumes a 2 ℓ + 1 value, i.e.

m= - ℓ, (-ℓ +1)....0......( ℓ - 1), + ℓ

Meanwhile, the frequency of a particular mode is given by the azimuthal eigenvalue m, and the meriodonal eigenfunction ℓ - together with n. Since m= 2 ℓ + 1, then the spherical surface is split into 2 ℓ + 1  regions.

We are most interested in the Lamb frequency, e.g.

L   =   Ö( ℓ(ℓ + 1) c _s ² / r² )

And the Brunt- Vaisala or B-V frequency:

N ² =  g [G1 (d ln p _o  /dr) - d ln r _o  /dr) ]

Which can be simplified to:

N ² =  g² / c _s²  (g  - 1)

(Which holds expressly for the isothermal case.)

Where  c _s ²  = G1 p _o  /r _o    and:

G1= d (n lnp )  / ln r

Example Problem (1):

Find the Brunt- Vaisala frequency for the isothermal case if the sound speed in the Sun is c _s = 900 m/s. (Take g =  5/3 and g =  273 ms ^-2  )

Solution:

N ² =  g² / c _s ²  (g  - 1)

N =  Ö  (273 ms ^-2 /900 m/s) [ 5/3 – 1] ) = 0.372 /s

 Example Problem (2):

Find the Lamb frequency for the same sound speed, if  ℓ = 100 and r = 2650 km.

Solution:

We use:  L   =   Ö ( ℓ(ℓ + 1) c ² / r² )

Where r = 2650 km   = 2.65 x  10 ⁶  m  

L   =    Ö100(100 + 1) (900 m/s) ² / (2.65 x  10 ⁶  m ) ²

L   =  3.4  x  10 ^-2 / s   = 340 microhertz

    It is of interest to note that only waves with the longest horizontal wavelength ℓ  reach the core of the Sun while high ℓ-modes do not penetrate the convective zone. Given  ℓ = 100 we can expect the example chosen will be near the solar surface.  Also of interest in this context is the acoustic cutoff frequency, defined:

w _ac   = g g/ 2c

     This should not be confused with the plasma cutoff frequency, e.g. w _c     for EM waves in plasmas. But there is one common attribute for both: a cutoff frequency is for any frequency for which the wave number k ® 0.

In typical k -w diagrams, with frequency  w along the ordinate and k along the abscissa. One would see the p-modes in the upper left lying above  w _ac     and the g-modes (gravity modes) at lower right below a dotted line for N. Another line given is for ck_h which represents the Lamb waves or f-modes.    How many total modes, with n, ℓ and m distinct operate in the Sun? This is not difficult to estimate. Let’s take n first. According to diagnostic diagrams showing “ridges" for oscillatory power at each frequency at least 20 have been observed. In the diagram shown below, the spikes or ridges for the p-mode represent the first harmonic and the baseline smooth curve from which they project is the fundamental.

This leads to a maximum radial order of n = 20 for the p-mode associated ridges.. Now, for each of these n values, at least 500 angular degrees ℓ have been observed. We also know that for each such ℓ there are at least 2 ℓ values (actually 2 ℓ + 1). So in this case: 2 ℓ = 2(500) = 1000. Then the total estimated modes at any given time works out to:

T _{n ℓm}   =  20 x 500 x 1000 =  10 ⁷

 Or, ten million modes, all overlapping in time and space. Because of this extreme multiplicity of modes, photographs of the solar surface appear featureless or more accurately like a disk of coarse sandpaper. Bear in mind also what we’d observe at the solar surface are reflections of the multitude of standing sound waves (p-waves) that fill the Sun’s interior. Not surprisingly, these match up quite well with the coarse solar granulation, e.g

However, it has been Dopplergrams making use of the Michelson Doppler Imager (MDI) that have given us the best observation portal on the rising and falling super cells known as supergranules.

It was Robert Leighton – of Babcock and Leighton solar dynamo fame- who first observationally isolated the motions of solar granules and tied their delay time for replacement with an average lifetime of five minutes. He found a given granule’s velocity was equal and opposite to  the velocity at the same location some 2.5 minutes later. He deduced that something on the Sun must be oscillating. The convective cells don’t normally do so thus it had to be explained in an alternative way.

Ultimately the problem of the solar 5-minute oscillations was resolved by treating the Sun as a resonant cavity.  In 1981, Leibacher and Stein showed that if one treated the Sun as a resonant cavity one could expect the relationship:

T = (n + ½)p / w

     In other words for the condition at which the sound speed equals the horizontal phase velocity (w/k _h ) one expects acoustic wave reflection.

Duvall and Harvey[1]  reinforced this work by measuring the frequency spectrum of this » 300s  oscillation and found it applicable for ℓ-modes less than 140, and radial modes R with order n = 2 to 26. Posing the degree ℓ- in terms of  the reflection radius r:

ℓ =  -1/2  +  [ ¼ +    4p² g² r² / c _s ² ]

The modes were thus established as being deep in the solar interior by matching all the modes in a series of data using the above equation.

Problems:

1) Find the acoustic cutoff frequency below which all waves would be reflected if a sound speed of 900 m/s applies.

2) Sketch the spherical surface with the mode values m = 0 and  ℓ = 2, and justify your sketch. Find the associated Legendre function:        (P _ℓm (q )).

3) Let a period T= 340.61 s be associated with the Lamb frequency. What would be the associated depth of reflection in the Sun, assuming ℓ = 20 and c= 900 m/s.

4) For the p-mode pattern shown find the associated Legendre function:  (P _ℓm (q )) if If q = p/4.

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[1] Duvall, T.L. and Harvey, J.W.: 1984, Nature, 310, 19.