A modified form of the ROEMER method is proposed
- using modern Orbital Information:
by Geoff Hitchcox, Christchurch, New Zealand, January 2002.
http://www.geocities.com/kiwi_36_nz/
This experiment uses a "modified" form of the classic Roemer
method to show the main sources of error and their magnitude in
the Roemer "Speed of Light" experiment.
It then compares this to a worked example of the classic Roemer
method. Conclusions may help others to improve accuracy when IO
eclipses are used to determine the "Speed of Light".
The modified and classic Roemer methods used in this experiment
use the same eclipse data. These are predictions by Jay Lieske
instead of using "observed" eclipses. The assumption being that
the Lieske predictions are more accurate than an "amateur" based
observation run. The intent being to find the weakness in the
Roemer method by a more controlled method than raw observations.
My modified Roemer method takes account of:
- Actual rotation rate of IO.
- Jupiter and IO eccentricity.
-------------------------------------------------------------------------
1. Reappearances at 2001 February 4.664819 (2001/02/04 15:57:20.36)
2001 March 12.065098 (2001/03/12 01:33:44.47)
=========
Interval = 35.400279 days (Lieske Prediction)
20 orbits of IO (see NOTE 1) = 35.38170255
Diff = 0.01857645 days = 1605.00528 secs
Jupiter Shadow change (see NOTE 2) of 3.1031 degrees = 1307.0004 secs
Residual = 298.00488 secs
2. Disappearances at 2001 September 17.349674 2001/09/17 08:23:31.83
2001 October 22.745097 2001/10/22 17:52:56.38
=========
Interval = 35.395413 days (Lieske Prediction)
20 orbits of IO (see NOTE 1) = 35.38371759 days
Diff = 0.01169541 days = 1010.483424 secs
Jupiter Shadow change (see NOTE 2) of 3.0109 degrees = 1289.000736 secs
Residual = -278.517312 secs
3. The difference in Residuals = 576.522 seconds
4. From JPL Ephemeris, distance between IO and geocentre:
2001-Feb-04 15:57 4.7063831667 AU
2001-Mar-12 01:34 5.2717261371 AU
Diff = 0.5653429704 AU
2001-Sep-17 08:24 5.3537521448 AU
2001-Oct-22 17:53 4.8171980071 AU
Diff = -0.5365541377 AU
5. Earth has moved 1.1018971081 AU distance w.r.t. IO in experiment.
6. Thus light appears to have taken 576.522 seconds to travel 1.1018971081 AU.
7. 576.522 / 1.1018971081 = 523.21 seconds to travel 1 AU
8. 1 AU = 1.496^8 km, so speed of light = 1.496^8 / 523.21 = 285,927 km/s
This value is within 4.6 % of the accepted value of "c".
This is 6 times more accurate than the "classic" Roemer method
that gives a value within 31 % the value of "c" (see NOTE 3).
This also shows that the "main" errors to take account of is
firstly the issue of IO orbital rate and secondly the eccentric
orbit of the Jupiter shadow (a tenth the effect of the IO
orbital error).
Note: The above method allows using each pair of eclipses to
determine "c". This gives the following values:
Feb - Mar = 283,805 km/sec = within 5.3 % of "c"
Sep - Oct = 288,199 km/sec = within 4.0 % of "c"
Showing that the residual "errors" are of the same magnitude!
In the above experiment, both IO orbital rate, and Jupiter
eccentricity are both calculated. However if just the IO orbital
rate is used (lots of maths saved) then the "speed of light" is:
(1605 - 1010.5) / 1.1018971081 = 539.524 secs / AU
"c" = 277,281 km/s
which is within 7.6 % of the correct value, showing that it is
indeed the orbital rate of IO that is the main error in the
Roemer method. Doing all the extra maths for the eccentricity
effect, only increases the accuracy by 3 % to 4.6 %
CONCLUSIONS:
If one attempts to emulate the ROEMER experiment to find "c",
the accuracy can be greatly improved by allowing for the
"actual" IO and Jupiter orbits. If one can time the eclipses to
a few seconds accuracy, this is still less than the other
"inherent" errors in the method!
The assumption that IO orbits Jupiter with isochronism (equal
time) is the main source of error in the Roemer method.
Although the above experiment used "predicted" values, the same
methodology can be used on "measured" data to increase the
accuracy over the "classic" Roemer method.
Only since June 1999 has the "public" had access to the JPL
ephemeris that give precise IO data with respect to Jupiter,
allowing calculation of the "actual" IO orbital rate for any
interval.
