Introduction to Astronomy (Coursera)

Introduction to Astronomy Notes from Olivier Henry ([email protected]), based on Coursera Introduction to Astronomy course by Prof. Ronen Plesser 1. 2. 3. 4. 5. 6. 7.

Coordinates on Earth: longitude/latitude.................................................................................6 Motion of the stars in the sky ..................................................................................................6 Seasons ....................................................................................................................................7 Horizontal coordinate system..................................................................................................8 Equatorial coordinate system ..................................................................................................9 Recap: Horizontal vs. Equatorial coordinate system.............................................................11 Localizing stars......................................................................................................................11 7.1. Using observer’s latitude, declination and altitude. ......................................................11 7.2. Exercise: maximum rise of a star based on latitude & declination ...............................12 7.3. Exercise: how high is the Sun at noon?.........................................................................12 8. Solar time vs. Sidereal time...................................................................................................13 8.1. Ignoring Precession and Nutation effects......................................................................13 8.2. Exercise: when will we see Vega at its highest point....................................................14 8.3. Precession effect............................................................................................................14 8.4. Nutation effect...............................................................................................................15 8.5. Equation of Time...........................................................................................................15 Part 1: Influence of Earth’s orbit around the Sun..................................................................16 Part 2: Influence of Earth’s inclination .................................................................................17 9. Motion of the Moon ..............................................................................................................18 9.1. The hidden side of the Moon.........................................................................................18 9.2. Eclipses..........................................................................................................................19 9.3. Partial, Total and Penumbral eclipses ...........................................................................20 10. Planetary Motions..............................................................................................................21 10.1. Synodic period...........................................................................................................21 10.2. Three laws of Kepler .................................................................................................22 a) The orbit of every planet is an ellipse with the Sun at one of the two foci.......................22 b)A line joining a planet & the Sun sweeps out equal areas during equal intervals of time.22 c) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit....................................................................................................23 Example: period of the ISS ...................................................................................................23 10.3. Centripetal force ........................................................................................................23 10.4. Newton's law of universal gravitation .......................................................................24 10.5. Law of conservation of momentum...........................................................................24 10.6. Potential energy.........................................................................................................25 10.7. Law of conservation of energy..................................................................................25 Exercise: finding the thermal velocity...................................................................................25 Exercise: finding the escape velocity ....................................................................................25 10.8. Tidal forces................................................................................................................25 11. Waves ................................................................................................................................27 11.1. General definitions ....................................................................................................27 11.2. Doppler effect............................................................................................................27 11.3. Light ..........................................................................................................................27 11.4. Heat transfer and radiation ........................................................................................27 Heat transfer ..........................................................................................................................27 Radiation ...............................................................................................................................28 Luminosity is in W (=J/s)......................................................................................................28 Page 1/91

Exercise: Flow produced by Sun at its surface......................................................................28 Exercise: Temperature of the Sun .........................................................................................29 Exercise: color of the Sun .....................................................................................................29 Exercise: finding the speed of an helium particle, knowing temperature .............................29 12. Electromagnetic force........................................................................................................29 13. Solar system ......................................................................................................................30 13.1. General ......................................................................................................................30 13.2. The Sun......................................................................................................................31 13.3. Interplanetary medium ..............................................................................................31 13.4. Age of the Solar system.............................................................................................32 13.5. Nuclear/radioactivity decay.......................................................................................33 13.6. Radiometric dating ....................................................................................................33 13.7. Radiometric dating (Uranium-lead) ..........................................................................33 13.8. The creation of the solar system................................................................................34 13.9. Kelvin–Helmholtz contraction: gravity contraction→ heat ......................................35 13.10. Terrestrial formation..................................................................................................35 13.11. Beyond the Snow line................................................................................................36 13.12. Orbital resonance.......................................................................................................36 Kirkwood gap ........................................................................................................................36 The Nice model .....................................................................................................................36 13.13. Timeline of the creation of the Solar system.............................................................37 14. The Earth ...........................................................................................................................38 14.1. General ......................................................................................................................38 14.2. Internal Heat ..............................................................................................................38 14.3. Finding temperature of Earth ....................................................................................38 Ignoring Earth’s atmosphere .................................................................................................38 Greenhouse model.................................................................................................................39 14.4. The Atmosphere ........................................................................................................39 14.5. Earth magnetism........................................................................................................40 15. The Moon ..........................................................................................................................41 16. Planets detection methods .................................................................................................43 16.1. Astrometry.................................................................................................................43 16.2. Radical velocity.........................................................................................................43 16.3. Transit method...........................................................................................................43 16.4. What have we found? ................................................................................................44 17. Analyzing Stars .................................................................................................................45 17.1. Laws of conservations of charge and of electrons ....................................................45 17.2. Chemical reactions to create heat..............................................................................45 17.3. Nuclear Fission to create heat ...................................................................................45 17.4. Nuclear fusion to create heat: PP chain.....................................................................46 Exercise: quantity of He produced since the Sun’s birth ......................................................46 17.5. Solar Structure...........................................................................................................47 The core.................................................................................................................................47 Inner Mantle ..........................................................................................................................47 Outer Mantle..........................................................................................................................47 Chromosphere .......................................................................................................................47 Corona ...................................................................................................................................47 17.6. Solar weather.............................................................................................................48 17.7. Parallax, or Finding distance from the Star...............................................................49 17.8. Astrometry, or finding the speed of the Stars............................................................50 17.9. Stellar statistics..........................................................................................................51 17.10. Binary stars................................................................................................................51 Page 2/91

