COMETARY PARALLAX StarFest 2005
Bays Mountain Preserve
October 22, 2005
John C. Mannone

Slide2:

Abstract
Planetarium software and PowerPoint slide utilities are engaged to graphically determine the parallax of a near object observed by amateur astronomers. This graphical method seems to favorably compare with spherical trigonometry methods (not discussed). Though applicable to some planets and our Moon, the technique will be demonstrated with comets on close approach (~1 au). This is useful for planned coordinated viewing/photography and for a classroom experiment to determine distance of approach. The technique can be extended to very close objects such as satellites and meteors, but video imaging and processing will be required.

Slide3:

Definition of Parallax
What is it?
When an object is viewed from two different positions, there is a shift in the apparent position of the object against a distant background.
Shift can be caused by several things, e.g.,
Change in refractive index which bends the light
Change in geometry (trigonometric parallax)
(Spectroscopic parallax applies to determination of distance from spectroscopically determined luminosity and spectral class)

Slide4:

Trigonometric Parallax
A simple example:
Look at me with one eye shut
Then the other
Note my apparent position against the backdrop is different

Slide5:

Trigonometric Parallax
Eyes are separated some base distance, b
The angular difference of my image perceived by each eye (each viewing position) is the parallax (angle) related to the base distance and the my distance to the observer. The further away, the smaller the angle:
Tycho Brahe tried to apply parallax in 1570’s, but Friedrich Bessel first successfully applied this to stars in 1838: 61 Cygni: 0.333” (modern result 0.289”). The closest star, Proxima Centauri, has largest p = 0.772”

Slide6:

Stellar Parallax

Slide7:

Stellar Parallax

Slide8:

Stellar Parallax

Slide9:

Stellar Parallax
parallax angle distance Parallax, p, and distance, d, are related through simple geometry, especially when the the parallax is small, as it is in the case of stars.
d (parsec) = 1/p (arcsec) 1 pc = 3.26 ly

Slide10:

Cometary Parallax Comets approach much closer than stars, so expect parallax angle be much larger.
Because of its rapid motion (relative to stars), a simultaneous observation will limit observation to different places on the Earth (instead of two different orbital positions of the Earth).
This limits the distance between observation sites to the chord through the Earth connecting the two locations.
A further reduction in the chord because of the comet’s perspective.
The parallax will be larger only by an order of magnitude over nearby stars.

Slide11:

Determination of Comet Approach Distance by Parallax Distance-Parallax Related through the Projected Chord
tan (p/2) = b/2d
d1 = d - R + (R2-b2/4)1/2 ~d for more distant objects
p is the parallax (angle), b is the projected chord distance A”B” between the 2 observing sites A and B (perpendicular to the zenith line d1 at a point on the surface of the Earth directly beneath the comet at C). R d p d1 b A” B” Comet’s apparent positions among background stars C

Why the Interest in Cometary Parallax? I purchased a personally autographed photograph of Hale-Bopp from Dr. Tom Bopp at UTC in March 2003.
It is one of his favorite photographs by Bill and Sue Fletcher.
I became interested in everything about the photograph: the photographic details, identification of the major stars. I reasoned others might have simultaneously photographed the comet, especially near closest Earth approach and wondered if the comet’s distance could be easily determined by comparing photographs.
Synchronizing time is easy with planetarium software.

Slide14:

Hale-Bopp Trajectory Near Perihelion Earth Closest Approach: March 22, 1997 (1.315 AU)
Sun Closest Approach: April 1, 1997 03:14 UT (0.914 AU)

Slide15:

“This is the beautiful Comet Hale-Bopp as it approached Earth in March of 1997.
The solid portion or nucleus of the comet is made up of ice, frozen gases, dust and small rock. Compared to most comets Hale-Bopp is very large - about 35 kilometers in diameter. As its orbit brought it closer to the sun, the frozen mass began to melt and a coma, which is a gaseous cloud, developed around the nucleus. This coma has grown to be hundreds of thousands of miles in diameter. Finally the tail developed which became millions of miles long.
This color photo reveals both the reddish cream-colored dust tail, and the many long blue streamers of the ion (gas) tail.” (photographers Bill & Sue Fletcher)

Slide16:

Joshua Tree National Park
"God just gave me a gift. I get to see things in the sky that the average person doesn't see…I think that what's out there is God's creation meant for our enjoyment." Wally Pacholka TIME Picture of Year 1997, TIME/LIFE Picture of the Century 2000

