Double Stars to Follow

(Adapted from a four-part series by Martin Gaskell)


Double stars. Every amateur astronomer has heard of them. Every star atlas lists them. We’ve probably all glanced at a few of them. We know, in theory at least, that they orbit each other, but what about really following them as they orbit each other with a telescope? When I started out in astronomy at Junior High School age I felt that actually following a pair orbiting was something totally out of my reach. I thought I needed something costly called a filar micrometer, costing $1000, and a big refractor on a solid German equatorial mount with an accurate clock drive (costing many thousands of dollars). When I was a freshman in college I did get access to such a setup, and making a few measurements satisfied me for a while.

More recently I’ve also learned that you don’t need a $1000 filar micrometer to measure double stars, and you can use a cheap home-made equatorial. I’m now measuring the separations and position angles of double stars to an accuracy several times greater than the resolution of the Hubble Space Telescope just using a piece of cardboard and a home-made paper degree scale on “Tel’Poke” our home-made 6″ f/8 Newtonian! Ask me if you’d like to know how to do this (the original article listed Martin’s phone number here, but it has been removed for this web page version — we hope to get Martin to write an article on how to build a “cardboard micrometer” in the future). A great thing about observing double stars is that light pollution is almost never a problem.

There was another problem though that I had as a beginning amateur in addition to lack of equipment, and this was knowing which stars were interesting to follow. I don’t think this is explained in any book, so I’m going to tell you about the best double star to follow in the late summer: 70 Ophiuchi.

70 Ophiuchi is the (eastern) left-most star of a triangle of 4th magnitude stars a few degrees to the east (left) of Beta and Gamma Ophiuchi below Hercules. It is shown on all star charts. At the beginning of August it is on the meridian at midnight (CDT).

70 Ophiuchi is one of the most interesting binaries in the sky for an amateur sized telescope for several reasons: both components are fairly bright, it has a large separation and it appears to move rapidly. All this is because 70 Ophiuchi is only 16.7 light years away from us. The closest the stars ever appear to be is 1.53″ (which happened in June 1989) and the widest they get is 6.75″ (which will occur in 2024). The A component is magnitude 4.3 and the B component is magnitude 6.0. The system has been followed since the first observations by William Herschel in 1779, so the stars have been followed for over two orbits.

The true orbit has a periastron distance (distance of closest approach) of 11.7 au (astronomical units; 1 au = the mean Earth – Sun distance) and an apastron distance (when they are furthest apart) of 35.0 au. The orbital period is 88.30 years with an uncertainty of about 6 months. To give a sense of the scale of the system, this corresponds to the B star orbiting the A star in an orbit which in our solar system would go from just outside the orbit of Saturn to between the orbits of Neptune and Pluto. This is an interesting fact to bear in mind when looking at the system, and a good thing to share with any friends looking at 70 Ophiuchi with you. 70 Ophiuchi A is a little less than half the luminosity of the Sun and 70 Ophiuchi B is less than 1/10 of the luminosity of the Sun. Both stars are redder and less massive than our Sun.

Stars orbit most quickly when they are near periastron and we are now (1994) closer to periastron than apastron, so the stars are still moving fairly quickly. The orbit is also inclined to our line of sight by 59.2 degrees and this tilt has been making the angular speed of the star appear faster the last few years. The results of this are that if you are careful you can detect the orbital motion of the system even during a single observing season!

As seen in an inverting equatorial telescope (no star diagonal), B is almost directly above A right now, almost “one o’clock” (PA 180 is straight up). If you don’t have a drive, or if you turn your drive off, you will see that B is approximately perpendicular to the direction the stars are drifting. The system is rotating clockwise (PA decreasing). By next (1995) spring the B star will be noticeably further to the right of vertical. In a year it is currently moving about the amount the hour hand on a clock moves in a quarter hour. This should be obvious even without making any sort of measurement of the angle. Just make your “first epoch” observations with a careful drawing in your observing book and wait about a year and repeat your observations and make a new sketch. You will have seen a double star move in its orbit! Of course, if you can make an actual measurement of the angle then the orbital motion will be even more obvious.

To get the best view of close double stars, leave your telescope outside for a couple of hours before you observe (reduce local seeing), make sure it is VERY well collimated, and use a VERY high magnification. With our 6″ Newtonian I like to use 500x. With a 10″ I like to use 700x. At these high powers an equatorial mount and drive are a big help.

