Patrick, Not that you need my help on the math, but below are three posts with background info on the analogous problem of computing the distance to Mars based on diurnal parallax. I will have to dig out the stored parallax construction that I did from the 2005 Mars opposition and put it back on the Utah Astro Gallery. The Yerkes Hands-On-Universe 2003 Mars opposition project to replicate measuring the distance to Mars by the diurnal parallax method, referenced in my post, is now at this location: http://www.handsonuniverse.org/activities/Explorations/MarsParallax/ but they seem to have deprecated the path rights to the math construction documentation. Here is Pete Lawrence's page from his 2005 attempt at the analogous problem of diurnal parallax: http://www.digitalsky.org.uk/Marsnight/marsnight-2005-10-23.html Some where in my archives I have Gill's 1878 paper on measuring the parallax of Mars - which I recall was useful - but it can probably be retrieved out of NASA-ADS with "Gill Acsension report 1878". My apologies in advance if this post is in the category of "too much information." Clear Skies - Kurt ================ From: "canopus56" <canopu...@yahoo.com> Newsgroups: sci.astro.amateur Subject: Historical astronomy - finding Mars parallax and distance Date: 19 Oct 2005 21:21:06 -0700 I just finished reading Arthur Berry's _A Short History of Astronomy_ (1898, 3d 1961). Berry reviews the major astronomical accomplishments through the end of the 19th century. Berry mentioned that in addition to the transit of Venus, the parallax of Mars at opposition was also used in the 1870s to establish an Earth-Mars distance and by Kepler's Third Law, the Earth-Sun distance. Berry at Chap. XIII, Sec. 281. Earlier measurements were made in the 1670s. Berry at Chap. VII, Sec. 161. This suggests that modern amateurs equipped with CCDs and web cams could replicate the experiment and measure the parallax of Mars using an Earth baseline during the Oct 30, 2005 greatest size and/or Nov. 7, 2005 opposition of Mars. Pete Lawrence restaged a similar historical experiment to find the parallax and distance of the Moon using the Nov. 8, 2003 and Oct. 28, 2004 lunar eclipses. Lunar Parallax Demonstration Project http://www.digitalsky.org.uk/lunar_parallax.html Two techniques have been used to measure the parallax of Mars. The first involves an Earth baseline and two observers. This method was first used by Cassini, Picard and John Richer during 1671-1673. Richer traveled to Cayenne (lat 5 deg) while Picard and Cassini observed from a newly completed observatory in Paris. Chap. VII, Sec. 161. This provided a north-south baseline on which the observers could make a position observation at the same time - opposition. The second technique, called the diurnal method, involves one observing station measuring Mars' position on the night of opposition at different times. The changed position of the OP over time resulting from the Earth's rotation provide the baseline. This technique was invented by Flamsteed (1646-1720) and was revived by Sir George Airy (of the Airy Disk) in 1857. In 1877, a Dr. "Gill" working at Ascension used the diurnal method to measure the parallax of Mars at opposition. Berry at Chap. XIII, Sec. 281. The current Mars opposition occurs on Nov. 7, 2005. At that time, Mars will be about 69,415,600 km (43,132,800 miles) from the Earth. Using the half-angle formula, those distances suggest a parallax at opposition on a 6,000 kilometer baseline of 2.8 arcsecs, or on a 12,000 km baseline of 5.6 arcsecs. A similar educatonal project was undertaken by the Univ. of Chicago Yerkes Observatory Hand-On-Universe outreach program for the 2003 Mars opposition. That project used the diurnal method. The project documentation is here - http://sunra.lbl.gov/~vhoette/Explorations/MarsParallax/ including a useful spreadsheet on computing the size of the baseline for the diurnal method. http://sunra.lbl.gov/~vhoette/Explorations/MarsParallax/docs/ ============================= From: "canopus56" <canopu...@yahoo.com> Newsgroups: uk.sci.astronomy,sci.astro.amateur Subject: Re: MARS PARALLAX NIGHT Date: 23 Oct 2005 13:02:38 -0700 canopus56 wrote:
For the diurnal method, the total parallax is computed from a baseline of - 1) the apparent motion generated by the relative distance traveled by Earth and Mars during the measurement period; 2) _less_ Mars's parallax along the baseline between the two diurnal measuring points.
