7. Project Goals
• Determine the masses of the Galilean moons
through astrometric observations.
• Kepler’s 3rd Law: simplified 2-body problem
• Knowing an object’s mass provides valuable
insight into many other properties (ρave, gs, etc.)
• Get as many measurements as possible to
precisely and accurately infer orbital period and
average orbital radius.
9. Tools: Pre-Observing
Sky & Telescope Online Observing Tool: shows which
moons will be visible at certain dates and times
10. Tools: Pre-Observing
HORIZON Interface: provided RA and Dec of
Jupiter, current distance between Jupiter and Earth,
actual values for calculated quantities (needed to
calculate error)
11. Observational Methods
• MDM Observatory: 1.3-meter +
4K Imager
• FOV: 0.31”/pixel, 21.3” span
• Took images with U, g, and Y
band filters ranging from 1- to 6-
second exposures.
• More frequent and numerous
observations allows us to better
infer p and a.
16. Periodic Regression Analysis
• Mathematica: r (“) = A sin(B t + C)
• r: the angular distance of the moon from Jupiter
• t: time (minutes)
• A: the max angular distance between Jupiter and
each moon (amplitude)
• B: used to find the orbital period (p = 2𝜋 / B).
• B and C are used in combination to find the time
of maximum elongation
17. Calculations
• r = dJupiter(A x π)/(3600 x 180)
• Since dJupiter varies slightly over the course of
our observations, we used B t + C = π/2 to
find t(dmax).
• Periodicity: 2π/B
20. Mass Discrepancy
• MJ >> m → our calculated orbital radii still
generated moon masses off by several orders
of magnitude
• Decided to use known values of orbital radii,
but our own calculated periods (which were
more accurate) to perform the calculations
21. Example: Europa
0.042% error
3.551181 days (accepted)
3.55 days (observed)
1.17% error
0.6709 x 106 m (mean accepted)
0.6632 x 106 m (observed)
1.898 x 1027 kg4.7998 x 1022 kg
22.
23.
24.
25.
26. Results
• Periods: all extremely accurate (within 0.05%).
• Orbital radii: Not precise enough to calculate masses.
Io Europa Ganymede Callisto
p (days) 1.77 3.55 6.10 16.69
Error in p (%) 0.002 0.042 0.008 0.006
a (106 km) 0.42113 0.66324 1.06920 1.87250
Error in a (%) 0.15 1.17 0.08 0.56
29. Sources of Error: Restrictions
• Very few observations
• short time span of observations
• imprecise periodic fit in Mathematica
30. Sources of Error: Random Error
• Jupiter’s saturation → imprecise zero point
• Clouds → improvised exposure times
?
31. Sources of Error: Systematic Error
• Instead of using barycenter, assumed a
center of mass at Jupiter’s center.
• Assumed circular orbits when they are
actually elliptical.
33. Summary
• Astrometry is important (gives us our bearings).
• Ellipticity, mass barycenter, orbit inclination, etc.
must be accounted for.
• Mass of orbiting bodies are often orders of
magnitude smaller than the bodies they orbit,
requiring very precise astrometry.
• Astrometry requires many precise observations over
long periods of time.
• Doppler Shift > Astrometry for mass calculation.
