1 / 35

Measuring the Solar System

Measuring the Solar System. What we want to determine the distance to the various planets, asteroids, and comets from the Sun Basic method: Take Earth-Sun distance = 1 unit of measure (the AU) Determine the distances to other planets in AU Measure the Earth-Sun distance directly:

bethany-duffy
Download Presentation

Measuring the Solar System

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Measuring the Solar System What we want to determine the distance to the various planets, asteroids, and comets from the Sun Basic method: Take Earth-Sun distance = 1 unit of measure (the AU) Determine the distances to other planets in AU Measure the Earth-Sun distance directly: During transits of inferior planets across Sun’s disk By radar (see later)

  2. Some terminology Superior planet - a planet with an orbit greater than Earth’s (Mars  Neptune) Inferior planet - a planet with an orbit smaller than Earth’s (Mercury and Venus) Conjunction - planet is directly lined up with the Sun as seen from the Earth Opposition - Sun and planet in line with Earth, but in opposite directions (180o apart) on the sky (as seen from Earth)

  3. Planet Sun Elongation Earth More Terminology Elongation: The angular separation of a planet from the Sun (as seen from the Earth)

  4. Inferior planet: 0o elongation  greatest elongation (GE) Mercury: GE = 23o Venus: GE = 46o Superior planet: 0o elongation  180o Measured quantities Conjunction Opposition Conjunction 180o Earth Opposition

  5. The orbit of an inferior planet Method of Copernicus Theory: At greatest elongation the observers line-of-sight is tangential to the planet’s orbit • Deduction: can determine the size of the orbit • relative to Earth’s (i.e., in AU) by measuring • the greatest elongation I wrote that theory in 1543…URW Angle measurement

  6. RE = radius of Earth’s orbit = 1 AURP = radius of planets orbit Tangent condition: Right-angle LOS Planet RP RE Earth Sun Greatest elongation (from observations)

  7. Method: Sin(greatest elongation) = RP / RE or, RP = RE x Sin(greatest elongation) Units: recall, RE = 1 AU OMG. It works “The answer lies in trigonometry”, sang out Holmes. “Gads… old boy…you’re right” replied Watson GE Measured

  8. Not bad for a beginner No need to copy table measured

  9. The first two of Kepler’s laws 1st law: The planets revolve around the Sun along elliptical orbits with the Sun at one focus WTF • 2nd law: • A line drawn from the planet to the Sun sweeps out equal areas in equal time

  10. Law: the first - the picture Planets have elliptical orbits, with the Sun at one focus Planetary orbit - exaggerated center Aphelion perihelion Sun “empty” focus

  11. The ellipse - the picture Two focal points Definition: Eccentricity (e) e = OF1 / a P r1 r2 O F2 F1 Distance OF1 = OF2 r1 + r2 = 2 a Semi-major axis (a) Major axis

  12. Eccentricty (e): Describes the ‘shape’ of the ellipse For a ‘closed’ orbit 0  e < 1 A circle = an ellipse of zero eccentricity Planets: Earth: e = 0.0167 Moon: e = 0.055 Mars: e = 0.094 Mercury: e = 0.206 – most eccentric planetary orbit Venus: e = 0.007 – most circular planetary orbit Halleys comet: e = 0.967 – ‘cigar’ shaped orbit No need to know numbers

  13. Comparison of orbits a = 17.94 AU e = 0.966 a = 3.124 AU e = 0.5 e = 0.05 a = 1700 AU e = 0.9998 e = 0.009 e = 0.252

  14. Law: the second - the picture The planet-Sun line sweeps out equal areas in equal time Time T D C B E Time T Time T F G A

  15. Kepler’s 2nd law says in pseudo-computer code: if area AFB = area CFD = area EFG then time (A to B) = time (C to D) = time (E to G) D C B E Time T Time T F G A

  16. So, K2 reveals…. Near perihelion - closest point to Sun large distance to travel in time T hence high velocity Near aphelion - greatest distance from Sun small distance to travel in time T hence low velocity The velocity of a planet (comet, asteroid…) varies according to its position in the orbit and distance from Sun

  17. Fast Slow aphelion perihelion The ellipse divided into 14 equal areas (blue and white triangles) and 14 equal travel times (red dot) along the orbital path ---- orbital period = T/14

  18. 1/4 area of ellipse as measured from the Sun Where is a planet after ¼ of its orbital period following perihelion ? Planet 1/4 of way around orbital path if velocity was constant Planet at 1/4 of orbital period P Aphelion Perihelion Sun Area of entire eclipse = one orbital period = perihelion to perihelion travel time Area of half ellipse = half orbital period = perihelion to aphelion travel time Area (Sun, P, Perihelion) = Area(Sun, P, Aphelion) = 1/4 area of ellipse

