# Ay 1 Lecture 2. Starting the Exploration

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1 Ay 1 Lecture 2 Starting the Exploration

2 2.1 Distances and Scales

3 Some Commonly Used Units Distance: Astronomical unit: the distance from the Earth to the Sun, 1 au = cm ~ cm Light year: c 1 yr, 1 ly = cm ~ cm Parsec: the distance from which 1 au subtends an angle of 1 arcsec, 1 pc = cm ~ cm 1 pc = 3.26 ly ~ 3 ly 1 pc = 206,264.8 au ~ au Mass and Luminosity: Solar mass: 1 M = g ~ g Solar luminosity: 1 L = erg/s ~ erg/s

4 The Scale of the Solar System Major planets: Up to ~ 50 au Not to scale! The Oort cloud: ~ 1000 au

5 Stellar Distances Nearest stars ~ a few pc Naked eye visible stars ~ up to a kpc Globular clusters ~ few kpc

6 Distances in the Galaxy Milky Way diameter ~ kpc

7 Our Extragalactic Neighborhood Magellanic Clouds ~ 50 kpc Andromeda galaxy (M31) ~ 700 kpc Virgo cluster ~ 16 Mpc

8 The Deep Universe: ~ 1 10 Gpc

9 Distances and Parallaxes Distances are necessary in order to convert apparent, measured quantities into absolute, physical ones (e.g., luminosity, size, mass ) Stellar parallax is the only direct way of measuring distances in astronomy! Nearly everything else provides relative distances and requires a basic calibration Small-angle formula applies: D [pc] = 1 / π [arcsec] Limited by the available astrometric accuracy (~ 1 mas, i.e., D < 1 kpc or so, now) π

10 How Far Can We Measure Parallaxes? Since nearest stars are > 1 pc away, and ground-based telescopes have a seeing-limited resolution of ~1 arcsec, measuring parallaxes is hard. 1838: Bessel measured π = arcsec for star 61 Cyg (modern value π = 0.29 arcsec) Current ground-based: best errors of ~ arcsec

11 How Far Can We Measure Parallaxes? Hipparcos satellite: measured ~10 5 bright stars with errors also of ~0.001 arcsec GAIA satellite: will measure positions of ~10 9 stars with an accuracy of micro-arcsecs - this is a reasonable fraction of all the stars in the Milky Way! Currently: measure D accurately to ~ a few 100 pc

12 A parsec is A. Radius of the Earth s orbit B. About cm C. Angle corresponding to the size of the Earth s orbit from 1 light year away D. About cm E. About 200,000 astronomical units

13 Distances to stars in our Galaxy range A. From ~ to ~ 50 kpc B. From ~ cm to ~ cm C. From ~ 1 to ~ 700 kpc D. From ~ 1,000 to ~ 50,000 astronomical units

14 2.2 Kepler s Laws, Newton s Laws, and Dynamics of the Solar System Kepler s nested Platonic solids

15 Kepler s Laws: 1. The orbits of planets are elliptical, with the Sun at a focus 2. Radius vectors of planets sweep out equal areas per unit time 3. Squares of orbital periods are proportional to cubes of semimajor axes: P 2 [yr] = a pl 3 [au] Derived empirically from Tycho de Brahe s data Explained by the Newton s theory of gravity

16 1. Inertia 2. Force: F = m a 3. F action = F reaction Newton s Laws e.g., for a circular motion in grav. field: centifugal force = centripetal force The law of gravity: } Conservation laws (E, p, L) m V 2 m M = G R R 2 Energy: E total = E kinetic + E potential m V 2 2 G m M R Angular momentum: L = m V R (gravitational) (point mass)

17 Motions in a Gravitational Field Motions of two particles interacting according to the inverse square law are conic sections: Marginally bound: E kin = E pot Unbound: E kin > E pot Bound: E kin < E pot Kepler s 1 st law is a direct consequence

18 Why Ellipses? A rigorous derivation (in polar coordinates) is a bit tedious, but we can have a simple intuitive hint: Decompose the total velocity v into the radial (v r ) and tangential (v t ) components v = v t, v r = 0 Consider the total motion as a synchronous combination of a radial and circular harmonic oscillator (recall that the period does not depend on the amplitude) v r v t v v = v t, v r = 0

19 Orbit Sizes and Shapes For bound (elliptical) orbits, the size (semimajor axis) depends on the total energy: E kin = 0, R = 0 E kin = E pot, R m 1 m 2 E kin E pot m 1 m 2 v The shape (eccentricity) of the orbit depends on the angular momentum: L max > L 0 Circular orbit: maximum angular momentum for a given energy L max Radial orbit: zero angular momentum L = 0