Here are two sets of predictions of the separation and position angle. They are based on two different calculations of the orbit. The second one is the most recent but a comparison gives you an idea of the uncertainty and also why double star measurements are still needed. The differences between the two predictions are several times the accuracy which can be achieved with a 10″ telescope and a piece of cardboard, so low budget amateur observations could be useful here (I usually get results to a fraction of a degree in PA and to a few hundredths of an arc second in separation).

Prediction #1 Prediction #2
Date Year PA Sep. PA Sep.
August 01, 1994 1994.58 168.6 2.44″ 173.2 2.25″
September 01, 1994 1994.67 168.1 2.46″ 172.6 2.27″
October 01, 1994 1994.75 167.6 2.48″ 171.9 2.30″
June 01, 1995 1995.42 163.7 2.66″ 167.5 2.47″
July 01, 1995 1995.50 163.3 2.69″ 167.0 2.49″
August 01, 1995 1995.58 162.9 2.71″ 166.5 2.51″
September 01, 1995 1995.67 162.4 2.73″ 166.0 2.54″
October 01, 1995 1995.75 162.0 2.75″ 165.5 2.56″

70 Ophiuchi has a long history of not conforming to the expected orbit. The orbit appears to have changed since 1878. This is a big mystery. Were the observations before 1878 simply systematically in error? Or was there an encounter with an intruder to the system, an interstellar Nemesis star? Only time and more observations will tell.


This is the second of what I hope will be a short series on the best double stars to follow. By this I mean the ones that are “actually doing something.” In the first article (page 4 of the July issue of “The Prairie Astronomer”) I described how I learned that you don’t need a filar micrometer and a big refractor on a massive German equatorial mounting to follow double stars. My mention that you could do it with a piece of cardboard on a 6″ Newtonian generated quite a bit of interest among members, and David Knisley has asked me to do a program on it at the April meeting (Tuesday April 25). I’ll give a full description of my methods there with a detailed description of my results, an audience demonstration, and the first annual (?!) return visit of “Tel’Poke” – the Gaskell family’s ultra-low-cost, home-made, 6″ equatorial Newtonian with a high precision clock drive (this year’s upgrades to Tel’Poke are almost a program in themselves!).

This month I want to talk about Castor (= Alpha Geminorum), perhaps the best-known double in the Winter/Spring sky, and according to some people the best double in the northern sky. The high declination of Castor makes it measurable all spring (it can still be done in late May.) Next month (if I get time) I want to write about Gamma Virginis, a star undergoing very exciting changes. Since Castor is a star you’ll probably look at often and show to friends I’ll describe the system and its history a bit.

Castor occupies a famous place in the history of astronomy — it was the first system beyond the solar system in which gravity was shown to be operating. We might take this for granted today, but in 1803 when Sir William Herschel announced that Castor A and Castor B were orbiting each other, this generated an excitement which was to last for decades.

Castor was probably first resolved way back in 1678 by G. D. Cassini but the duplicity was not rediscovered until 1718 by James Bradley (the Astronomer Royal in England) and James Pound. Bradley and Pound made a fairly good estimates of of the position angle (PA = angle of the secondary relative to the secondary measured counter-clockwise from north) but unfortunately did not measure the separation (double star micrometer work didn’t really begin in earnest for another 100 years). This is a pity as a good measurement by them (accurate to 0.25″, say) would have helped eliminate much of the uncertainty in the orbit. The English minister, Rev. John Michell proposed (on statistical grounds which would seem overwhelming convincing today) that Castor, and close binaries like it, were true gravitationally bound systems, but the theory was not accepted until Sir William Herschel’s famous announcement in 1803 that Castor A and Castor B were orbiting each other.

To complete the story of the Castor system, a third star, Castor C, a 9.1 magnitude star 72.5″ away in PA 164, visible in any small telescope, has also been shown to be a distant member of the system. It is probably orbiting A and B with a period of 10,000 years. Each of A, B and C has been shown to be a spectroscopic binary (periods of 9.2, 2.9 and 0.8 days respectively) so there are six stars total in the system. Castor C is interesting because the orbit of the components is edge on causing eclipses of amplitude about 0.5 magnitudes (easily detectable visually). Castor C is given the variable star designation YY Gem.