  19. “Eight of thirteen” Discovered in 1618 and published in Harmonice Mundi (Harmony of the World) as number 8 in a list of 13 points necessary for the contemplation of celestial harmonies Kepler argued: “if the Sun has the ‘power’ to govern the motion of the planets, then there must be some relationship (law) between planetary periods and distance” 7 of 9 - of course Tertiary Adjunct of Unimatrix 01 Do your assignments resistance is futile

  20. The third law Say what? The square of a planets orbital period (P) is proportional to the cube of its orbital semi-major axis (a) P2 = K a3  OOTETK • where, P = orbital period of planet • a = semi-major axis of planets orbit • K = a constant

  21. Slope = K P2 Pluto straight line Mercury a3 OOTETK A clever dodge: if P(years) and a(AU) then K = 1 and P2(yr) = a3(AU)

  22. Example: The first asteroid (now dwarf planet), Ceres, was discovered on January 1st, 1801. Observations reveal that it has an orbital radius (semi-major axis) a = 2.77 AU So Hence

  23. Periodic comet Encke has an orbital period of 3.3 years (the shortest of any known active comet) From K3 Hence

  24. The orbit of Halley’s Comet Given this information Orbital period = 76 yrs Aphelion distance = 35.3 AU

  25. Questions: How close does HC approach the Sun? What is the orbital eccentricity? From Kepler’s 3rd law: P2(yr) = a3(AU) So, a3(AU) = 76 x 76 = 5776 Hence a(AU) = (5776)1/3 = 17.94 By definition: perihelion distance + aphelion distance = 2a So, we have: Perihelion distance = 2a – aphelion distance = 2x17.94 – 35.3 Which gives Perihelion distance = 0.58 (AU) Closer than orbit of Venus to the Sun

  26. OF Aphelion (35.3 AU) perihelion a Sun 0.58 AU Definition: Eccentricity (e) e = OF / a Perihelion distance a = 17.94 AU By construction: a = perihelion distance + OF Hence: OF = a – perihelion distance = 17.94 – 0.58 = 17.26 AU And, accordingly, eccentricity e = 17.26 / 17.94 = 0.96

  27. OF Aphelion perihelion a Sun Last seen in 1986 – back in 2061 Perihelion distance Summing up – For Halley’s comet Orbital period = 76 years Semi-major axis a = 17.94 AU Perihelion distance = 0.58 AU Aphelion distance = 35.3 AU Sun displacement from center OF = 17.26 AU eccentricity e = 0.96

  28. Newton’s genius Same “physics’ everywhere in the lab, in the Solar System and anywhere else in the Universe – in other words we can measure and understand the physics of the cosmos around us He argued: The rules describing the acceleration of objects falling on the Earth can also describe the motion of the planets • Hypothesis: Newton, 1687: • There is a gravitational attraction between all of the planets and the Sun

  29. Keeping the planets in their place Newton’s 1st law of motion A body will remain at rest or in constant motion along a straight line path unless acted upon by an external force In reality, a planet is continuously accelerated towards the Sun by a gravitational force It is this continuous gravitational interaction that causes a planet to follow an elliptical orbit rather than a straight line path through space

  30. In each second the Moon ‘falls’ 1.4 mm towards the Earth (away from straight line path) and moves 1 km around its orbit – the Moon is continuously falling towards Earth, but still stays in orbit Fg Path of Moon with gravity (orbit) Moon Path of Moon without gravity Earth See ASTRO page 50

  31. Kepler’s 3rd law…. Newton style Cutting to the chase - Newton showed: P2/a3 = K = 4p2 / G(MSun + MPlanet) In other words, Newton found that the constant K in K3 is related to the system mass Units are now kilograms, meters and seconds (The SI units of measure) G is the universal gravitational constant

  32. Phobos Period = 7.656 hours Orbital radius = 9400 km Moon diameter is about 20 km Discovered by Asaph Hall in 1877

  33. Weighing Mars MMars + Mphobos = (4p2) a3 / G P2 Can safely assume MMars >> MPhobos so, using SI units (meters, sec., kg) MMars = (4p2) (9.4x106)3 / G (7.656 x 3600)2 = 6.6 x 1023 kg = 1/10th MEarth

  34. Not just for planets Provided a measure of the size of the orbit (a) and the orbital period (P) can be made K3 as formulated by Newton can be used to find the masses of astronomical objects….. Later on we will weigh the Milky Way Galaxy using K3

  35. We now have a set of tools and laws to describe: Position on the sky – the celestial sphere (ecliptic) The distances to the planets and the scale of the Solar System – Copernicus’s method for inferior planets Orbital shape – semi-major axis and eccentricity Planetary motion – Kepler’s three laws The mass of a planet – if it has a Moon – Newton’s refinement to K3 Our next task is to take an inventory of the Solar System – what exactly is it and what kind of objects does it display ?

More Related