Slide17:

Date and Time: April 4, 1997, 8 PM PST
Camera: 50mm Minolta lens f/1.7 on a tripod;
Film/Exposure: Fuji 800 film (35 mm)/ 30 seconds
Length/Width Ratio: 1.36 => picture cropped Joshua Tree located with the help of digital desert and aviation charts: Coordinates 34N, 116W
Elevation 3000 to 4000 ft
f = 50 mm, f/ = 1.7, D = 29.4 mm
Approximate FOV:
2arctan [(36 x 24 mm/2)/50 mm]
FOV = 27.0o x 39.6o

Slide18:

“Comet Hale-Bopp photographed on the morning of March 8, 1997, from Stedman, N.C. This 10-minute exposure was made with a 12.5-inch reflecting telescope (f = 1200 mm) and exposed on Fujicolor SG-800 Plus film. The telescope tracked the comet during the exposure, rendering the stars as short lines. Hale-Bopp is moving northward against the stars at the rate of 1.5 degrees per day*. The comet continues to be visible to the naked eye in the predawn northeastern sky.” (Jim Horne, photo 33)
~50,000 mph Calculated FOV 1.15o x 1.72o

Slide19:

Asagio, Vincenza, Italy Cathedral City, CA, USA HALE-BOPP March 8, 1997 9-hour time difference means
photos taken at different local times

Slide20:

Joshua Tree National Park, CA, USA HALE-BOPP March 8, 1997 (actually March 7) This Fletcher photograph was made with the special Schmidt camera/telescope.
An 8-inch Celestron equivalent to a super fast (f/1.5) 305 mm telephoto lens.
Equipped with curved film holder => no distortion along width.
Wide field of view 4.5o x 6.75o

Slide21:

An 8-inch Celestron equivalent to a super fast (f/1.5) 305 mm telephoto lens.
Equipped with curved film holder => no distortion along width HALE-BOPP March 7, 1997 4:40 AM This Fletcher photograph was made with the special Schmidt camera/telescope. Wide field of view 4.5o x 6.75o

Slide22:

Software Simulation
Photographic Analysis
Parallax is determined by superposition of images with the same field of view or scale.
Both views are aligned. The transparency can be adjusted with the picture editing feature. This facilitates the correct overlapping.
Angular separation between the comet and the star is determined (a standard feature on Starry Night Backyard software).
The parallax is determined by comparison with the scaled comet-star distance. Parallax by Graphical Methods

Slide23:

Hale-Bopp 100 degree field of view from Joshua Tree, California

Slide24:

Hale-Bopp 30 degree field of view from Joshua Tree, California

Slide25:

Hale-Bopp 15 degree field of view from Joshua Tree, California

Slide26:

Hale-Bopp 1 degree field of view from Joshua Tree, California

Slide27:

Hale-Bopp 1 degree field of view from Asagio, Italy

Slide28:

Hale-Bopp 1 degree field of view overlays
68% transparency of top slide Italian Italian USA USA

Slide29:

Hale-Bopp 1 degree field of view overlays
Overlap Background Stars Italian Italian USA USA

Slide30:

Hale-Bopp 1 degree field of view overlays
Rotate to align along RA/Dec lines

Slide31:

Hale-Bopp 1 degree field of view overlays
Re-establish Overlap Italian

Slide32:

Hale-Bopp 1 degree field of view overlays
Re-establish Overlap Italian Measure angular separation on Starry Night; relate to scale length Measure length; use ratio and proportion to obtain parallax

Slide33:

Comet Hale-Bopp
March 8, 1997 11:40Z Parallax, p = (.30 inch) (249”/10-7/8 inch) = 6.87’’ +/- 10% 4’9’’ arc between indicated star HIP 09881
and comet measured 10-7/8 inch 0.30 inch parallax Asiago, Italy Joshua Tree National Park, CA Using a different star, the results are summarized below

Slide34:

Parallax by Analytical Methods

Slide35:

R p d1 b A” B” C CB CA Celestial Sphere C Projected Geographic Positions Apparent Comet Positions Projected on Celestial Sphere

Slide36:

C CA CB Apparent positions of comet from projected A and B Actual position of comet: C p Parallax seen on a Spherical Triangle DRA DDec

Slide37:

