Sometimes the rings of Saturn are wide and sometimes thin. Are they really getting thinner?
At another viewing angle, the same flat circle changes from a wide ellipse to a thin line.
Planets and Small Solar System Objects
When we look at Saturn from the Earth, the angle between our line of sight and the ring plane changes. The ring itself is not shrinking because the photo is narrower; separating the direction of the Earth and the direction of the sun can explain which side is seen.
Guess first
The rings of Saturn appear thinner in the photo, does that mean the real rings suddenly narrowed?
Show answer and explanation
Not necessarily, the viewing direction will also change the projection — Planar rings have different short and long axis ratios when viewed at different inclination angles.
Then connect experience and science
First make sure it is the projection of a plane ring
It looks like a circle when viewed from the front, but looks like an ellipse when viewed diagonally; it looks very narrow when viewed from the side.
Short to long axis ratio = |sin B|
Where do the Earth and the Sun stand?
B gives the direction of the earth, and B′ gives the direction of the sun. They are not always of the same sign, and the side we see cannot always be regarded as the fully illuminated side.
Find B and B′ respectively
Distance also controls the corner size of the image
Even if the ring opening angle is the same, the total angular size of the rings in the sky will be different as Saturn changes its distance from the Earth.
a = 375.35″ / Δ
Finally, turn the entire image to the direction of the sky
P gives the tilt of the circumpolar pole relative to celestial north. The ring-to-shaft ratio and rotation angle in the figure are both calculated at the same time.
Date position + light travel time + light travel time + nutation
Expand the calculation
aarc seconds = 375.35/ΔAU barc seconds = aarc seconds × |sin B| 短长轴比 = b/a = |sin B|
a and b are the angular diameters of the major axis and minor axis of the outer ring. B is the latitude of the earth's direction relative to the torus, and the sun's direction is calculated separately with B'; the above is the projection and scale chain, and the direction B itself also depends on the position algorithm of this lesson.
give it a try
Compare the 1992 calculation example in the original book with the 2000 date. The ratio of the minor axis to the major axis comes from observing the opening angle, rather than manually flattening the pattern.
No need to memorize symbols
- B / B′
- Earth/Sun latitude above the ring plane
- a / b
- Angular diameter of major axis/minor axis of the outer edge of the ring projection, in arcseconds
- P
- Saturn's North Pole Position Angle
- ΔU
- The longitude difference between the direction of the Earth and the Sun on the ring plane
Algorithm, accuracy and scope of application
Implement Chapter 45: ring plane inclination and intersection, multiple light time iterations, date FK5 Earth/Saturn ecliptic position, B/B′, outer projection axis a=375.35″/Δ and b=a|sinB|, ΔU, and meridional light travel The difference/nutation corrected P. The position is from the original book truncation VSOP87, the FK5 longitude/latitude correction is in the same order as AA+; the tolerance is consistent with the solar mean ecliptic angle length difference. The original book 45.a is 1992-12-16. UT, add 59 seconds to TD. The ring width, slits, and Saturn color are for reference, and the outer edge axis ratio and direction are driven by real calculations; the ring particles, shadow details, or all disappearing moments are not directly determined according to the formula. When B and B′ have different signs, the dark surface situation can be pointed out, and false accurate luminosity is not given.
my understanding check
What do B and B′ describe respectively?
Earth direction and sun direction — Both are referenced to the ring plane, but from different viewpoints.
What happens to the projected short axis when B is close to 0°?
close to 0 — b=a|sinB|。
Are the ring slits and ring particles drawn here a complete physical simulation?
no — The calculations are based on the overall observation geometry, and the width and appearance of the annulus are only indicative.
The width of the ring you see depends on the viewing angle and distance; taking into account which side the sun shines in can explain the light and dark.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 45 chapter
Planets and Small Solar System Objects · All interactive courses