1. Eduardo Manuel Alvarez Observatorio los Algarrobos, Salto, Uruguay
Diurnal Parallax Determinations of Asteroid Distances Using Only Backyard Observations from a Single Station
Eduardo Manuel Alvarez
Observatorio los Algarrobos, Salto, Uruguay
Most of you know me – I’m Bob Buchheim – I’d like to introduce to you Eduardo Alvarez. Eduardo did all of the hard work on this project, and invented the new ideas in it, but he asked me to do the presentation since English isn’t his first language.
Before describing our project, let me show you a picture of Eduardo’s observatory. It is fully equipped for photometry and astrometry; and he also has a program of public viewing and outreach activities.
In addition to optical astronomy, Eduardo has also made and experimented with a radio-telescope system.
Robert Buchheim
Altimira Observatory, Coto de Caza, CA USA
SAS XXXI Symposium – Big Bear Lake, CA, USA – May 23-24, 2012

2. Subject New method for measuring asteroid parallax from a single site
simple, accurate, self-contained
three mathematical insights
Technical base supporting the new method
Some practical examples
Our project related to the determination of asteroid distances, using a “self-contained” method in which all of necessary data is determined at a single observing location. It is “new” in that three insights that Eduardo developed greatly simplify the observations, and eliminate any need for an ephemeris or other “external” data.
I’ll describe how it works, and show some example results.

3. Single-site “Diurnal Parallax”
Earth’s rotation provides a “baseline” so that a single observer can measure the parallax.
Parallax angle
φ(t) = [ RAtopocentric – RAgeocentric ] cosδ
In case you’ve forgotten my talk from last year, here is how “diurnal parallax” works.
Suppose that you observe an asteroid early in the evening,
and then again later at night,
the rotation of the Earth carries you along a circular path, and provides a baseline with which to measure the parallax angle. the parallax angle is the difference between the topocentric RA position, and the geocentric RA of the asteroid.

4. Objectives & Challenges
Demonstrate measurement of distance to asteroid
single-site diurnal parallax
“Self-Contained” method
All necessary data observable from single site
no need for ephemeris or external data
Challenges:
Infer “geocentric” position from “topocentric” observations.
Asteroids move …
rapidly!
“back-out” secular motion, leaving only parallax motion
Accuracy vs. Simplification:
measurements & calculations
Our goal was to demonstrate the measurement of the distance to an asteroid, from a single observatory, in a way that does not require an accurate ephemeris or any other “external” data about the asteroid.
This is an astrometric project, but it presents some special challenges:
First, since the parallax angle is the difference between the observed RA position and the geocentric RA position of the asteroid, we need to find a way to determine the geocentric position, using only measurements made from the surface of the Earth. It’s difficult to “observe” the geocentric position, partly because of the difficulty of travelling to the center of the Earth, and partly because the solid Earth isn’t transparent.
Second, asteroids move pretty rapidly across the sky (the net combination of Earth and asteroid orbital motion), so we need to find a way to “back out” the secular motion, leaving only the effect of parallax.
Third, we wanted to see how far we could simplify the observations and calculations without hurting the acuracy of resulting distance estimates.

5. Geometry of Model x y z α δ R
R∙cosδ
Z= R∙sinδ D
observer ν μ
Geocentric RA = Topocentric RA when target is at transit
use sequential transits to measure geocentric RA rate (ν)
+ Assume:
Geocentric distance ≈ constant
geocentric Dec rate ≈ topocentric Dec rate
Secular Geocentric rates are (approx.) constant
The math is described in some detail in our paper. This figure illustrates the idea: the asteroid is moving relative to the center of the Earth (along a path that is not known a-priori, and which we’ll need to determine with our observations), and the Observatory is moving relative to the center of the Earth, carried along by the Earth’s rotation. The Geocentric vector (from the center of the Earth to the asteroid) is shown in blue, and the topocentric vector (from the Observer to the asteroid) is shown in Red. Both of these vectors are functions of time.
The diurnal parallax is the difference between geocentric and topocentric RA. We can measure topocentric RA, but how do we know the geocentric RA? That is the first insight – when the asteroid is at transit, geocentric RA equals topocentric RA So, once per night we can accurately measure the geocentric position of the target. (We know the time of transit by comparing RA to LST).
Some additional assumptions are inevitably required. The assumptions in our model are:
the geocentric distance doesn’t change (over the interval of a few nights when measurements are made)
the geocentric rates in RA and Dec don’t change over the few nights when measurements are made.

6. Estimate Geocentric RA rate from Consecutive Transit Observations
RAgeo rate estimated by
RA(t2) – RA(t1)
ν ≈
t2 – t1
night #2
“RA rate ν”
(measured)
night #1
RA position
If you measure the target RA at transit on two consecutive nights, you can estimate the geocentric RA rate. Assuming a “constant geocentric rate in RA” is a reasonable first approximation. It turns out to be quite accurate when the asteroid is at opposition. You subtract out this secular motion, and what remains is the parallax motion, from which you calculate the distance.
For example, here are the results from observations of 8106 Carpino on two consecutive nights, right at opposition …
time

7. Example: 8106 Carpino Found parallax = 7.52 arc-sec
Found distance = AU
True distance = AU
error = 2.5%
Pretty good!
parallax, arc-sec
... Its diurnal parallax was 7.5 arc-sec. Our distance determination was only 2.5% different from the known value. Because it was at opposition, the geocentric rate was indeed almost exactly constant over these two nights, and the assumption of “linear motion in RA” was essentially correct. So using this assumption to “back out” the secular motion, leaving only the parallax motion, worked nicely. But the farther away from opposition, the less correct this assumption is…
elapsed time, hr