Spherical Geometry Parallax is calculated from object’s equatorial coordinates from both locations using the law of cosines for spherical triangles
cos c = cos a cos b + sin a sin b cos C = sin a' sin b' +cos a' cos b' cos C
c parallax, a and b equatorial colatitudes, C equatorial longitude difference, a' and b' are the corresponding latitudes = 90-a and 90-b (degrees) C A B a b c The 3-dimensional Exact Calculation of Parallax Symbols in this graphic have different meanings

Slide38:

Three-Dimensional Exact Solution- Celestial Sphere
Spherical Trigonometry
Parallax by Analytical Methods cos p = sin latA sin latB + cos latA cos latB cos (lonB-lonA)

Slide39:

Need chord length to calculate distance
and an understanding of the celestial rotating coordinate system

Slide40:

Courtesy of Scott Robert Ladd, “Stellar Cartography” Equatorial and Horizon Coordinates

Slide41:

Greenwich Mean Sidereal Time
Hale-Bopp March 8, 1997 11:40Z “Sidereal time is the measure of the earth's rotation with respect to distant celestial objects.
By convention, the reference points for Greenwich Sidereal Time are the Greenwich Meridian and the vernal equinox (the intersection of the planes of the earth's equator and the earth's orbit, the ecliptic). The Greenwich sidereal day begins when the vernal equinox is on the Greenwich Meridian. Greenwich Mean Sidereal Time (GMST) is the hour angle of the average position of the vernal equinox, neglecting short term motions of the equinox due to nutation.”
Rick Fisher NRAO Green Bank, WV Calculator by AstroJava

Slide42:

Projected Chord Determination
Vector Analysis
or
Coordinate Rotation
Using Transformation Matrices
or
Graphically using a Celestial
Sphere model and string
Not reviewed here

Slide43:

Projected Chord by Vector Analysis

Slide44:

C B A x y z Prime meridian through
Greenwich, England Observing Stations A and B; Comet is at Zenith at point C Position Vectors A, B, and C

Slide45:

C B A x y z Position Vectors A, B, and C
Chord Vector L = A-B
Translate the Chord Vector
L = A-B to the Origin and note angle between L and C L Vector Analysis for Projected Chord

Slide46:

C B A L b C L C = (LxCx + LyCy + LzCz) =LC cos b Vector Dot Product

Slide47:

L b C The projection of the chord perpendicular to the line of sight from the comet (along C):
b = L sin b The LC-plane b

Slide48:

Projected Chord by Coordinate Transformation
Using Rotation Matrices

Slide49:

C B A x y z Prime meridian through
Greenwich, England Observing Stations A and B; Comet is at Zenith at point C

Slide50:

C B A x y z Equatorial Coordinates
Comet RA = a and Dec = d Locations A and B latitude and longitude

Slide51:

C B A x y z View from comet of chord AB is at an oblique angle.
Therefore, it only sees its projection.

Slide52:

C B A x y z Rotate the coordinate axes in the xy-plane about the z-axis through the hour angle H to change reference from H = 0 on the prime meridian to H where the comet is located on the celestial sphere. y’ x’ z’ H H = a -GST

Slide53:

C B A x y z Rotate the coordinate axes x’z’ about the y’ axis through the polar angle. This reference changes from z = z’ =0 through the north pole to z’’ through the zenith of the comet. y’ x’ z’ 90-d y” x” z”

Slide54:

C B A Parallel line to z’’ axis and perpendicular to the x”y”-plane. The view from the comet now shows the projected chord and the parallax calculation can proceed as normal. y” x” z” B” A” 90

Slide55:

x’’ sin d 0 -cos d cos H sin H 0 x
y’’ = 0 1 0 -sin H cos H 0 y
z’’ cos d 0 sin d 0 0 1 z
Double Rotation Matrices
x’ = R1 x x’’ = R2 x’ x’’ = R2R1x b2 = [(xB”- xA”) 2 + (yB”- yA”) 2

Slide56:

Coordinate Information for Comet Hale-Bopp
March 8, 1997 11:40Z
Simultaneously Viewed from USA and Italy

Slide57:

Joshua Tree Asiago Joshua Tree Comet Coordinates
J (now) Epoch from Starry Night Backyard v 3.1
RA 22h 15.348m = a
Dec 39o 49.504’ = d
GST = 22h 44m 51.7s
Lat comet = d = 39.825067o
Lon comet = H = a - GST = -29.514m
@15o/hour H = -7.378417o
Hale-Bopp March 8, 1997 11:40Z