The most recently available photographic parallax measurements give the distance to the system as 45 light years with an uncertainty of about 5 light years. The separation between the A components is 1/25th of an AU. The B components are about 90 AU away from the A components on average, or three times further away than Pluto is from the sun. The B components have a separation of 1/30th of an AU. The C components are more than 1000 AU away (300 times further apart than the sun and Pluto are) but are only 1/60th of an AU apart. Robert Burnham Jr. in his famous “Celestial Handbook” gives an interesting scale model of the Castor system. If A1 and A2 are 2.3 inches apart then B1 and B2 are 340 feet away and have a separation of 1.7 inches. C1 and C2 are 4500 feet away and have a separation of 0.9 inches. It would be fascinating to be on a planet in such a system! If the 45 light year distance is correct, the A and B stars are less massive than stellar evolution theory would predict. The A stars, for example, each seem to be similar to Sirius in spectral type and luminosity, but their masses come out to be about the same as the mass of the sun (Sirius is about twice as massive as the sun).

When Pound and Bradley discovered the duplicity of Castor, the companion was almost due north of the primary. When the famous Wilhelm Struve began micrometric measurements in 1826 the companion was almost due west of the primary. In my lifetime it has gone from being due south of the primary in 1954/55 to due east in 1982. Thus the system has been seen to rotate three-quarters of the way round.

You might think that there is nothing more to be learned about such a famous and well-studied system as Castor, especially not from visual measurements, but this is not true. The orbit of A and B is uncertain by an amount which is detectable using my cardboard micrometer. The main reason is that A and B have not yet been observed to go through a full orbit. As we saw in the previous article on 70 Ophiuchi, even when a system has been observed to go through several orbits there can be still be significant discrepancies when one tries to predict future behavior. We will see this to be the case again with Gamma Virginis. For Castor even the orbital period of A around B is quite uncertain. It has been calculated many times. In 1933 it was calculated to be 340 years, in 1940 to be 380 years, in 1956 to be 511 years, in 1958 to be 420 years and most recently (in 1988) to be 467 years. This is a 50% uncertainty! The semi-major axis of the orbit (the average AB distance) is also uncertain with values ranging from 6.3″ to 7.4″. This would be much less uncertain if the early 18th century observers had measured the apparent separation. It is agreed that the orbit of A and B is tipped towards us, making an angle of about 65 degrees with the plane of the sky (90 degrees would be seeing the orbit edge-on). A and B appeared to be at their closest in 1969/70, but because of the tilt of the orbit we are not sure when the real closest approach of the stars (“periastron”) took place. Orbital calculations place it between 1950 and the late 1960’s.

What do these uncertainties mean for the amateur observer in the mid- to late-1990’s? In the table below I have calculated the predicted positions from three ephemerides due to P. Muller in 1956, W. Rabe in 1958 and W. D. Heintz in 1988 (the later is the one used in Norton’s 2000.0). It will be easy to distinguish between the predictions using a cardboard micrometer. Castor is one of the best stars in the sky to measure in this way. It is so bright and so far apart that you can do it under seeing conditions which would make measuring other stars impossible. If you have a Celestron Microguide eyepiece with a degree scale this should also let you distinguish between the PAs. You should be able to see a change between now and 1996 if you are careful. If you don’t go in for quantitative measurements you will have to wait longer, but the change in PA will be obvious even from eyeball estimates over a 5 to 10 year period. This should also be true for the separation as well. Have a look at Castor this month, and even if you can’t make an accurate measurement, try to estimate the PA as accurately as you can. Write it down in your observing book for your future enjoyment! I note with pleasure that the estimated PA I scribbled down in the margin of my “Norton’s” in 1972 is now off by a full 100 degrees!

PREDICTED SEPARATIONS AND POSITION ANGLES OF CASTOR AB Date Muller 1956 Rabe 1958 Heintz 1958 PA Sep. PA Sep. PA Sep. Mar 1, 1995 = 1995.08 72.1 3.54" 66.6 3.44" 70.5 3.45" Jun 1, 1995 = 1995.42 71.8 3.57" 66.2 3.46" 70.1 3.48" Oct 1, 1995 = 1995.75 71.4 3.60" 65.8 3.49" 69.7 3.51" Mar 1, 1996 = 1996.08 71.1 3.63" 65.4 3.52" 69.4 3.54"



This series is about double stars “that are actually doing something” and Gamma Virginis, is certainly a star starting to do something. The two components of Gamma Virginis are rushing towards periastron (closest approach) in only ten years time and this provides a rare opportunity to see rapid orbital motion in even the smallest of telescopes.