8. Estimating Geocentric RA rate Using Consecutive Transit Observations … but not at opposition
night #2
night #1
RA position
“true” geocentric RA motion (not measurable)
As you get farther from opposition, the RA rate changes noticeably from night to night. The nature of the problem is illustrated here. Suppose that the “true” motion of the asteroid – geocentric RA versus time -- over a couple of nights is shown as the yellow line. Obviously we can’t actually measure this curve.
time

9. Constant-RA-rate linear approximation of “true” RA(t) curve
night #2 culmination
“RA rate ν”
(measured)
night #1 culmination
RA position
“true” geocentric RA motion (not measurable)
if we estimate the geocentric RA rate by drawing a line between the two “transit” data points, we get the dashed line.
Look closely at what goes on during a single night …
time

10. Compensating errors On night #1, ν over-estimates the RA rate …
night #2 culmination
night #1 culmination
RA position
Considering the first night, the short red line is the “true” geocentric motion of the asteroid, and the dotted line is our estimated geocentric motion. On the first night, the dotted line over-estimates the rate …
time

11. Compensating errors On night #1, ν over-estimates the RA rate …
On night #2, ν under-estimates the RA rate
night #2 culmination
night #1 culmination
RA position
… and on night #2, the dotted line under-estimates the “true” geocentric RA rate.
Of course, if the true RA-vs-time line curved downward instead of upward, then the situation would be converse, but it would still be the case that our dotted-line would be a mis-estimate for both nights, and the error would be in the opposite direction for night 1 vs night 2.
time

12. Compensating errors Wonderfully,
Using both nights, the errors in calculated parallax / distance tend to cancel out, resulting in a very accurate (average) distance estimate.
night #2 culmination
night #1 culmination
RA position
Eduardo recognized a subtle feature of this situation. We can use the dotted-line RA rate to get two estimates of distance (one from each night), and these two will be in error, but with opposite sense – one will over-estimate the distance and the other will under-estimate the distance. The average of the two estimates will be very close to correct, because the errors are always of opposite sign, and cancel out almost exactly. This is mathematical insight number 2.
It further turns out that we don’t need to follow the asteroid all night. If the observations are taken on the same side of transit – on both nights – then the errors induced by the difference between “true” vs. “estimated” geocentric rate still cancel out almost exactly. That is, if you make measurements before transit on both nights, the calculations will work out better than if you followed the asteroid all night (from before transit to after transit) on a single night.
To show just how will this works, here are some examples:
time

13. Distant target: 414 Liriope
Found parallax = arc-sec
Found distance = 2.67 AU
True distance = AU
Error = 5.1%
parallax, arc-sec
414 Liriope was a distant main belt asteroid, with parallax of less than 3 arc-sec, and (importantly) it was observed well after opposition. By making observations only “before transit” on both nights the average found distance was within 5% of the “true” value.
elapsed time, hr

14. Fast-mover, Near-Earth Asteroid (162421) 2000 ET70
Found parallax = 145 arc-sec
Found distance = AU
True distance = AU
Error = 6.1%
parallax, arc-sec
Near-earth asteroid (162421) 2000 ET70 presented a difficult target, moving rapidly in both RA and distance. Measurements were made on two consecutive nights, before transit on both nights, at oopposition. The resulting distance estimate was accurate to 6%.
It is significant that the calculated distance on night 1 was 10% “low”, and on night 2 was 20%“high” compared to the ephemeris; but by averaging the two nights’ results the average was only 6% different from the ephemeris.
elapsed time, hr

15. How many data points do you really need?
Each night, many data points to fit sine-curve
Two nights, two short sets of observations each night:
early in night
at transit
“Four-Point Shortcut”
The standard routine in using diurnal parallax is to get many data points in the course of the night – maybe as many as one measurement every few minutes – to map out the whole parallax curve. A least-squares curve-fitting routine then determines the best-fit parallax. Having many data points is good because one or two discrepant measurements are overwhelmed by the large number of reliable measurements … but it also means that a large investment of measuring- and calculating-effort is needed.
Eduardo wondered if we could get away with simplifying the project, but (a) taking only a few images early in the night, and a few images near transit, and then (b) doing the same on the subsequent night. He named this the “four-point shortcut”, since it uses only four sets of observations (each “observation” being a few images).

16. How many data points do you really need?
Each night, many data points to fit sine-curve
Two nights, two short sets of observations each night:
early in night
at transit
With good astrometric images, “four-point shortcut” gives excellent distance estimates
“Four-Point Shortcut”
It turned out that with “backyard-level” astrometric accuracy, this four-point shortcut was surprisingly accurate: not quite as good as using a dense array of measurements, but good enough to determine main-belt asteroid distances to ±5%. That is “mathematical insight #3” – most of the information in the “sine curve” of parallax angle is contained in the data points measured (1) far from transit and (2) close to transit. The other intermediate data points aren’t really all that important.

17. Conclusions: Measurement of Asteroid Parallax
A feasible “backyard” project
Modest equipment, standard software
Replicate historically-important observations and calculations:
observe diurnal parallax effect
measure distance to solar-system object
determine scale of Solar System (“Astronomical Unit”)
Mathematical insights simplify:
Observations early in evening, and at transit
then go to bed …
Two consecutive nights provide sufficient data
In conclusion, we have shown that measurement of asteroid parallax is a quite feasible project, with modest equipment and standard software.
It enables a student to replicate a historically-important observation – the existence of diurnal parallax, the measurement of the distance to an asteroid, and hence the scale of the solar system.
Eduardo’s insights simplify the observations, by enabling the observer to take data during only half of the night, and gathering all of the necessary information on two consecutive nights.
All in all, a successful project!

18. Questions?
Any Questions?
Thank you for your attention!