Stellar Parallax. Actual parallax was finally observed by Frederich Bessel in The target was the obscure star 61 Cygni.

Transcription

1 1 Stellar Parallax Actual parallax was finally observed by Frederich Bessel in The target was the obscure star 61 Cygni. 61 Cygni has a huge proper motion 5 arcseconds per year making it a likely candidate for being one of the nearest stars. The observed parallax was 0.3, so 61 Cygni was 3.3 parsecs (600,000 AU) away. d parsecs = 1 parallax angle (arcsec) Parallax angle is defined as ½ the annual angular displacement of the star. A 1 AU displacement of the Earth produces an angular shift of one arcsecond for a star at 1 parsec distance.

2 What's a Parsec? The parsec is a unit of measure defined by the Astronomical Unit (Earth s orbit) and an arcsecond. Consider a star's parallax motion observed as the Earth shifts in position by 1 AU (half of the maximum shift that can be observed across the diameter of the Earth's orbit) If that star is at a distance where the shift due to 1AU displacement of the Earth is one arcsecond then the star is, by definition, one parsec away. One parsec = 0665 AU there's that number again, the number of arcseconds in a radian... Some distances: It is 1.3 parsecs, more than 50,000 AU, to the nearest star system Alpha Centauri. The dozen or so nearest stars are a few parsecs away. The Orion nebula star forming region is 500 pc away. The center of the Galaxy is 8,500 pc (~30,000 light years) away. The Andromeda galaxy is 600,00 parsecs away. The Virgo Cluster is 10 million parsecs (10 megaparsecs) away. One parsec is 3.6 light years.

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5 5 Eppur Si Muove Coriolis Force Rotation of the Earth is evident from effects due to observation in a rotating frame of reference (although the effects are subtle). Gaspard-Gustav Coriolis first characterized this rotating frame effect in A resident of Earth near the equator is carried along at about 450 meters/s by the rotational motion of the Earth. Rockets launch to the East from equatorial latitudes (French Guyana / Florida) to get a boost from Earth rotation. Heading north (or south) from the equator an individual moves to a latitude where the rotation velocity of the Earth is cos(latitude) smaller but the individual retains their original velocity/momentum. They end up being ahead of where they thought they should go. They also conserve angular momentum as they are closer to the Earth s rotation axis. Those living in the rotating reference frame explain that the apparent acceleration must be due to a velocity dependent (Coriolis) force.

6 6 Eppur Si Muove Coriolis Force F rot = m a m ω v ( ω r ) m ω a coriolis = v ω

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8 8 Foucault Pendula At the north pole the Coriolis Force will be just enough to rotate a pendulum exactly once per Earth rotation. It s not a conspiracy, the Earth is just turning under the pendulum from the distant inertial reference frame. Since the mean Coriolis acceleration varies as sin(lat), the precession period is 1 day / sin(lat) F rot = m a m ω ( ω r ) m ω v

9 9 Hadley Circulation Cell boundaries produce jet streams

10 10 Jet Streams Hadley Cell Boundaries

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12 1 Jupiter The Coriolis force is 3 times greater on Jupiter than on Earth. The rotation period is just over 8 hours. Given a radius 10 times larger than Earth, the equatorial rotation velocity is 30 times larger. Consequence: A dozen Hadley Cells and not much buckle. Juno

13 13 Tycho and Kepler

14 14 Tycho and Kepler Ironically, Tycho Brahe was an adherent to the Earth-centered universe. His assistant, mathematical genius, and mystic Johannes Kepler, eagerly waited for the opportunity to analyze Tycho's data (which Tycho kept close and secret). Kepler discovered remarkable properties of planetary orbits: 1. All planetary orbits were ellipses with the Sun at one focus.. Planets moved faster when closer to the Sun in a way that a line between the Sun and planet swept out equal area in equal time. 3. The orbital period of a planet was related to its average distance from the Sun. P=a3

15 15 Kepler's First Law Elliptical Orbits An astounding discovery Planets followed detailed mathematical relationships. All planetary orbits were ellipses with the Sun at one focus. A circle is a special case of an ellipse where the foci are on top of each other. The more separated the foci the more eccentric the ellipse.

16 16 Conic Sections

17 17 Small Angles D x θ =tan ( ) D x 1 Explicitly correct but... No! D x x θ= D The result is in units of radians arcseconds in a radian. Yes!