SUMMARY
Assuming one obtains accurate eclipse timings (having overcome
the fact the eclipse takes place over many minutes) then post
processing of the data (as outlined by this paper) allows a many
fold improvement in the accuracy of the determination of "c".
Jovian eclipse timings have been used by Dr Jay Lieske to
improve the currewnt ephemerides at JPL. These can now be used
by the public to obtain data (like the orbital rate of IO) to an
accuracy not available before. This paper shows the means and
method to obtain that data and hence improve the final figure of
"c".
The method discussed here also shows the main error in the
Roemer method is the lack of Isochronism of the orbit of IO
around Jupiter leading to the largest error term in the final
calulation.
HISTORICITY: (Yes, it's a word)
The value quoted for Olaus Roemer in 1676 is 214,000 km/s! This
seems a very curious figure, it is within 29% of the true value
of "c". Considering that an accurate figure for the AU was not
known for another 100 years after Roemer, where does this number
214,000 come from? Perhaps someone using Roemers method with
modern figures, or was it originally given in AU units (without
knowing what AU units equated to in Km)?
****************************************************************
NOTE 1:
To find the Actual time the 20 orbits took of IO around Jupiter.
Taking the first eclipse (in each pair) as a "reference", use
the JPL Ephemeris to find the time where the RA (Apparent Right
Ascension of IO with respect to Jupiter's true-equator and the
meridian containing the Earth equinox of J2000.0) is the same at
second eclipse. This can be found in 2 steps using iteration, a
very simple task once setup. The accuracy of this can be to 0.5
seconds of time. See NOTE 4 for the macro I used to obtain the
ephemeris data from JPL.
Date__(UT)__HR:MN:SC.fff Date________JDUT R.A. (degrees)
2001-Feb-04 15:57:20.400 2451945.16481944 91.5045061
2001-Mar-12 01:06:59.500 2451980.54652199 91.5049499
Diff = 35.38170255 days
2001-Sep-17 08:23:31.800 2452169.84967361 92.8908578
2001-Oct-22 17:36:05.000 2452205.23339120 92.8907551
Diff = 35.38371759 days
First Diff = 35.38170255
Second Diff = 35.38371759
Diff = 0.00201504 (days) = 174.1 seconds
The above IO orbital variation represents the greatest "error"
in the Roemer method (assuming timing is within a few seconds).
The above also shows that the IO orbit cannot be calculated with
great accuracy using an "average" orbital period. For greatest
accuracy need to use the JPL ephemeris which takes into account
the major IO perturbations.
This 174.1 second difference is 57ppm (parts per million) over
35.38 days, making IO not a very good "clock" in the short term!
****************************************************************
NOTE 2:
Account for the Jupiter (reference) shadow change, due to
Jupiters orbit around the sun during the experiment:
Eccentricity
------------
Jupiter = 0.04839266
Io = 0.0041
Using JPL Ephemeris to give the heliocentric ecliptic longitude
of Jupiter:
Date__(UT)__HR:MN hEcl-Lon hEcl-Lat
****************************************
2001-Sep-17 08:23 91.6435 -0.2011
2001-Oct-22 17:52 94.6544 -0.1331
diff = 3.0109 degrees
Now use the JPL ephemeris to give the time required for IO to
move in its orbit to cover this angle:
2001-Oct-22 17:36:05.000 2452205.23339120 92.8907551
plus 3.0109
= 95.9016551
2001-Oct-22 17:57:34.000 2452205.24831019 95.9017593
Diff = 1289.0 secs
Mar-Feb 3.1031 degrees 1307.0 (secs for IO to move shadow)
Sep-Oct 3.0109 degrees 1289.0 "
N.B. For the two sample periods, the angle moved by Jupiter
around the Sun is not quite equal, and so it follows the time
for IO to cover the extra shadow is not quite the same:
The eccentricity of Jupiter and IO give an 18 second difference
in the time for IO to cover the shadow change of Jupiter, this
is about 1 tenth the error of the IO orbital rate issue ( see
Note 1).
I have made the "assumption" that Jupiters shadow rotates the
same angle as does the sun around Jupiter. This is of course not
exact because the shadow is subtended from the limb not the
centre of Jupiter (that IO orbits). This geometric error and the
shape of the shadow cone (and IO's orbit through it), and
Jupiters local weather (remember it is a GAS planet!) are
probably the remaining sources of error. But they are smaller
than the IO rotation error and the eccentricity of Jupiter!
****************************************************************
NOTE 3:
The following is Roemers classic method.
It results in a figure 31 % higher than the accepted value of "c".