Gamma Virginis is the third magnitude star about 15 degrees to the “upper right” of Spica in the direction of Leo. For the deep sky folks, if you can find the Virgo cluster of galaxies, Gamma is the third magnitude star directly south of the cluster! The system is quite nearby, only 32 light years away. The two stars are almost identical F-type main sequence stars, a little hotter than our own sun and each about 3.5 times brighter. The identical magnitudes makes resolution of the system easier and this will be a plus when they start to get very close in a few years time. They orbit each other in a very eccentric, comet-like orbit that takes them as far apart as 70 astronomical units (about twice the distance from the earth to Neptune or Pluto) and as close as 3 au (closer than we get to Jupiter).

Gamma Virginis was one of the first star ever discovered to be double. A missionary in India, named Richaud, discovered the duplicity in 1689. It was rediscovered by Bradley, Pound and Cassini in 1718-1720. The first useful measurement (of the position angle only) was made by Sir William Herschel in the fall of 1781. In the 1820’s and early 1830’s its motion was so rapid and obvious that in 1833 his son, Sir John Herschel, calculated its orbit. This was only the second time that the orbit of a double star had been calculated. John Herschel predicted periastron in the spring of 1836. At apastron (maximum separation) the stars are 6 arc seconds apart (this last happened in the 1920’s) but the orbit is very eccentric so that at closest approach the stars are only half an arc second apart. In the 1830’s this generated considerable excitement. Here was a star which, in only a decade, went from being a fairly easily resolvable close pair to being unresolvable in every telescope in the world except for the great Dorpat refractor used by the famous Otto Struve! The verification of John Herschel’s reduction helped fuel the double star fever of the latter 19th century.

It is now 1995 and you get a chance to relive the excitement of the 1830’s! According to the most recent (1990) orbital calculation of Dr. W. D. Heintz, Gamma Virginis has a 168.68 year period and the next periastron will occur less than ten years away, in late February 2005. Between now and then Gamma Virginis is closing rapidly. The very eccentric orbit makes Gamma Virginis different from most other double stars. For the majority of double stars the main change is in the position angle. For Gamma Virginis, now only will the position angle (PA) be changing rapidly (about 3 degrees per year right now; increasing to 70 degrees per year in 2004/2005) but the separation can be seen to be changing rapidly at a rate of about 0.14 arc seconds per year (easily detectable with a cardboard micrometer) and this rate will increase slightly towards periastron. In the table below I give the predicted separation and position angles for the next few years. The cumulative change in separation between now and 2005 is so large that it will be obvious in a small telescope without making any quantitative measurements. Just note the position angle in your note book relative to the east-west drift in the telescope, estimate the separation relative to the resolution of your telescope and write down your impressions in your observing book so that you can refer back to them in future years. When 2005 comes, unless you have a very big telescope (more than 10″ aperture), you won’t be able to resolve it at all! Then even a very big telescope will only resolve it in extremely good seeing.

As with the stars I mentioned in the first two articles, there is some uncertainty in the orbit of Gamma Virginis. This is because it is only now completing its first well-studied orbit. If you use the older ephemeris given in Norton’s 2000.0 (a period of 171 years) you will get positions which are off by quite a lot — over 2 degrees in PA and over a quarter of an arc second in separation. The differences between the old and the new Heintz ephemeris have been easily detectable with my cardboard micrometer. How good is the Heintz ephemeris itself? I gave Gamma Virginis particularly intensive study last spring (1994) with our 6″ Newtonian (“Tel’Poke”) and came up with a PA in agreement with the 1990 ephemeris but separation about 0.05″ too large. This is about the same as my calibration uncertainties, but 5 years of observations by more experienced amateur observers with larger telescopes gave a mean separation 0.08″ larger than the ephemeris and speckle observations with the 26″ at the US Naval Observatory in 1994 gave a separation 0.04″ wider than the ephemeris so it looks like the Heintz ephemeris is under predicting the separation slightly. Any discrepancies will be larger as periastron approaches so I’m going to be keeping a close eye on Gamma Virginis with my cardboard micrometer. From these numbers you can see, that at least for a bright pair like Gamma Virginis, the $0.25 cardboard micrometer on Tel’Poke compares quite favorably with a $100,000 piece of equipment on a 26″ refractor at the US Naval Observatory! Come the spring of 2000 however I am going to need to have built a slightly larger telescope!