18 18 Stellar Parallax Actual parallax was finally observed by Frederich Bessel in The target was the obscure star 61 Cygni. 61 Cygni has a huge proper motion 5 arcseconds per year making it a likely candidate for being one of the nearest stars. The observed parallax was 0.3, so 61 Cygni was 3.3 parsecs (600,000 AU) away. d parsecs = 1 parallax angle (arcsec) The further the star the smaller the wiggle. Parallax angle is defined as ½ the annual angular displacement of the star. A 1 AU displacement of the Earth produces an angular shift of one arcsecond for a star at 1 parsec distance.

19 19 Kepler's Second Law Equal Areas Another quantitative relationship... Planets move faster when they are closer to the Sun in such a way that a line between the planet and the Sun sweeps out an equal area in the same time interval. The Law of Equal Areas

20 Apparent/Local Solar Time vs. Mean Solar Time Local Solar Time is sundial tme. Noon = the tme hen the Sun crosses the Meridian Because of the Earth s elliptcal orbit (more exactly because of the varying angular speed as the Earth moves around its elliptcal orbit (Kepler s nd La in acton)) the tme of Solar Noon drifs throughout the year. The obliquity of the Earth is also a factor in this drif. Mean Solar Time averages the day length throughout the year. Apparent - Mean There are four dates during the year here mean = apparent solar tme The Equation of Time The offset between mean and apparent solar time

21 The Analemma

22 Kepler's Third Harmonic Law Relates A planet's average distance from the sun - a specifically ½ of the long axis of the elliptical orbit for a circular orbit this semi-major axis is just the radius of the circle The time it takes the planet to orbit the Sun the orbital period, P P = a 3 This equation works as written if P, the orbital period, is expressed in years, and a, the semi-major axis, is expressed in astronomical units. The Earth's orbital period is 1 year and it is 1 A.U. from the Sun. 1 = it works.

23 3 An Example Using Kepler's Third Law Consider an asteroid discovered in a circular orbit 4 A.U. from the Sun. How long does it take this asteroid to orbit the Sun? P = a 3 3 P = 4 = 64 a = 4 AU P= 8 years This asteroid takes 8 years to complete an orbit around the Sun. Kepler's Laws in Motion

24 4 An Example Using Kepler's Third Law Jupiter's orbit has a semimajor axis of a=5. A.U. 3 3 P = a = (5.) = P = = 11.9 years Kepler's Laws in Motion

25 5 Deriving Kepler's Third Law Recall the derivation of circular orbital velocity mv Fc = R GM m F g= R GM v orbit = R

26 6 Deriving Kepler's Third Law Express velocity on the left side in terms of orbital period, P πr GM v= = P R 4π 3 MP = R G This result generalizes to the elliptical orbit with semi-major axis, a, in place of the radius, R.

27 7 Deriving Kepler's Third Law If the equation is calibrated/scaled using units of years and AU then all of the constants multiply out to 1 and you are left with the elementary expression of Kepler's Third Law P=a3. That way of expressing Kepler's equation implicitly assumes the mass of the star being orbited is 1 solar mass. The general simplified expression for any given star (or planet!) is... MP = a 3 Where M is in units of solar masses.

28 8 Deriving Kepler s Second Law Consider an object traveling at velocity, v, at some point in an elliptical orbit around the Sun. Dt 1 1 A = r v t Δ T + (v r Δ t )(v t Δ t )

29 9 Deriving Kepler s Second Law Consider an object traveling at velocity, v, at some point in an elliptical orbit around the Sun. Dt 1 1 A = r v t Δ T + (v r Δ t )(v t Δ t ) =L/m a constant

30 30 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. Fixing the focus locations while changing this distance leads to ellipses of different eccentricity. The long axis is the major axis Major axis

31 31 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. Fixing the focus locations but changing this distance leads to ellipses of different eccentricity. The semimajor axis (a) is one half the long axis of the ellipse For a circle (a special case of an ellipse) the semimajor axis is the radius. semi-major axis a

32 3 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. Fixing the focus locations but changing this distance leads to ellipses of different eccentricity. The semimajor axis (a) is one half the long axis of the ellipse The minor axis is the shortest chord across the ellipse. minor axis

33 33 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. Fixing the focus locations but changing this distance leads to ellipses of different eccentricity. The semimajor axis (a) is one half the long axis of the ellipse the semiminor axis (b) is one half the short axis. semi-minor axis b