Brian Loader of New Zealand calculated and commented the following:
-------------------------------------------------------------------
1. Reappearances at 2001 February 4.664819 and
2001 March 12.065098
=========
Interval 35.400279 days
2. Disappearances at 2001 September 17.349674
2001 October 22.745097
=========
Interval 35.395413 days
As expected the second is less than the first - the Earth was moving away
from Jupiter early in the year and towards Jupiter later.
3. The difference in time is 0.004866 days = 420.422 seconds
4. From the Astronomical Almanac, the distances of Jupiter are
February 4.664819 ... 4.7037196 AU
March 12.065098 ... 5.2690456 AU
=========
Increase 0.5653260 AU
September 17.349674 ... 5.3511576 AU
October 22.745097 ... 4.8146081 AU
=========
Decrease 0.5365495 AU
5. Add these two together 1.1018755 AU
6. Thus light appears to have taken 420.422 seconds to travel 1.1018755 AU.
7. 420.422/1.1018755 = 381.552 seconds to travel 1 AU
(Roemer could have got this far, except he would have had to observe and
time the eclipses, and he would not have had the distances in AU to such an
accuracy!)
8. 1 AU = 1.496^8 km, so speed of light = 1.496^8/ 381.552 = 3.92^5 km/s
Although the above does not use the orbital period of Jupiter, implicit in
it is the assumption that the orbital period is not changing - in fact it
was. Also it assumes the eclipses are constant length, in fact at present
they are getting slightly longer as the path of Io through the shadow of
Jupiter gets to be more central. The effect is to make the interval between
one disappearance and the next slightly low.
--------End of the worked example by Brian Loader--------
Thanks to Brian for the above "classic" Roemer example and comments.
*******************************************************
NOTE 4:
Ephemeris used:
Solar System Dynamics Group,
Horizons On-Line Ephemeris System
4800 Oak Grove Drive, Jet Propulsion Laboratory
Pasadena, CA 91109 USA
http://ssd.jpl.nasa.gov/
The following is the MACRO I used to obtain the orbital info for
IO from JPL.
I used the macro to firstly find the time to the nearest minute,
and then knowing that used the macro to find the nearest 0.5
second, so that 2 uses of the macro would give the required
equivelent RA time.
You may wish to use it, so here are some short form instructions:
Copy and Paste the following lines starting with:
!$$SOF
and ending with:
!$$EOF
And send as an email to:
horizons@ssd.jpl.nasa.gov
With the SUBJECT line containing the word: JOB
Send the following exactly as it is to make sure you have
everything correct.
If you make a mistake, the HORIZON system does not tell you what
you did wrong - it simply does not send any answer back :-(
When I used it, JPL provided the answer emails within a matter of
seconds - amazing service from Pasadena to New Zealand!
To make use of the following for your own experiments, you will
need to change the "start" and "stop" times, and change the
"step size" accordingly. Everything else can remain the same!
*******************************************************
!$$SOF (ssd) JPL/Horizons Execution Control VARLIST
! send to: horizons@ssd.jpl.nasa.gov
! with SUBJECT line containing: JOB
! Macro to give the RA of IO with respect to Jupiter.
! Used to help find Rotation Rate of IO to nearest half second
! and the angle the "Jupiter Shadow" has moved between eclipses.
! Macro by Geoff Hitchcox, Christchurch, New Zealand. JAN 2002
EMAIL_ADDR = ''
COMMAND = '501'
OBJ_DATA = 'NO'
MAKE_EPHEM = 'YES'
TABLE_TYPE = 'OBS'
CENTER = '500@599'
START_TIME = '2001-MAR-12 01:06:00.0 UT'
STOP_TIME = '2001-MAR-12 01:07:00.0'
! N.B. Step size cannot result in time steps LESS than 0.5 sec in time
STEP_SIZE = '120'
QUANTITIES = '2'
REF_SYSTEM = 'J2000'
OUT_UNITS = 'KM-S'
VECT_TABLE = '3'
VECT_CORR = 'NONE'
CAL_FORMAT = 'BOTH'
ANG_FORMAT = 'DEG'
APPARENT = 'AIRLESS'
TIME_DIGITS = 'FRACSEC'
RANGE_UNITS = 'AU'
SUPPRESS_RANGE_RATE= 'NO'
SKIP_DAYLT = 'NO'
VEC_LABELS = 'NO'
ELM_LABELS = 'YES'
R_T_S_ONLY = 'NO'
TIME_ZONE = '+00:00'
EXTRA_PREC = 'YES'
!$$EOF
*******************************************************