After thinking on it some more, looks like I got it wrong in the last post. This post contains what I feel is the right construction and estimation for Mars parallax within apparent diurnal motion. Attached is a geometric construction for the diurnal motion of Mars at opposition on 11/7/2005 at 8:30 UTC as seen from 43 deg N latitude using two astrophotos taken about 3 1/2 hours before and after opposition: http://members.csolutions.net/fisherka/astronote/observed/20051107Mars/20051... The apparent diurnal motion in this example - 418" across 7 hours - is predicated to be the sum of: 1) the astrometric change in position of 396". 2) Mars's parallax of 37.5", adjusted to 21.74" when the parallax is measured from 43 deg North. The astrometric position change is caused by the Earth's revolution around the Sun and a baseline from the Earth's 7 hour motion at its orbital velocity of 20358 km/hour, or a total baseline of 142506km. The half-angle forumla result on this baseline is 396" which is similar to the 393" arcsec astrometric position change per the MPC epheremis generator. At first glance, apparent diurnal motion of Mars would simply be based on the distance travelled by the Earth because of its rotation between taking the photographs plus the observer's Earth baseline movement as a result of the Earth's rotation during that same period. See diagram. However, part of the rotation distance travelled duplicates the revolution baseline distance travelled. For example in the geometric construction, the observing point moves as a result of the Earth's orbit over 7 hours by 142506km. The observing point also moves as a result of the Earth's rotation a total of 7410km. However, 1/2 the total baseline movement caused by the Earth's rotation is duplicated by the motion caused the Earth's revolution. So, in the geometric construction, the observer does not move 149916 km. The observer moves 149916km minus 1/2 the Earth's rotation baseline movement of 7410km, or 3705km. This gives a baseline perpendicular to Mars over the 7 hours of 146211km. On a 146211km baseline, a 43 deg North observer should see 418" of apparent movement of Mars across a 7 hour period centered on the opposition at 11/7/2005 8:30UTC. This post, data in prior posts in this thread, and the above geometric construction schematic illustrates how astrophotographers can prepare geometric constructions to confirm if Martian apparent motion seen in their astrophotos measured by the diurnal method (two photos taken at the same latitude at differing times) is consistent with a Mars parallax of 37.5". - Canopus56 ============================== From: "canopus56" <canopu...@yahoo.com> Newsgroups: uk.sci.astronomy,sci.astro.amateur Subject: Re: MARS PARALLAX NIGHT Date: 21 Oct 2005 19:43:43 -0700 Brian Tung wrote:
Err, sure--if you can measure those 12,000 km to an accuracy of a few cm, and the time to an accuracy of a millisecond or so.
Here's the predicted values I get for three dates and times using data from the Minor Planet Center Ephemeris generator, the half-angle forumla and Duffet-Smith's simplified equation for parallax. The MPC ephemeris generator gives the one-way-light distance to Mars. The half-arcsecond variance on 38" is much less than what can measured with a typical amateur's CCD setup, but still it's almost two Mars angular diameters, or even using a 6,000 km baseline, it's about one-Mars diameter. Half-angle formula: Parallax= 2 * arctangent ( ( Earth baseline distance / 2 ) / Mars distance ) Duffett-Smith's simplified parallax equation (Practical Astronomny with Your Calculator, 3ed at Sec. 39, p. 70), for the Sun, planets and comets (other than the Moon): Parallax = 8.794 arcsecs / r where r is the ratio of the distance to the object divided by one astronomical unit. Constants - The speed of light is 17987547.48 kilometers per second. Earth equatorial diameter baseline: 12756.27 kilometers ================= 10/23/2005 6:00 UTC Approx. date of Parallax of Mars Project Light-speed distance: 3.8922 minutes Distance kilometers: 70011132.3 km Half-angle formula parallax Earth diameter baseline: 37.58215967 arcsec Duffett-Smith method parallax: 37.58155686 arcsec Half-angle formula parallax 6,000 km baseline: 17.6770293 arcsec ================= 10/30/2005 7:30 UTC Maximum angular size; Minimum light speed distance Light-speed distance: 3.859126 minutes Distance kilometers: 69416212.16 km Half-angle formula parallax Earth diameter baseline: 37.90425134 arcsec Duffett-Smith method parallax: 37.90364337 arcsec Half-angle formula parallax 6,000 km baseline: 17.82852735 arcsec ================= 11/7/2005 8:30 UTC Opposition - Sun-Earth-Angle = 180 deg Light-speed distance: 3.909013 minutes Distance kilometers: 70313556.94 km Half-angle formula parallax Earth diameter baseline: 37.42051558 arcsec Duffett-Smith method parallax: 37.41991537 arcsec Half-angle formula parallax 6,000 km baseline: 17.60099888 arcsec To find the parallax for any Earth baseline, take the value for the "Half-angle formula parallax Earth diameter baseline" and multiply by the ratio of your baseline in kilometers to the Earth diameter of 12756.27 kilometers. - Canopus56 P.S. - Other minor variations include using the Earth's polar diameter (12713.5 km) or mean diameter (12745.59 km).