Sorting out synodic and sidereal years for planets

1
Sorting out synodic and sidereal years for planets
sidereal period of Earth 1 y sidereal period of
planet 3 y
one year later Earth has completed 1
orbit Planet has completed 1/3 orbit
½ year still later Earth has completed 1 ½
orbits Planet has completed ½ orbit
2
Whats happening at 9 mo 12 mo?
Start

• sidereal year (period) of earth
• PE 12 mo 1 y
• sidereal year (period) of planet
• P 36 mo 3 y
• they start at the top, when planet and sun
  are in opposition
• 9 months later planet and sun are in conjunction
• 12 months later planet and sun are 120º apart in
  the sky and see caption at left

Start 12 mo
9 mo
9 mo
12 mo
3
After a total of S 18 mo 1.5 y planets
synodic year is complete, since planet and sun
are finally again in opposition
12 mo

• In general S synodic year P sidereal year
  PE 1 y
• Superior 1/P 1/PE 1/S
• Inferior 1/P 1/PE 1/S

12 mo
18 mo
Here 1/P 1 1/S 1/P 1 - 1/1.5 1/P 1 -
2/3 1/P 1/3 P 3 y
18 mo
4
The idea of greatest elongation inferior planets
only
5
Fig.04.11
Tycho Brahe looked but did not see parallax for
stars!!

• if Earth is NOT fixed at the center of the
  celestial sphere
• some stars are further away than others
• over one years time, nearby stars wiggle back
  and forth against the backdrop of the distant
  stars
• he failed to see it

6
What is a parsec?

• with baseline of 2 AU, angle would be 2 for d
  1 parsec
• Brahes equipment was good enough to observe
  parallax if stars were no more distant greater
  than d 7000 AU
• q D/d 2 AU/7000 AU .29 mrad
• .29 mrad(180º/p rad)(60 /º) 1 very
  closely
  
• 7000 AU ? .1 ly
• 1 parsec 206,000 AU
• 1 parsec 3.26 ly 206,000 AU (parallax angle
  2 )
• nearest star is 4 ly distant

1 AU
1
d 1 parsec
7
Johannes Keplers Laws of Planetary Orbits

• Kepler used Tycho Brahes incredibly good data
• There is no theory behind his laws they fit the
  data
• Planets orbit in ellipses, with sun at one focus
  
• -- planar, but not coplanar (or co-major-axial)
• -- circles are really too perfect non-eccentric
  ellipses
• Equal areas swept out in equal times
• -- slowest at aphelion, fastest at perihelion
• -- (period)2 (semi-major axis)3
• P2 a3 if P in years, a in AU

8
Drawing an ellipse Keplers 1st law
sun is at one focus, nothing is at the
other actually, at the focus is the
center-of-mass of the pair
9
Some ellipses, all with identical semi-major axis
abut different focal length f, (eccentricity e
f/a)
e 0 circle e 1 barely open orbit
f
a
10
Equal areas in equal times Keplers 2nd law
really a consequence of angular momentum
conservation
11
period P increases as semi-major axis a
increases, in a quantifiable way Keplers 3rd
law
if P is expressed in Earth years (y) and
semi-major axis a is expressed in astronomical
units (AU) then P2 a3 or P a3/2

• A AU P y v AU/y
• 1 6.3
• 2.9 4.4
• 5.2 3.6
• 8 3.1
• 11.2 2.8

• how is speed related to semi-major axis?
• v distance/time 2pa/P or P 2pa/v
• so 2pa/v a3/2 or v 2p/?a

12
The Analemma

• at center of time zone (latitude 0º, 15º, 30º .
  . )
• asymmetric shape has to do with eccentricity of
  Earths orbit (speed varies)
• Locate a given day on the figure 8. June 21 is
  at very top, and progress clockwise from there
  Dec 21 is at very bottom
• Horizontal time, in min, sun crosses your
  meridian (noon) before (to L early) or after
  (to R late) clock noon, on that day. Ranges
  from 16 min early, to 14 min late
• Vertical latitude for which sun strikes zenith
  at noon, on that day (i.e. suns declination)

13
23.5
0º
-23.5
-16 min (early) 0 min 14 min (late)