Slide58:

Joshua Tree Asiago Observer Coordinates (estimated)
A- Joshua Tree
LatA 33o 44.4’ N
LonA 116o 25.2’ W
Time Zone -7 hr => 4:40 am March 8, 1997 local daylight time
B- Asiago
LatB 48o 22.809’ N
LonB 9o 37.331’ E
Time Zone +1 hr => 12:40 am March 9, 1997 local standard time
Lat comet = d = 39.825067o
Lon comet = H = -7.378417o
(from Joshua) Hale-Bopp March 8, 1997 11:40Z

Slide59:

Joshua Tree Asiago A- Joshua Tree
LatA 33o 44.4’ N
LonA 116o 25.2’ W
Time Zone -7 hr
B- Asiago
LatB 48o 22.809’ N
LonB 9o 37.331’ E
Time Zone +1 hr
Lat comet = d = 39.825067o
Lon comet = H = -7.378417o
(from Joshua) Hale-Bopp March 8, 1997 11:40Z Rotation Matrix 1
Rotate about z-axis through
the hour angle H
R1 = [cos H sin H]
[-sin H cos H]
Rotation Matrix 2
Rotate about y-axis through
the angle 90-d
R2 = [sin d -cos d]
[cos d sin d ]

Slide60:

Joshua Tree Asiago A- Joshua Tree
LatA 33.74o
LonA -116.42o
B- Asiago
LatB 48.38o
LonB 9.622o
Lat comet = d = 39.825067o
Lon comet = H = -7.378417o
(from Joshua) Hale-Bopp March 8, 1997 11:40Z Rotation Matrix 1
Rotate about z-axis through
the hour angle H
R1 = [cos H sin H]
[-sin H cos H]
Rotation Matrix 2
Rotate about y-axis through
the angle 90-d
R2 = [sin d -cos d]
[cos d sin d ]

Slide61:

Joshua Tree Asiago A- Joshua Tree
LatA 33.74o
LonA -116.42o
xA = cos lonA cos latA
yA = sin lonA cos latA
zA = sin latA
B- Asiago
LatB 48.38o
LonB 9.622o
xB =cos lonB cos latB
yB =sin lonB cos latB
zB = sin latB Hale-Bopp March 8, 1997 11:40Z

Slide62:

Joshua Tree Asiago The results of the double rotation are:
xA’’ = -0.600323
yA’’ = -0.786065
zA’’ = 0.555425
xB’’ = -0.167348
yB’’ = 0.194196
zB’’ = 0.966583
The projected chord is calculated:
b = [(Dx”)2 + (Dy”)2]1/2 = 1.07162 earth radii units
R = 6378 km => b = 6834.82 km Hale-Bopp March 8, 1997 11:40Z

Slide63:

Joshua Tree Asiago Actual distance to Earth 1.382 AU
From orbital parameters in Starry Night
b = 6672.88 km from spherical trigonometry (compare to Earth radius of 6378 km)
p = 6.87” from graphical method
d1 =1.372 AU (0.72% high)
Accurate, but imprecise (10%)
p = 6.8319” from spherical trigonometry
d1 = 1.385 AU (0.19% high)
Accurate and precise
d1 = (b/2)cotan(p/2) Hale-Bopp March 8, 1997 11:40Z

Conclusion:

Conclusion 1) Graphical determination of parallax is effective with planetarium software, such as Starry Night, and PowerPoint picture options. Scanned photographs of simultaneous photographs would be analyzed in the same way.
2) Results are very accurate, though more difficult to reproduce than with spherical trigonometry. This was applied to Comet Hyakutake with superior results.
3) Procedure is sufficiently simple for secondary educational outreach and amateur astronomy, yet easily extended to collegiate level.
4) Extension to Lunar parallax using solar system objects like Jupiter as background is very effective.

Conclusion:

Conclusion 5) Extension to ISS is possible with the help of Heaven-Above website for satellite position and altitude. Video imaging and processing would be required to synchronize simultaneous observations. This would be a good calibration technique since the distance to the satellite would be known.
6) Extension to Meteoritic parallax is an advanced experiment similar to satellite tracking except for the uncertainty of when a rapidly moving meteor will appear. It’s height is unknown, but is in the ionosphere and could be determined.

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