20 Kepler s 2nd Law: A quick and simple derivation Angular momentum, at any time: L = M pl V r = const. Thus: V r = const. (this is also an adiabatic invariant ) Element of area swept: da = V r dt Sectorial velocity: da/dt = V r = const. Independent of M pl! It is a consequence of the conservation of angular momentum. Planets move slower at the aphelion and faster at the perihelion V r

21 Kepler s 3rd Law: A quick and simple derivation F cp = G M pl M / (a pl + a ) 2 G M pl M / a 2 pl (since M pl << M, a pl >> a ) F cf = M pl V 2 pl / a pl = 4 π 2 M pl a pl / P 2 (since V pl = 2 π a pl / P ) F cp = F cf 4 π 2 a 3 pl = G M P 2 (independent of M pl!) Another way: E kin = M pl V 2 pl / 2 = E pot G M pl M /a pl Substitute for V pl : 4 π 2 a 3 pl = G M P 2 It is a consequence of the conservation of energy

22 It Is Actually A Bit More Complex Kepler s laws are just an approximation: we are treating the whole system as a collection of isolated 2-body problems There are no analytical solutions for a general problem with > 2 bodies! But there is a good perturbation theory, which can produce very precise, but always approximate solutions Discovery of Neptune (1846) Comet impacts on Jupiter Relativistic effects can be used to test theory of relativity (e.g., precession of Mercury s orbit

23 It Is Actually A Bit More Complex Dynamical resonances can develop (rotation/revolution periods, asteroids; Kirkwood gaps; etc.) If you wait long enough, more complex dynamics can occur, including dynamical chaos (Is Solar System stable?)

24 Kepler s 3 rd law is A. Cubes of orbit sizes ~ squares of orbital periods B. Squares of orbit sizes ~ cubes of orbital periods C. A consequence of the conservation of energy D. A consequence of the conservation of angular momentum

25 The shape of a closed orbit depends on A. Total energy B. Total angular momentum C. Angular momentum for a given energy D. None of the above

26 2.3 Celestial Coordinate Systems, Time Systems, and Earth s Rotation

27 The Celestial Sphere Think of it as an outward projection of the terrestrial long-lat coordinate system onto the sky the Equatorial System

28 The Equatorial System The coordinates are Right Ascension (RA, or α) and Declination (Dec, or δ), equivalent to the georgaphical longitude and latitude RA = 0 defined by the Solar position at the Vernal Equinox

29 The Seasonal Change of the Solar Declination

30 Annual Solar Path

31 The Alt-Az Coordinate System It is obviously location-dependent Most telescopes nowadays are built with Alt-Az mounts

32 Other Common Cellestial Coordinate Systems Ecliptic: projection of the Earth s orbit plane defines the Ecliptic Equator. Sun defines the longitude = 0. Galactic: projection of the mean Galactic plane is close to the agreed-upon Galactic Equator; longitude = 0 close, but not quite at the Galactic center. (α,δ) (l,b)

33 Ecliptic (Blue) and Galactic Plane (Red)

34 Synodic and Sidereal Times Synodic = relative to the Sun Sidereal = relative to the stars As the Earth goes around the Sun, it makes an extra turn. Thus: Synodic/tropical year = (solar) days Sidereal year = sidereal days = solar days Universal time, UT = relative to the Sun, at Grenwich Local Sidereal Time (LST) = relative to the celestial sphere = RA now crossing the local meridian (to the South)

35 The Precession of the Equinoxes The Earth s rotation axis precesses with a period of ~ 26,000 yrs, caused by the tidal attraction of the Moon and Sun on the the Earth s equatorial bulge There is also nutation (wobbling of the Earth s rotation axis), with a period of ~ 19 yrs Coordinates are specified for a given equinox (e.g., B1950, J2000) and sometimes epoch

36 Earth s Orbit, Rotation, and the Ice Ages Milankovich Theory: cyclical variations in Earth-Sun geometry combine to produce variations in the amount of solar energy that reaches Earth, in particular the iceforming regions: 1. Changes in obliquity (rotation axis tilt) 2. Orbit eccentricity 3. Precession These variations correlate well with the ice ages!

37 The change of seasons is due to A. The tilt of the Earth s rotation axis relative to the celestial equator B. The tilt of the Earth s rotation axis relative to the plane of the ecliptic C. Eccentricity of the Earth s orbit D. Precession of the equinoxes E. Human sacrifices

38

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