Lunar Parallax
On Friday, August 4, at 10 PM CDT (2006-08-05 3:00 UT), Rick, in Bowie, MD,
and I, in Roeland Park, KS, simultaneously took photographs of the Moon
expressly for the purpose of measuring the lunar parallax.
Except for the distances involved, this is no different than holding a
pencil vertically at arm's length and watching it appear to jump back and
forth relative to the far wall as you close one eye and then the other.
Because the distance between KS and MD is sufficiently large, the Moon
appeared to be in slightly different position, against the background
stars, in the photographs. This slight shift in the apparent position of
of a object is called parallax and can be used to compute the
distance to the object. Measuring the lunar parallax is a great
project for amateur astronomers.
I summarize my computations and results here. Rick's results can be
found on his
web site.
Creating the KS and MD photos
The associated animation shows these photographs taken in KS and MD.
In truth, a series of ISO 200 photos with exposure times from 8 s to
1/1000 s were taken from each location while tracking the Moon.
In post-processing, I could see that my 8 s photo over-exposed
the Moon but captured several brighter stars. Conversely, the 1/250 s
photo nicely captured the Moon but no stars. By comparing the 1/250 s
photo to the 1/125 s photo, the 1/125 s to the 1/60 s and
so on, I was able to align the 8 s and 1/250 s photos removing
any tracking errors. Next, I removed the background from the 8 s
photo, then masked and replaced the over-exposed image of the Moon with
that from the 1/250 s photo thereby creating a photo showing both the
Moon and the surrounding brighter stars. After Rick and I exchanged
photos, I performed exactly the same operations on his photos with the
addition step of adjusting the intensity slightly so make the appearance
of the photos as similar as possible. Lastly, using the brighter stars
common to both photographs, I scaled and aligned the (composite) Moon photo
taken in MD to the one taken in KS.
Measuring the lunar parallax from the KS and MD photos
There were five bright stars visible in both photographs. They are listed
here with J2000 coordinates from the Bright Star Catalog.
| |
|
RA |
Dec |
|
RA (hours) |
DEC (°) |
| 42 Oph (theta) |
|
17h 22.0m |
-25°00' |
|
17.366667 |
-25.000000 |
| 21 Sco (alpha) |
|
16h 29.4m |
-26°26' |
|
16.490000 |
-26.433333 |
| 22 Sco |
|
16h 30.2m |
-25°07' |
|
16.503333 |
-25.116667 |
| 23 Sco (tau) |
|
16h 35.9m |
-28°13' |
|
16.598333 |
-28.216667 |
| 26 Sco (epsilon) |
|
16h 50.2m |
-34°18' |
|
16.836667 |
-34.300000 |
Using the star coordinates plus measurements from the photos,
the scale of the photos was computed.
| star 1 – star 2 |
|
arcsec |
/ |
pixels |
= |
arcsec/pixel |
| 26 Sco – 23 Sco |
|
24503.03 |
/ |
776 |
= |
31.58 |
| 26 Sco – 42 Oph |
|
41672.90 |
/ |
1322 |
= |
31.52 |
| 42 Oph – 23 Sco |
|
38841.01 |
/ |
1233 |
= |
31.49 |
average scale = 31.53 arcsec/pixel
Next, I measured separations between a distinct feature on the Moon and
several stars. These were used to compute the apparent shift of the Moon
as seen from MD relative to KS.
| Moon – star |
|
KS photo |
|
MD photo |
|
KS – MD shift of Moon |
| |
|
sep (pixels) |
angle (°) |
|
sep (pixels) |
angle (°) |
|
pixels |
arcsec |
| S Mare Crisium – 23 Sco |
|
374.1 |
7.37 |
|
347.7 |
7.10 |
|
26.45 |
834.2 |
| S Mare Crisium – 21 Sco |
|
590.0 |
25.39 |
|
563.1 |
26.02 |
|
27.64 |
871.4 |
| S Mare Crisium – 22 Sco |
|
652.2 |
38.28 |
|
626.8 |
39.30 |
|
27.83 |
877.6 |
| Average KS – MD shift |
|
|
|
|
|
|
|
27.31 |
861.1 |
KS – MD parallax = 861 arcsec.
Computing the distance to the Moon using the lunar parallax
The lunar parallax is the angle formed between the line-of-sight from
MD to the Moon and from KS to the Moon. It was determined by
measuring the apparent shift of the Moon relative to the background
stars as seen from each site. This is shown in the simplified schematic.
To convert the lunar parallax to the distance to the Moon, only a few
other pieces of information are needed: the distance between the MD
and KS sites and the orientation of that baseline relative to the
line-of-sight to the Moon. For the highest accuracy, there are other
corrections that can be applied, but a quite good value can be found
from a few values and simple trigonometry.
| KS lat, lon, hgt |
|
39 : 02 : 09.3 |
|
-94 : 37 : 58.8 |
|
305 m |
|
from GPS |
| MD lat, lon, hgt |
|
39 : 00 : 38.4 |
|
-76 : 45 : 36.9 |
|
64 m |
|
from NGS PID JV5890 for Bowie, MD |
| |
|
|
|
|
|
|
| KS XYZ |
|
-400724 m |
|
-4944846 m |
|
3995604 m |
| MD XYZ |
|
1136573 m |
|
-4830727 m |
|
3993277 m |
| |
|
|
|
|
|
|
| KS - MD |
|
-1537297 m |
|
-114119 m |
|
2327 m |
KS – MD distance = 1541.528 km. This is the length of the yellow
KS – MD line in the schematic.
| KS – MD XYZ unit vector |
|
-0.997255 |
|
-0.074030 |
|
0.001510 |
| KS – MD NEU unit vector |
|
0.099581 |
|
-0.987703 |
|
-0.120526 |
| KS – MD az & el from MD (°) |
|
275.76 |
|
-6.97 |
|
|
| Moon az & el relative to baseline @ 3:00 UT (°) |
|
-69.6 |
|
23.5 |
More succinctly, the angle between the MD – KS line and the
MD – Moon line is 73.41°.
In turn, projected baseline length, KS – P, must be 1477.367 km.
With the projected baseline length plus the lunar parallax of 861 arcsec
in hand, my computed KS – Moon distance is 353,892 km.
The true distance, from the AA
website was 377355 km; therefore, my computed value is -6.8% in error.
Extending this just a little further, my computed Earth – Moon,
more formally call the Moon's geocentric distance, is 356,296 km which is
also 6.8% too small. Again turning to the photos, I measured the Moon's
diameter as 1907.6 arcsec which implies the Moon's diameter is 3,273 km, or
5.8% smaller than the true value of 3,476 km.
With a little more preparation, it should be possible for Rick and I to
improve upon these results and measure other necessary values, such as
the Moon's azimuth and elevation at the time of observation, and we plan
to do so in the future. Recalling that Rick used the same data that I
did but performed his processing and calculations completely independently,
if one averages his measured lunar parallax with mine, one gets gets a
value of 808 arcsec which implies a distance of 377,355 km to the Moon;
an error of only 0.7%.
Last modified: 2006-08-13