COMETARY PARALLAX
StarFest 2005Bays Mountain Preserve
October 22, 2005John C. Mannone
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.
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 lightChange in geometry (trigonometric parallax)
(Spectroscopic parallax applies to determination of distance from spectroscopically determined luminosity and spectral class)
Trigonometric Parallax
A simple example:
Look at me with one eye shut
Then the other
Note my apparent position against the backdrop is different
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”
Stellar Parallax
Stellar Parallax
Stellar Parallax
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
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.
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
pd1
b
A”
B”
Comet’s apparent positions among background stars
C
GraphicalSoftware SimulationPhotographic Analysis
Image Overlap/Scaling
AnalyticalThree-Dimensional Exact Solution- Celestial Sphere
Spherical Trigonometry
Determination of Cometary Parallax
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.
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)
“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)
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
Date and Time: April 4, 1997, 8 PM PSTCamera: 50mm Minolta lens f/1.7 on a tripod;Film/Exposure: Fuji 800 film (35 mm)/ 30 secondsLength/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 ftf = 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
“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
Asagio, Vincenza, ItalyCathedral City, CA, USA
HALE-BOPP March 8, 1997
9-hour time difference meansphotos taken at different local times
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
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
Software SimulationPhotographic 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
Hale-Bopp 100 degree field of view from Joshua Tree, California
Hale-Bopp 30 degree field of view from Joshua Tree, California
Hale-Bopp 15 degree field of view from Joshua Tree, California
Hale-Bopp 1 degree field of view from Joshua Tree, California
Hale-Bopp 1 degree field of view from Asagio, Italy
Hale-Bopp 1 degree field of view overlays68% transparency of top slide
Italian
Italian
USA
USA
Hale-Bopp 1 degree field of view overlaysOverlap Background Stars
Italian
Italian
USA
USA
Hale-Bopp 1 degree field of view overlaysRotate to align along RA/Dec lines
Hale-Bopp 1 degree field of view overlaysRe-establish Overlap
Italian
Hale-Bopp 1 degree field of view overlaysRe-establish Overlap
Italian
Measure angular separation on Starry Night; relate to scale length
Measure length; use ratio and proportion to obtain parallax
Comet Hale-BoppMarch 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
Parallax by Analytical Methods
Rp
d1
b
A”
B”
C
CB
CA
Celestial Sphere
C
Projected Geographic Positions
Apparent Comet Positions Projected on Celestial Sphere
C
CA
CB
Apparent positions of comet from projected A and B
Actual position of comet: C
p
Parallax seen on a Spherical Triangle
RA
Dec
Spherical Geometry
Parallax is calculated from object’s equatorial coordinates from both locations using the law of cosines for spherical trianglescos c = cos a cos b + sin a sin b cos C = sin a' sin b' +cos a' cos b' cos Cc 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
Ba
bc
The 3-dimensional Exact Calculation of Parallax
Symbols in this graphic have different meanings
Three-Dimensional Exact Solution- Celestial SphereSpherical Trigonometry
Parallax by Analytical Methods
cos p = sin latA sin latB + cos latA cos latB cos (lonB-lonA)
Need chord length to calculate distanceand an understanding of the celestial rotating coordinate system
Courtesy of Scott Robert Ladd, “Stellar Cartography”
Equatorial and Horizon Coordinates
Greenwich Mean Sidereal TimeHale-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
Projected Chord Determination
Vector Analysis
or
Coordinate RotationUsing Transformation Matrices
or
Graphically using a CelestialSphere model and string
Not reviewed here
Coordinate Information for Comet Hale-Bopp
March 8, 1997 11:40Z Simultaneously Viewed from USA and Italy
Joshua TreeAsiago
Joshua Tree Comet CoordinatesJ (now) Epoch from Starry Night Backyard v 3.1
RA 22h 15.348m = Dec 39o 49.504’ = GST = 22h 44m 51.7s
Lat comet = = 39.825067o
Lon comet = H = - GST = -29.514m@15o/hour H = -7.378417o
Hale-Bopp March 8, 1997 11:40Z
Joshua TreeAsiago
Observer Coordinates (estimated)
A- Joshua TreeLatA 33o 44.4’ N
LonA 116o 25.2’ WTime Zone -7 hr => 4:40 am March 8, 1997 local daylight time
B- AsiagoLatB 48o 22.809’ NLonB 9o 37.331’ E
Time Zone +1 hr => 12:40 am March 9, 1997 local standard time
Lat comet = = 39.825067o
Lon comet = H = -7.378417o
(from Joshua)
Hale-Bopp March 8, 1997 11:40Z
Joshua TreeAsiago
Actual distance to Earth 1.382 AUFrom orbital parameters in Starry Night
b = 6672.88 km from spherical trigonometry (compare to Earth radius of 6378 km)
p = 6.87” from graphical methodd1 =1.372 AU (0.72% high) Accurate, but imprecise (10%)
p = 6.8319” from spherical trigonometryd1 = 1.385 AU (0.19% high)Accurate and precise
d1 = (b/2)cotan(p/2)
Hale-Bopp March 8, 1997 11:40Z
Conclusion1) 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
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.