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.