17.11. Eclipsing binary stars ................................................................................................51 17.12. Recap: Analyzing Stars .............................................................................................52 By visual observation ............................................................................................................52 Using Doppler effect .............................................................................................................52 Particular case of Eclipsing binaries .....................................................................................53 17.13. Mass-luminosity relation...........................................................................................53 17.14. Main-sequence stars ..................................................................................................53 This is our Sun.......................................................................................................................53 CNO cycle for MS stars > 1.3 Rsun ........................................................................................53 Radiation and Convection effects..........................................................................................54 Expansion by contraction ......................................................................................................54 18. Star evolution ....................................................................................................................55 18.1. Star creation process..................................................................................................55 18.2. T-Tauri stars ..............................................................................................................55 18.3. Main Sequence ..........................................................................................................56 18.4. From MS to Subgiant stars........................................................................................56 18.1. From Subgiant to Red giant branch stars ..................................................................56 18.2. From Red giant branch to Helium core flash ............................................................57 18.3. From Helium core flash to Horizontal branch...........................................................57 18.4. From Horizontal branch to Asymptotic Giant branch...............................................58 18.5. From Asymptotic Giant branch (AGB) to Thermal Pulse AGB ...............................58 18.6. From Thermal Pulse AGB to White Dwarf...............................................................58 18.7. White Dwarf Nova ....................................................................................................59 18.8. Supernova (type Ia) ...................................................................................................59 18.9. Instability branch: Variable stars as Standard candles ..............................................60 18.10. Blue stragglers...........................................................................................................61 18.11. From MS to Red Supergiant......................................................................................61 18.12. From Red Supergiant to Helium flash to Blue Supergiant........................................61 18.13. From Blue Supergiant to Massive star AGB.............................................................62 18.14. From MS to Wolf-Rayet stars ...................................................................................63 18.15. From MS to LBV stars ..............................................................................................63 18.16. From Core collapse to Supernova type-Ib/Ic/II.........................................................63 18.17. From Supernova type-Ib/Ic/II to Neutron (pulsar) star .............................................64 18.18. Recap: Stars on HR diagram .....................................................................................65 19. Relativity ...........................................................................................................................66 19.1. Principle of Relativity ...............................................................................................66 19.2. Spacetime ..................................................................................................................66 19.3. Lorentz transformations ............................................................................................66 19.4. Relativistic Spacetime ...............................................................................................67 19.5. Length contraction.....................................................................................................67 19.6. Time dilation .............................................................................................................67 19.7. Doppler effect due to high speed...............................................................................67 19.8. Velocity addition .......................................................................................................67 19.9. Lorentz metric ...........................................................................................................67 19.10. The Invariant Interval................................................................................................68 Time-like interval ..................................................................................................................68 Light-like interval..................................................................................................................69 Space-like interval.................................................................................................................69 19.11. Conservation laws .....................................................................................................69 19.12. Lorentz transformations applied to Energy and Momentum.....................................69 19.13. Principle of Equivalence ...........................................................................................69 19.14. Gravitational redshift.................................................................................................70 Page 3/91

