Well placed on the meridian at around 9:00 pm, EDT, tonight, April 17th (01:00 UTC, 4/18) is the stunning spectacle of the waxing gibbous moon and mighty Jupiter within 1/2 degree of each other.
Having just passed Regulus, the heart of Leo, the lion, the duo is within 5 degrees of Denobola, the lion’s tail. For best effect, it is best seen with a pair of binoculars or a wide field, low-power eyepiece-telescope combination. To the left in this low-power, wide-field view of Jupiter and its 4 Galilean moons is the NE limb of the moon. Note that all non-stellar objects are located near to or aligned with the ecliptic, included in this view for reference. Pay particular attention to the alignment of the ecliptic with the orbital plane of Jupiter’s moons.
If you do own a telescope, choose a higher magnification and zoom in on Jupiter; even a 50x magnification will clearly show the moons’ tiny disks while you also observe the NE limb of the moon at the same time, thus providing quite the foreshortened depth of field, a field depth of 1500-Jupiter is 1500x the distance to the moon! Through a tremendous leap of insight, Galileo suggests Jupiter, and its family of satellites that appear to transit to and fro, is a model of the larger solar system. Through your binoculars or low-power telescopic view, you will see the the 4 Galilean satellites, named in honor of their discoverer.
Not only can we get a sense of discovery, seeing for the first time ourselves what Galileo saw in an epic leap of insight, something that would prove to be a key discovery, a tremendous empirical observation in support of the Heliocentric model of the solar system, we also gain a deeper insight into the simple but elegant relationship between any object in orbit around another. Kepler’s three laws of orbital motion teach us that all orbits are elliptical, that equal areas are traced out in equal times and that an object’s period varies as the 3/2 power of the semi-major axis or the distance between them. This elegantly simple but powerful relationship is universally true and is a consequence of the law of gravity and the laws of motion, both mathematically formalized by Newton in his “Philosophiæ Naturalis Principia Mathematica”, published in 1687, 57 years after Kepler’s death in 1630.
A timing of Jupiter’s Galilean satellites reveals a particular resonance between their periods and their distance from Jupiter, a resonance that is in perfect accord with Kepler’s Third law. The inner three moons, Io, Europa and Ganymede have a 4:2:1 orbital resonance. This means that for every four times that Io orbits Jupiter, Europa orbits twice and Ganymede orbits once. Callisto, the most distant from Jupiter, is expected to join in several hundred million years establishing a 8:4:2:1 orbital resonance.
Given that Io’s mean distance from Jupiter is 422,000 kilometers, we can compute Europa’s distance from Jupiter using Kepler’s third law. We know that Europa’s orbital period is twice that of Io. Using Kepler’s Third law, we substitute 2 for the distance in the following equation to find Europa’s distance relative to Io: P = A^(3/2) – for relative periods where A is the semi-major axis (distance) and P is the period. Since we know Europa’s orbital period is twice that of Io’s, we use the same equation to find the distance: A = P^(2/3) or 2 to the (2/3) power: 2^(2/3) = 1.58. The distance from Jupiter to Europa = 1.58x the distance to Io or 670,000 km. For a skilled observer, you can time the transits of these moons with a telescope and use Kepler’s laws to discover these simple but elegant relationships on you own! There is nothing like the thrill of discovery while, at the same time, acquiring a deeper insight into the workings of nature. Nature does not give up her secrets easily but with a telescope and a little patience we can unravel them one at a time. Whether you enjoy the simple beauty of Kepler’s laws or the visceral beauty of the night sky – or both – let tonight’s waxing gibbous moon be your guide.
Note: with the exception of the above animation, all images were produced with Stellarium.
Imagination is more important than knowledge
An index of all articles in this blog can be found here.