EAON circular # 9

(Technical note by Raymond Dusser)

B a s i c s   o n

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F R I N G E S   or   P E N U M B R A

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                We assume here that for main belt common asteroids their angular diameter is always much larger that the occulted star angular diameters, so the limb can be considered as a straight line where it crosses the small star disc.

Part 1 :  the star is assumed pointlike

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         Any shadow is then lined by diffraction fringes. The basic parameter is the Fresnel's unit  u :

      u = SQR[ dist * lambda / 2]                            where :

"dist" is the distance from the occulting edge to the place of observation

"lambda" is the wavelength, in the same unit as  dist  and thus  u .

SQR stands for square root, /  for division, and  *  for multiplication.

                   For  a celestial body, the formula above gives  u  in meters if  dist  is in  Gm (Gigameters, 1 astronomical unit AU is 149.59787Gm) and lambda is in nm (nanometers). One can also compute  u  from the parallax  p  in arc seconds (") :

      u = SQR[ 657.8 * lambda / p]   where lambda is in nm and  u  in meters.

The angular size of the Fresnel unit in " is (with p in " and lambda in nm) :

      u" = 206 264.8 * u / dist = SQR[ p * lambda ] / 248 684      

         One can then compare  u"  directly with the angular diameter S" of the occulted star (if and when it is known) :

         If S" is clearly smaller than u"/2, the diffraction fringes can be seen  (part 1).

         If S" is clearly greater than u"/2, the fringes are blurred by penumbra (part 2).

ALL the fringes are  OUTSIDE  the geometrical shadow.

MORE ADVANCED :

Thus, starting outside from the edge of geometrical shadow, one can find:

- at 1.22 u B1 with 1.37*Io, an increase of 37% above the regular star light intensity.

- 0.65 u farther, at 1.87 u, D1 with 0.78*Io, 22% under the regular star light intensity.

- 0.47 u farther, at 2.35 u, B2 with 1.20*Io, 20% above mean light intensity.

- etc...                  

         The light curve (monochromatic) can thus be sketched :

1.37                            **

                                 *   *

                                *    

                                      *         **

                                                         **

                              *        *       *  *

                                       *               *   *

 1.00---------------------*------------------------------

                                        *     *        *   

                                                             *

                           *                       *  *

                                                              **

 0.84                   *             *   *       **

 0.78                *                **

                        *

                       *

 0.50

                      *

                     *

 0.25            *

                   |

                 * |

                   |

               *   | 

             *     |            1             2            3

 0 -*--*----------|------------|-------------|------------|--  

                   |           |          |       |

                   0          1.22        2.35   3.08

             GEOMETRICAL           1.87    2.74

             SHADOW EDGE

NOTE  In white light, all the fringes are coloured since the Fresnel unit  u  is some 32% larger at 700nm (red) than at 400nm (violet), and as near from the geometric edge as the second bright orange-yellow fringe, a chromatic blurring effect appears.

         WHERE can we find the edge of the geometrical shadow ? It is easily demonstrated that  ALL  the light curves for ALL wavelengths cross at one point : the edge of geometrical shadow. THERE, for all wavelengths, the intensity is :

I = 0.25 * Io  

    If fringes do exist, the observer must find the point where the light intensity is 1/4 of regular light intensity. That means a 1.5 mag drop of the starlight (instead of 0.75 mag drop when penumbra smears the fringes. See below, part 2). For a given wawelength, the distance from 0.50 level to 0.25 level is as short as 0.3536 u .

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Part 2 :  the star is not pointlike

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         When the star is NOT POINTLIKE as assumed above, its disc generates a PENUMBRA. Knowing the angular width of the star, it is easy to compute the equivalent diameter of the star at occulting body's distance (dist). This is the width P of the penumbra. If  r  is the angular radius of the star (diameter S/2), one has P :

P  =  2*dist* tan r

P  =  4.848 * dist * S  =  725.27 * DIST * S

{P is in meters if S is in mas (milliarcseconds) and either dist in Gm, or DIST in AU}.

MORE ADVANCED

When   2% of the diameter is occulted, only 0.5% of the light is occulted.

When 10% of the diameter is occulted, only   5%  of the light is occulted.

When 25% of the diameter is occulted, only  20%  of the light is occulted.

When 98% of the diameter is occulted, already 99.5% of the light is occulted.

         WHERE can we find the edge of the geometrical shadow ? It is at the center of the stellar disc, when 50% of the diameter and of the light flux is occulted : for all wavelengths, the intensity is

I = 0.5 * Io  

Comparing penumbra and Fresnel fringes width

         How does the penumbra smear the fringes ? At the very place of the 1st dark fringe D1 for one side of the stellar disc, we can find a bright fringe B1 or B2 for the other side of the stellar disc.