34 34 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. The semimajor axis (a) is one half the long axis of the ellipse Fixing the focus locations but changing this distance leads to ellipses of different eccentricity. the semiminor axis (b) is one half the short axis. The length of the string is twice the major axis. r' r + r' = a r

35 35 Ellipse Details Geometrically, an ellipse is the set of all points that are equidistant from the two foci. The semimajor axis (a) is one half the long axis of the ellipse Fixing the focus locations but changing this distance leads to ellipses of different eccentricity. the semiminor axis (b) is one half the short axis. In cartesian coordinates the formula for an ellipse is: x + y =1 a b

36 36 Ellipse Details In polar coordinates (the origin is at a focus) the equation defining an ellipse is a(1 e ) r= 1 + e cos θ where q is called the true anomaly and defined to be zero in the direction of positive x. e is the eccentricity Things to note: The focus is ae from the center e is the ratio of the separation of the foci to the major axis. r + r' is a constant equal to a (the length of the major axis)

37 37 Getting to Polar Coordinates Note the triangle with sides r, r', and ae. One interior angle of this triangle is p-q Apply the Law of Cosines r ' = 4a e + r aer cos(π θ) p-q note cos(p-q) = -cos(q) and use r + r' = a to substitute for r' above. rearrange to recover a(1 e ) r= 1 + e cos θ

38 38 Ellipse Basics Relating axis ratio to eccentricity In polar coordinates the equation defining an ellipse is a(1 e ) r= 1 + e cos θ e is the eccentricity Things to note: where q is called the true anomaly and defined to be zero in the direction of positive x. The focus is ae from the center r + r' is a constant equal to a (the length of the major axis) consider the right triangle formed by a focus and a point on the y-axis b + (ae) = r = a b e= 1 a

39 39 Periapse and Apoapse In polar coordinates the equation defining an ellipse is a(1 e ) r= 1 + e cos θ where q is called the true anomaly and defined to be zero in the direction of positive x. e is the eccentricity Things to note: a particle is closest (periapse/ perihelion/perigee) to the focus when cos(q)=0, farthest when cos(q)=p (apoapse) periapse=a(1 e) apoapse=a(1+e)

40 40 Deriving Kepler s First Law Orbits are Ellipses It s just math.

41 41 Deriving Kepler s First Law Orbits are Ellipses NO.. it s not. It is a beautiful interplay between the force of gravity and the conservation of angular momentum. Kinetic and Potential energy slosh back and forth as nature tries to (and always succeeds at) conserving angular momentum in real time. Orbits are falling combined with some tangential velocity relative to the source of gravity. L=mv T r GMm = m a = F r 3 r

42 4 Why should L be conserved? Noether s Theorm Postulated by one of the great (largely unrecognized) mathematical geniuses of the 0th century Emmy Noether. Physical quantities obey conservation laws if the physics is invariant to coordinate transformation. Take the Solar System, instantaneously rotate the reference frame 90 degrees about the Sun s rotation axis. The planets continue on the same orbits. This particular invariance requires conservation of angular momentum! This mathematical construct was fundamental in resolving inconsistencies in General Relativity. Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff. A. Einstein.

43 43 Periapse and Apoapse In polar coordinates the equation defining an ellipse is a(1 e ) r= 1 + e cos θ where q is called the true anomaly and defined to be zero in the direction of positive x. e is the eccentricity Things to note: a particle is closest (periapse/ perihelion/perigee) to the focus when cos(q)=0, farthest when cos(q)=p (apoapse) periapse=a(1 e) apoapse=a(1+e)

44 44 Orbital Velocity A complete derivation of Kepler's first law (section 3.1.) yields L r= GMm (1 + e cos θ) a(1 e ) r= 1 + e cos θ but so L = GMa (1 e ) = r v t m since L = mr v tangetial velocity is fully tangential at periapse and apoapse, so v peri GM 1+e = a 1 e [ ] 1/ v apo GM 1 e = a 1+e [ ] 1/ note these equations yield circular orbital velocity when e=0.