19.15. Relativistic Potential energy......................................................................................71 19.16. Gravitational lensing .................................................................................................71 19.17. Gravity is geometry ...................................................................................................72 19.18. Gravitational waves...................................................................................................72 20. Black holes ........................................................................................................................73 20.1. Horizon......................................................................................................................73 20.2. Singularity .................................................................................................................73 20.3. Emission of X-rays....................................................................................................73 20.4. No Hair ......................................................................................................................74 20.5. Cosmic censorship conjecture ...................................................................................74 20.6. Hawking radiation .....................................................................................................74 20.7. Wormholes ................................................................................................................74 20.8. Example: compute wavelength of X-ray emission of the accretion disk surrounding black hole ..................................................................................................................................75 21. Galaxies .............................................................................................................................76 21.1. The Milky way ..........................................................................................................76 21.2. Tracking matter .........................................................................................................76 21.3. The Milky way disk structure....................................................................................76 21.4. The Milky Buldge and Core......................................................................................77 21.5. The Milky Halo .........................................................................................................77 21.6. Weighting the Milky way..........................................................................................77 21.7. Dark matter................................................................................................................78 21.8. Spiral galaxies ...........................................................................................................78 21.9. Galactic evolution......................................................................................................78 21.10. Measuring distance to galaxies: Redshift..................................................................79 21.11. Cosmic expansion......................................................................................................79 21.12. Recap on formulas.....................................................................................................80 21.13. Galaxy clusters ..........................................................................................................81 22. Cosmology.........................................................................................................................82 22.1. The cosmological principle .......................................................................................82 22.2. Robertson-Walker model ..........................................................................................82 22.3. Angular size distance (k=0).......................................................................................82 22.4. Luminosity distance (k=0).........................................................................................83 22.5. Correcting the temperature for redshift .....................................................................83 22.6. Correcting the galaxy speeds for redshift..................................................................84 22.7. Einstein field equations .............................................................................................84 22.8. Isotropic Homogenous Matter...................................................................................85 22.9. Friedmann equations .................................................................................................85 22.10. Cosmological parameters ..........................................................................................86 22.11. The Early universe: radiation era ..............................................................................86 22.12. Matter-dominance era................................................................................................86 22.13. Dark-energy-dominance era ......................................................................................86 22.14. The Particle horizon ..................................................................................................87 22.15. The event horizon......................................................................................................87 22.16. Cosmic microwave background ................................................................................87 CMB ......................................................................................................................................87 Angular Power Spectrum of the CMB ..................................................................................88 22.17. Big Bang Nucleosynthesis.........................................................................................88 22.18. LCDM Cosmology ....................................................................................................89 22.19. Inflation .....................................................................................................................89 22.20. Exercise: compute the distance when the light was emitted, and the distance now, from a galaxy.............................................................................................................................90 Page 4/91

22.21. Exercise: compute the distance angular radius of an object......................................90 22.22. Exercise: compute the brightness of an object, knowing its luminosity ...................90 22.23. Exercise: compute the observed luminosity period, knowing its real luminosity period (e.g. when light was emitted) .........................................................................................90 22.24. Plasma and Ionization ...............................................................................................91

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1.

Coordinates on Earth: longitude/latitude

Longitude is going east.

2. Motion of the stars in the sky Because of the Earth’s rotation, starts are moving with an α angle which depends on the latitude of the observer (Stars – except the Sun - are so far from Earth that they seem fixed): α = 90° - latitude. Hence: Highest point, from • For an observer on the pole, observer’s view starts go on a path parallel to the celestial equator • Stars rise and descend • The Sun moves along the Celestial sphere from West to East (RA increases), because The Earth rotates in the same direction as it orbits around the Sun. Within a year we see all stars.

Spring/Summer on Northern Hemisphere

Spring/Summer on Southern Hemisphere Page 6/91

Special case when α=90°: • We can use the starts to find our latitude: the angle above our head to which we see the Polaris star corresponds to our latitude on Earth. Polaris (or α UMi, or α Ursae Minoris, or Alpha Ursae Minoris) is a North Star (also called Pole Star) in the constellation Ursa minor (“petite ourse” in French), very close to the celestial pole. • Also, stars located at the observer’s Zenith have declination = latitude. Zenith The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. The zenith angle is the angle between a direction of interest (e.g., a star) and the local zenith.

3. Seasons Seasons are due to the inclination of Earth (24°) vs. the Sun. • At solstice, the day is the longest or shortest. “Solstice” comes from Latin “sol” (sun) and “stilium” (stoppage) because from one day to the next, the Sun seems to stay at the same place vs. Earth. • At Equinox, day and night are about the same length: 12 hours, except on poles. On poles, days are 6 months, nights are 6 months, changing at solstice. In fact, this would be true if the Sun was just a point. But since the Sun is seen from Earth as a sphere, in practice the day is longer by a few minutes (depending on the latitude). At equinoxes, Sun rises vertically. Vernal equinox is ~ March 21, Autumnal equinox is ~ September 21. June solstice is ~ June 21, December solstice is ~ December 21.