         If the penumbra erases fringes, the geometrical shadow edges are at 50% of the star light flux, a drop of 0.75 magnitude (instead of 1.5mag when there are fringes and no penumbra, see part 1).

[see Françoise Roques & al., integral (12), Icarus 147, page 533 (2000)]

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Examples

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1/         (483) Seppina occults TYC 0420-01701-1 (mag 11.5) on 2005 Jun 19

2/         (483) Seppina occults UCAC2  32899392  (mag 11.4) on 2005 Jun 24

         Comparing the published magnitudes of the targets in various wavelength bands, one may tentatively fit a F5 dwarf at 320 parsecs as the star of the first event, and a K1 giant at 1050 parsecs as the star in the second event (Jean Lecacheux). (483) is said 69km for its diameter, and was at mag 13.4 .

FIRST EVENT.

         (483) was at 2.377 AU (Astronomical Units) from earth, or 355.594Gm.

         u = SQR[(355.594*10^9) * (600*10^-9) / 2] = 326.6 meters.

            Ø = 65 meters.

         This apparent diameter is really small (only 20 %) compared to the Fresnel unit  : so we may assume that the star is pointlike, the shadow is lined with Fresnel fringes. Occulting body was ~1000 times larger, the limb can be taken as a straight line. In this case, the edges of the geometrical shadow are crossed when the star light intensity is dimmed by 75%.

         Caution ! Does it mean a 75% = 1.5 mag light drop ? Not at all : it would be true only if the light coming from the asteroid was negligible, but it cannot be neglected in this event. And of course, the light coming from the occulting body is not diffracted !

         The shadow speed was about 12.717km/s, that means running a Fresnel unit in 0.0257 second. On the central line, the dip from the 1st B1 bright crest to the 1st D1 dark trough lasted less than 0.0167 s : it was necessary to time faster than that, or 60f/s (frames/second) to perceive any fringe. From B1 to 25% level, the duration was 0.0313 s : 32f/s were enough. But to discriminate between 50% and 25% levels, a time resolution of 6ms only was necessary.

         The S/N ratio must be high enough to record this variation without any doubt. And also to discriminate between 50% and 25% levels, the S/N ratio must be still better :

If a light drop of 6.280 SU = 20% of the combined star and asteroid lights has to be safely detected at 2.6 sigma level, the combined light should be detected with a S/N ratio about 13.

NOTE Of course, if the observer is not on the central line, he will be crossing the shadow edge and the fringe system with a slant that increase the travel time. At more than 70% from the center line to the outside limit, the travel time is increased by more than 40 % : 42 f/s were enough to detect 1st fringe, and 0.0085s resolution to discriminate the 25% level.

SECOND EVENT

         (483) was about the same distance from earth 2.377 AU, and the Fresnel unit was the same :

             u = 326.6 meters.

         The star diameter should be ~17 solar ones, some 23 600 000km. The distance of the star is 1050 pc, or 216 580 000 AU. Hence 23 600 000km is to be multiplied by the ratio 2.377/216 580 000 . We find :

            P = 259 meters.

            This is more than 0.65 u : the fringes were blurred by the penumbra, the shadow was lined by a zone of fading starlight about 260 m wide.

         In this case, the edges of the geometrical shadow are crossed when the star light intensity is dimmed by only 50%.

         Caution ! This does not always mean a 0.75 mag drop, since the light from the occulting planet (Seppina, here) stands steady.

         The shadow speed was about 12.66 km/s, that means running half of the penumbra in 0.1295/12.66 = 0.0102 second. On the central line, it was necessary to time as fast as that (100f/s) to discriminate this light level from full light. Here also, if the observer is at some 70% from the center line to the outside limit, the travel time is increased by more than 40 % and the half crossing time becomes 0.0145 s  : 69 f/s were enough.

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Other examples

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1/ The case of gamma GEM occulted by (381) Myrrha.

         On 1991 January 13, it has been observed from Japan by many observers, one of them (Hiroshi Oya) used a 18cm Cassegrain telescope with a photoelectric receptor (see The Astronomical Journal, 105 #4, April 1993, pp 1553 sq). It has been alleged (for instance at the European Symposium on Occultation Projects 2003) that Fresnel fringes had been observed.