45 45 Orbital Velocity In general, the orbital velocity is given by 1 v =GM r a ( ) note this equation yields circular orbital velocity when a=r (i.e zero eccentricity)

46 46 Interplanetary Travel

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48 48 Orbital Energy Bound orbits have negative energy 1 GMm E total = K + U = m v r Negative total energy corresponds to a bound orbit. Zero energy corresponds to a parabolic trajectory (velocity reaches zero at infinity) Positive total energy corresponds to a hyperbolic (unbound) trajectory.

49 49 Gravity and Tides Tides result from the differential gravitational force across a finite sized object. simply put, the near side of the object feels a stronger tug of gravity than the far side. Horribly not to scale!!!

50 50 Gravity and Tides Tides result from the differential gravitational force across a finite sized object. consider the gravitational force acting on a parcel of mass at the center of the object. Fgrav GMm = r different r for different parts of the Earth.

51 51 Gravity and Tides Tides result from the differential gravitational force across a finite sized object. consider the gravitational force acting on a parcel of mass at the center of the object. Fgrav GMm = r subtract the force at the center from all the other forces...

52 5 Gravity and Tides Tides result from the differential gravitational force across a finite sized object. subtract the force at the center of the object from all of the other force vectors and the resultant forces represent the tidal stresses on the object.

53 53 Gravity and Tides Tides result from the differential gravitational force across a finite sized object. Quantify tidal forces by differentiating the gravitational force law with respect to r F grav GMm = r d F grav = GMm dr r3

54 55 Consequences of Solid Body Tides Europa's Ocean Volcanoes on Jupiter's Io

55 56 How Much Do Things Stretch in Response A completely rigid object will not change shape in response to tidal forces. A fluid will adjust to fill a surface of constant potential energy. a fluid on a more rigid sphere (Earth's oceans) will flex differentially to the solid surface. solid body lunar tides in the Earth are about 0.3 meter on average ocean tides are about 1 meter on average this 1 meter is in excess of the solid body tides

56 57 Ocean Tides High ocean ties occur approximately every 1.5 hours as the Earth rotates beneath the slowly orbiting Moon. A given location experiences two high tides a day as Earth rotation carries the observer through each of the tidal bulges. The Moon orbits slowly enough relative to Earth's rotation rate (9.5 day synodic month vs. 1 day-long day) that we can consider it occupying a fixed location in space Direction of Earth Rotation

57 58 Ocean Tides vs. Latitude The Moon's orbit lies close to the ecliptic plane and thus is significantly inclined to the Earth's equator. Add to that the 5-degree inclination of the Moon's orbit to the Ecliptic and the Moon can reach a declination of nearly 30 degrees. A location's two high tides can be quite different in a given day depending on the Moon's declination. The image above shows the significant difference between the two daily tides when the Moon is at high declination (if the Moon is on the celestial equator the simple tides would be uniform at a given latitude).

58 59 Real Ocean Tides Landmasses substantially complicate the simple situation. The image below shows the tidal range at different ocean locations.

59 60 Bay of Fundy, Canada

60 61 Solar vs. Lunar Tides In the force equations M is the mass of the tide-causing object, r is the separation between the two objects. dr is the size of the object on which the tides are being raised. The Sun is 30 million times the mass of the Moon, but the Moon is 400 times closer than the Sun. The Sun has about 1/3 the Moon's tidal influence on the Earth. d FSun M sun r moon = d F moon M moon r sun ( )( ) 3 d Fgrav = GMm dr 3 r

61 6 Superposition of the Solar and Lunar Tides When the Sun and Moon align (New and Full Moon) tides are higher than when they raise tides in different directions (First and Last Quarter Moon).

62 63 Superposition of the Solar and Lunar Tides When the Sun and Moon align (New and Full Moon) tides are higher than when they raise tides in different directions (First and Last Quarter Moon). Note that planetary tides, often invoked by nutcase theories of global doom from planetary alignment, are vanishingly insignificant compared to the Sun and Moon. Jupiter's tidal force on Earth is 1/100,000th that of the Sun's.

63 64 The Sun's Role in Tides The Sun also has significant gravitational influence on the Earth. It is much further away than the Moon, but also much more massive. Solar tides are about 1/3 the strength of Lunar tides. When the Sun and Moon align (New and Full Moon) tides are higher than when the raise tides in different directions (First and Last Quarter Moon). Note that planetary tides, often invoked by nutcase theories of global doom from planetary alignment, are vanishingly insignificant compared to the Sun and Moon. Jupiter's tidal force on Earth is 1/100,000th that of the Sun's.