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Earth is inclined 23.5°, which 1. Creates seasons 2. Creates the impression from Earth that the Sun is orbiting in an elliptic path, called the Ecliptic.

The Earth rotates in the same direction as it orbits around the Sun.

4. Horizontal coordinate system This system is using: 1. azimuth (angle from magnetic North) 2. altitude (height of the star in the sky)

The star’s altitude and azimuth change through the night and depend on the observer’s position.

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5. Equatorial coordinate system This system is using: 1. Right ascension is the longitude (going east). But not starting from Greenwich, rather from the Vernal Equinox. Degrees are converted in hours minutes seconds (24h = 360°) 2. Declination is the latitude of the star if projected on Earth 3. RA always remain the same for a Star, except for close Stars like the Sun. RA varies between -23.5° and +23.5° for the Sun, through the year. This system does not depend on the observer’s position or time. • • • •

1 degree = 1hour of arc 1 hour of arc = 60 minutes of arc = 3600 seconds of arc Thumb is ~ 1 degree at arm’s length Hand is ~ 20 degrees at arm’s length

20°

-23.5° RA +23.5° RA Hour

°

Hour

°

0° RA For two stars one hour of right ascension apart, you will see one star cross your meridian one hour of time before the other. The start of the RA (Vernal equinox) can be easily visualized using the Pisces (“poisson”) constellation:

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Remember that from Earth, in this coordinate Stars seem fixed, because celestial sphere very big.

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6. Recap: Horizontal vs. Equatorial coordinate system Horizontal coordinate system (azimuth, altitude)

Equatorial coordinate system (RA, Declination)

7. Localizing stars 7.1. Using observer’s latitude, declination and altitude.

ZA = Zenith Angle (angle between star and observer’s zenith), seen from observer

Latitude = →

declination + ZA Declination-ZA

ZA = |latitude – declination|

or depending on position of star vs. observer and Altitude = 90°-ZA

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7.2. Exercise: maximum rise of a star based on latitude & declination If observer’s latitude is 38° and Star’s declination is 30°, then ZA= 8° and therefore Star will raise at maximum at 90°-8° = 82° above the observer’s horizon.

7.3. Exercise: how high is the Sun at noon? From Athens, how high is the Sun at noon? Athens is a latitude 37.7N. At Equinox Declination = 0, therefore ZA = 37.7° since ZA = |latitude – declination| Therefore, Altitude = 52.3° since Altitude = 90°-ZA. At Summer solstice Declination = +23.5°, therefore ZA = 37.7-23.5 = 14.2° and altitude = 90-14.2 = 75.8°. At Winter solstice Declination = -23.5°, therefore ZA = |-37.7-23.5| = 61.2° and altitude = 90-61.2 = 28.8°.

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8. Solar time vs. Sidereal time 8.1. Ignoring Precession and Nutation effects By definition, 24h is the time needed for the Earth to see the Sun come back at the same place. This is called the solar time. But that ignores the fact that during this time, the Earth has been orbiting around the Sun. Hence the Earth rotates in less than 24h. This is called the sidereal time.

1. Earth orbits around the Sun in 365.25 days  α24 = 360/365.25 ° per day, or per 24h. This is the degrees by which the Earth orbits the Sun in 24h 2. Earth rotates

360° + α24 in 24h 360° in 360/(360 + α24) x 24h  Earth rotates 360° in Tsideral = 360/(360 + α24) x 24 = 23h 56’ in

Tsideral = 23h 56’ Time for Earth to do 360° rotation Tsolar = 24h Time for Earth to view Sun at the same place

This is equivalent to Tsolar - Tsideral = 1/366.25 days = 24x60/366.25 ≈ 4 min (3.83 min) since in one year, the Earth rotates 365 times relative to the Sun, but 366 times relative to the stars. That is why Stars (including therefore the Sun) rise 4 minutes earlier every day. Hence in one year, the Earth has rotated 365.25x24/23.93356 = 366.2 times. Hence stars shift slowly with every year. This effects adds to the precession effect. • Solar time is called local time • On Sep 21, solar time and sidereal time are the same • ST = LT +/- 4minutes Sidereal time= 0 at vernal equinox (June 21st). Any celestial body is crossing the local meridian at its right ascension. • •