Gamma Virginis

Date Year PA Sep. PA(obsd. by MG) Sep. (obsd. by MG)
June 21, 1994 1994.48 278.1 2.29″ 277.4 +/-0.6 2.34″ +/-0.03
May 01, 1995 1995.33 276.1 2.17″
May 01, 1996 1996.33 273.4 2.03″
May 01, 1997 1997.33 270.5 1.89″
May 01, 1998 1998.33 267.0 1.73″



For those who already have a larger telescope I must briefly mention Xi Ursa Majoris, the most southerly obvious naked eye star of Ursa Major (it’s just above the east end of Leo). This system is only 26 light years from the sun. The stars orbit each other with a period of 60 years so they’ve been seen to go through three complete orbits since they were discovered by William Herschel in 1780. Xi Ursa Majoris is famous for being the first star to have its orbit determined, by M. Savary in 1828, 5 years before John Herschel calculated the orbit of Gamma Virginis. The stars went through a closest approach in the sky of 0.85″ in 1992 and will be at periastron this (1995) year (the difference is due to perspective caused by the tilt of the orbit as seen from the Earth). Between now and the year 2000 the system will widen from 1.10″ to 1.77″. At present it can just be resolved by a 6″ but an 8″ or 10″ would be better. It is certainly possible to measure it with a cardboard micrometer on a 10″ — I measured it in 1992 when the separation was 0.89″. If you can’t measure it, this is a system to just watch as the stars swing round each other. The change in PA is rapid — well over a degree per month. Since I measured it in 1992 the PA has changed from 13 degrees to 312 degrees this (1995) spring, a change of 61 degrees! This is the sort of change you can see in the sketches in your observing notebook.

Here is a star which you might be able to follow through an entire orbit if you live long enough! Below is a brief ephemeris due to W. D. Heintz. It was calculated 30 years ago, so I would not be surprised if there are deviations from it at a level which could be detected by a cardboard micrometer and a 10″ – 12″ telescope.

Xi Ursa Majoris

Date Year A Sep.
January 01, 1995 1995.0 316.9 1.10″
January 01, 1996 1996.0 303.6 1.29″
January 01, 1997 1997.0 293.7 1.47″
January 01, 1998 1998.0 285.8 1.61″
January 01, 1999 1999.0 278.9 1.71″


In the earlier articles in this series I have concentrated on double stars that you can watch “doing something” over a few years even with a small telescope. These were stars that were easy for a 6″ Newtonian and should be detectable in a 4″ or less. In this article (and maybe in a sequel or two), I want to focus on some stars that are more of a challenge, but some of which have another property: you might live through an entire orbit!

If you are only starting now, how short an orbit has to be for you to live through depends on how old you are! Actually the number of bright stars that can be resolved in a moderate size amateur telescope and which have short orbits is not very large. One of the laws of orbital motion is that the shorter period orbits are smaller (Keppler’s third law), so the shorter the period the harder the stars are going to be to separate. According to a list of bright doubles on “The Constellations Web Page” by Richard Dibon-Smith, the shortest period stars include the following:

NAME Period Separation (years) (2000.0) Burnham 395 25 0.5" 85 Pegasi 26.3 0.7" Beta Delphini 26.6 0.5" Zeta Herculis 34.4 0.7" Eta Cor Bor 41.6 0.7" Kruger 60 44.6 3.0" (mags 9.8/11.4) Xi Scorpii 45.7 0.4" Sirius 50.1 4.6" (mags -1.6/8.4) Zeta AB Cancri 59.7 0.8" Xi Ursa Majoris 59.8 1.8" Gamma 2 And 61.1 0.4"

Of these only Xi Ursa Majoris can be considered easy for a small telescope, and don’t count on following it through a whole orbit, unless you’re under 20! With two exceptions, all the others have separations of less than an arcsecond most of the time. The exceptions are Sirius, where the companion gets lost in the glare and Kruger 60, a very nearby, wide binary, but one with very faint components. Let’s revisit Xi Ursa Majoris and then talk about the two high up in the late spring sky: Eta Coronae Borealis and Zeta Herculis.