         But for a 600nm wavelength the Fresnel unit was ~340 meters, whereas the now assumed ~1.35mas diameter of the star generated a 2512 m wide penumbra (about 8 times the largest interfringe) . The misunderstanding comes from a tentative caption of a figure which says "Fresnel diffraction convolved by the equipment time constant of about 200 milliseconds produced the gradual disappearance and reappearance." Of course the attempted deconvolution only gave an upper limit of 2.6mas for the star diameter and no Fresnel fringes have been detected with a 0.2s time constant and a 15.2km/s shadow speed, the whole penumbra being crossed in less than 0.16s.

2/ The case of TYC 0304 00965 (Mv10.3) occulted by (287) Nephthys (M12.3).

         On 2000 June 6, a visual observation with a 207mm telescope was thus described : "A blink of full amplitude, i.e. starting from the normal star level and falling to the sky level, then returning. Possibly all the event was a passing train of diffraction fringes...Estimated duration = 0.8s +0.2/-0.3 (with black sky during about 0.3s)." Interviewed for more details, the observer (Jean Lecacheux) said that he saw a fast "twinkling" before and after the blink.

         Disappearance UT 21 40 01.73 ± 0.35     (PE 0.7s subtracted)

         Reappearance  UT 21 40 02.52 ± 0.25     (PE 0.7s subtracted)

         The observed timings are in good agreement for the predicted time for a grazing occultation at the west limb of Nephthys.

(Note : Here under, all computations are done with full accuracy, and rounded afterwards).

         From 1.68 AU = 251.3 Gm, the Fresnel unit was 255.6 m for a 520 nm wavelength. The Nephthys shadow speed was only 5.62 km/s, unusually slow.

         The star, with high galactic latitude, is probably a dwarf F2 spectral type, some 200 parsecs away, with 1.4 the Sun's size. Then its angular diameter would be ~0.06 mas, yielding an equivalent diameter of 73 m (0.3 u) at Nephthys distance.

         All the best conditions were gathered to observe Fresnel fringes !

         Now one may tentatively make some hypothesis to adjust the observed timings with the existing fringes system.

1st adjustment Assuming the geometric shadow limb crossed during 0.3 second :

         Henceforth the observer was at a distance from the center of the shadow :

34 km * cos â = 33.9895 km, that is to say only 10.5 meters inside the limb !

         Henceforth the timing of events would be :

UT 21 40 00.97        crossing B²            mag 10.00     (not observed)

UT 21 40 01.10        crossing D¹            mag 10.38     (not observed)

UT 21 40 01.29        crossing B¹            mag   9.84    (not observed)

UT 21 40 01.66        crossing ¤2            mag 10.75     (not observed)

UT 21 40 01.73        star at 40%             mag 10.93    

UT 21 40 01.98        ingress   ¤4            mag 11.27

UT 21 40 02.13        mid-event               mag 11.33

UT 21 40 02.28        egress    ¤4

UT 21 40 02.52        star at 40%

UT 21 40 02.59        crossing ¤2                               (not observed)

UT 21 40 02.96        crossing B¹                               (not observed)

UT 21 40 03.16        crossing D¹                               (not observed)

UT 21 40 03.28        crossing B²                               (not observed)

2nd adjustment assuming that the 0.3 second "black" chord is from 50% flux level

(circle ¤2 around the shadow).

         Henceforth the timing of events would be :

UT 21 40 01.06        crossing B²            mag 10.00     (not observed)

UT 21 40 01.20        crossing D¹            mag 10.38     (not observed)

UT 21 40 01.42        crossing B¹            mag   9.84    (not observed)

UT 21 40 01.98        ingress  ¤2             mag 10.75

UT 21 40 02.28        egress   ¤2

UT 21 40 02.83        crossing B¹                               (not observed)

UT 21 40 03.06        crossing D¹                               (not observed)

UT 21 40 03.19        crossing B²                               (not observed)

         At 80 meters outside the limb, the star light was dimmed at 46% or mag 10.82 for the combined light of the star and (287). The timings would have been done at 34.1521 km from the center, 0.595 u outside the limb. There the starlight is at 76.5%, and the combined magnitude 10.387. One cannot understand why an observer should trigger a stopwatch in such a manner...

3rd adjustment assuming now that the timings were made at crossing the 1st dark fringe :

         Henceforth the timing of events would be :

UT 21 40 01.48        crossing B²            mag 10.00

UT 21 40 01.73        crossing D¹            mag 10.38

UT 21 40 02.13        midtime                 mag 10.11

UT 21 40 02.52        crossing D¹

UT 21 40 02.78        crossing B²

         The obvious problem is that there is no possibility of "black sky during about 0.3s". This 3rd hypothesis cannot be reconciled with the observation.