Back in Part III, I described Xi UMa as being for those who had larger telescopes (meaning larger than a 6″). When I measured it in 1992, the separation was 0.89″. Back then the position angle was changing at over a degree per month. However, the stars were also appearing to fly apart by almost 0.2 arcseconds per year, so, in just a couple of years since 1995, the system has become much more accessible with smaller telescopes. Actually, I was too pessimistic back in 1995; I successfully measured it back then with our 6″ (at almost 1.2 arcseconds separation). Between my first measurement of Xi UMa in 1992 and 1998 I will have seen the pair rotate through almost 90 degrees. Here is an updated ephemeris for the next few years.

Epoch PA Separation 1997.42 290.1 1.53" (June 1) 1998.42 282.7 1.66" 1999.42 276.2 1.74" 2000.42 270.1 1.79" 2001.42 264.3 1.80" 2002.42 258.5 1.80"


Zeta Boo is not a very short period star (P=123.4 years), and it is not changing much in separation or PA this decade, but it’s a good “warm-up” for the next two stars, it’s close – about 0.85″ in the spring of 1997 – but the stars are reasonably bright and almost exactly equal in brightness (magnitude 4.6 each). Try Zeta Boo after Xi UMa to see how much of a challenge these stars are going to be. If you can’t resolve Zeta Boo, then the next couple of short period ones won’t be possible.

Although not a star of very short period, Zeta Boo is nonetheless quite interesting: the orbit is extremely eccentric (e=0.957) and in the year 2021 the pair closes to 0.03″. At the moment the stars are just past their furthest apart, and moving relatively slowly, but after the turn of the millennium, they will close rapidly. Take a look at Zeta Boo each spring in good seeing to see if you can still resolve it.

Epoch PA Separation 1997.42 300.7 0.85" (June 1) 2000.42 299.6 0.79"


Now let’s try one of the shorter period ones (P=41.6 years), but one that is a shade harder than the last. The combined magnitude of this pair is only 5.02, so it barely makes naked eye visibility in the city, but it’s easy to find in the northern crown. Like all the binaries on my lists that “do something”, it’s relatively nearby in galactic terms: about 50 light years. At its widest, in 1993, the stars were only 1.0″ apart, but with individual magnitudes of 5.7 and 6.0 the stars are very similar and this helps resolve them. In our new 8″ (Tel’Poke II) they are easy in good seeing, Timothy, Daniel and I are building a new ultra-high resolution micrometer, just for measuring these very close pairs, and a recent trial run with a prototype of the new micrometer suggested that Eta CrB was not hard to measure, so long as the seeing was good. I did make some measurements in 1995 of Zeta Bootis, a binary of similar separation, with my old micrometer on Tel’Poke I (6″ aperture) and, even though the old micrometer was not designed for really close pairs, I got an answer very close to the predicted position. Here is the ephemeris for the next few years. You can see a drawing of the orbit in Burnhams. Burnham comments that the pair can be followed throughout the entire orbit with a good 6″.

Epoch PA Separation 1997.42 51.8 0.85" (June 1) 1998.42 56.2 0.80" 1999.42 61.3 0.75" 2000.42 67.1 0.69" 2001.42 73.9 0.64"


Now we come to another challenge. The problem with Zeta Her (P=34 years) is the big magnitude difference, 3.1 and 5.6. That’s a factor of ten in brightness. I first measured Zeta Her back in 1992 using a 10″ Newtonian. It was tricky to measure then at about 1.57″ because of the magnitude difference. The pair is now starting to sweep round rapidly towards periastron in February 2002. It is sweeping out about a degree per month for the next few years and it is currently closing very rapidly at a rate of about 0.2″ per year. In June 2000 the separation will be only 0.63″, in 2002 it will be 0.49″. This will be very hard to resolve, given the magnitude difference, but if you’re middle-aged and you want to follow a star around it’s entire orbit, this is one of the ones to try!

Although I measure separations and position angles with my micrometers, you can have the experience of watching Zeta Her and other pairs go round, just by making an accurate drawing in your note book.

Epoch PA Separation 1997.42 43.6 1.20" (June 1) 1998.42 34.4 1.04" 1999.42 21.3 0.84" 2000.42 359.3 0.63" 2001.42 320.2 0.49" 2002.42 275.8 0.56"