Conclusion

         Of course, many other parameters could be adjusted, such as timings given at ±0.25s, the 520nm choice for the wavelength, the Nephthys diameter, the fact that its shape could be far from a sphere, etc. Anyhow, without these further adjustments, there is some coherence between observation and computation.

         Amongst thousands of reports, this is the only one worth of consideration. But the assumption of such a grazing observation is somewhat extraordinary.

         There is yet no surely known observation of fringes from an asteroidal occultation.

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Lunar occultations

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            This is the only kind of occultation when fringes HAVE been observed. The Fresnel unit is in the range of one decameter. But the Moon motion is slow, in the range of 1km/s. And over all, the stars have a small equivalent diameter. At 0.4 Gm from the observer, a large enough 1 mas star has an equivalent diameter of 2 meters only. Hence fringes can be recorded, all the better when the occultation is grazing.

            The first record was made by Albert Arnulf from Paris on 1933 April 6th at 20h 55m 16s GMT, the Moon occulting Regulus at about 85km from the limb. Arnulf recorded two fringes on a rapidly moving photographic plate (Comptes-rendus de l'Académie des Sciences 1936, Paris, 202, 115) and deduced tentatively a 1.8 mas diameter for Regulus.

            The main target of this method is Regulus (a bright star occulted by the Moon is a requisite. Antares, which is also occulted by the Moon, is a red giant much too large to produce fringes).

            For instance, R. R. Radick observed the lunar occultation of Regulus on 1980 March 28 (at UT 03 15 40.23) with a 90cm telescope from Cerro Tololo (the Astronomical Journal #86, November 1981). He used mainly a photomultiplier tube behind a yellow interferential filter with a ~4 nm passband, to avoid the chromatic blur of the fringes. The time resolution was 1 ms, and the limb slope was ~5°, increasing the crossing time by a factor ~11.5 .

            With a S/N ratio of 33.5, Radick recorded 7 fringes in the yellow narrow band. From that he deduced a 1.32 ± 0.12 mas diameter for Regulus. Obviously, this kind of observation is out of reach for an amateur astronomer system.

NOTE. The latest data for Regulus (the Astrophysical Journal, #560, July 2005) came from an interferometric array on Mount Wilson. Like Altair and Achernar, Regulus should be an oblate fast rotator : the equatorial speed is about 320km/s ; at a distance of 32.5 parsecs, the rotation period is less than 16 hours ! Due to the centrifugal weakening of gravitation, the star exhibits a large equatorial darkening :

   One can then understand why the estimated diameter derived from Fresnel fringes in a lunar occultation is strongly dependent on limb slope (12.7° for Arnulf, 5° for Radick).

            An asteroidal occultation of Regulus by a small mainbelt asteroid, (166) Rhodope, was observed on 2005.10.19 . The Fresnel unit was 370m for a 600nm wavelength, whereas the penumbra was 2800 m to 3400 m according to the position of observers, and even 9600 m (0.3s) for a grazing observation.

            Once more, there were NO fringes at all.

For EAON, Raymond DUSSER

Augmented, 2006, May 1st

Acknowledgments to Jean Lecacheux & Jean Schwaenen, who helped me constantly.

ANNEX 1

CHROMATIC BLURRING OF FRINGES

ANNEX 2

(SOME MORE ACCURATE VALUES)

(FOR 6 BRIGHT or DARK FRINGES)

"distance" from the geometric shadow edge, in Fresnel units  u  (positive values outside)

"intensity" with source = 1 (reflected light by the occulting body NOT taken into account)

         MORE ADVANCED

ANNEX 3

NEPHTHYS FRINGES

ANNEX 4

FRESNEL  UNIT  FOR

FINITE DISTANCES

         When the experiment is made in a room, let be :

S the source of light (assumed pointlike : for instance a small hole in a sunlighted screen)

E the occulting edge (for instance, any opaque body)

O the place of observation (for instance a white paper sheet)

SE the distance from the source to the edge

EO the distance from the edge to the place of observation

SO = SE + EO

         The Fresnel unit is given by :

u = SQR[ SE * EO * lambda / 2 * SO ]

                        All computations given above are still valid. For instance, if SE = EO = 2 meters, for a yellow 577nm = 0.000000577m wavelength (the one used by Radick), one finds  u  = 0.000537m = 0.537mm. The first bright fringe is then  1.22 * u = 0.66 millimeter from the geometric shadow edge, and the fringes are easy to see with the naked eye.