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Earth rotates 366.25 times @ 23h56’, or 365.25 times @ 24h. Therefore, sidereal time for Earth is 365.25 days (of 24h). Local Sidereal Time Sidereal Time (“Greenwich Sidereal Time”) is the time such as a day is 23h56 min, and starts (ST=0) at Vernal equinox. Local Sidereal Time is the Sidereal time, but depending on the observer’s location. Local Sidereal Time = Greenwich Sidereal Time + observer’s longitude in hours (360° = 24h) Local Sidereal Time vs. local (Solar) time • On September 21: local time = Local Sidereal Time. Then ∆ = 4 minutes per day • On December 21: local time = Local Sidereal Time - 6h on Dec 21 0:00 ST, it’s not • On March 21: local time = Local Sidereal Time - 12h yet Dec 21 LT because • On June 21: local time = Local Sidereal Time - 18h Tsideral = 23h 56’ Each star is at its highest point (= on our meridian) when Local Sidereal Time = RA.

8.2. Exercise: when will we see Vega at its highest point When is Vega as high as possible (from our point of view) at midnight? Vega’s RA is 18h36’. Vega is on our meridian at Local Sideral Time = 18h36’. (ignoring latitude and time zone effects) 1. On June 21st, LST = LT + 18h therefore at LT = 0 on June 21st, LST= 18h and Vega is nearly on our meridian. Hence Vega is at its highest point at midnight LT, on June 21st + 9 days (36’ = 9 days x 4’) = July 1st. 2. We could also do: 18h36 = 18+36/60 = 18.6 h. Since 24h ↔ 365.25 days, 18.6h ↔ 283 days. September 21 + 283 days = July 1st. 3. We could also use the daily gap between Local Sidereal and Local Time (3.83 minutes). But we need to remember that this value is based on 366.25 days, not 365.25 days, so we need to withdraw 1 day: 18h36 = 1116’ = 284 x 3.83 minutes → September 21 + 283 – 1 days = July 1st.

8.3. Precession effect When the Earth is orbiting around the Sun, and rotating around itself like a spinning top, the gravity from the Sun attracts the weight excess located around the Earth equator closer to the Ecliptic. This cause the Earth’s axis to slowly move in a cone shape, as a spinning top would do (rotation of the axis in the opposite direction of the spinning top rotation). Rotation of the axis rotates 360° in 26000 years, hence 1.4° every 100 years. Page 14/91

Because of this effect, Vernal equinox advances slightly every year, hence the name “precession of the equinoxes”.

8.4. Nutation effect The Moon also creates such effect, called Nutation. Period is 18.6 years. Oscillation is 17,2" = 17.2/3600° = 0.005°

Precession + Nutation effect

8.5. Equation of Time •

•

The Earth’s orbit around the Sun is not a circle, for several reasons. Because of this, the Sun is not moving at a regular speed around the Ecliptic. The equation of Time indicates the difference between the time viewed from a sundial (“real”) and the official time (or “apparent” because based on the assumption that the Sun is moving at regular speed on the Ecliptic). The sundial indicates the real time, whereas our clocks indicate the apparent time (= average)

Real time = Apparent time - ∆T(d) where d is the day (d=1 for Jan 1st).

The real equation of Time in minute is: ∆T(d) = 4 x [C(d) + R(d)]

where C and R are expressed in degree, such as:

C (d) = 1.918° sin(d) + 0.02° sin(2d) + 0.0003° sin(3d) R(d) = -2.468° sin(2d) + 0.053 sin(4d) – 0.0014° sin(6d)

This can actually be approximated as: ∆T(d) = ∆Tc(d) + ∆Tr(d) ∆Tc (d) = 7.678 sin (B+1.374) ∆Tr (d) = -9.87 sin (2B)

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Part 1

Where B(d) = 2π (d-81)/365

Part 2

Equation of time = sum

(d=81 is the Spring Equinox)

Part 1: Influence of Earth’s orbit around the Sun • • •

Earth’s orbit is not really circle, it’s a little elliptic Earth’s speed is not constant on the elliptic path This effect account for up to 9 minutes difference in the real and apparent time.

Earth is fastest

Earth is slowest

(Earth’s orbit is greatly exaggerated on the drawing. Speed varies from 30287 km/s to 29291 km/s) Perihelion happens around Jan 4, Aphelion happens around July 4. ∆Tc (d) = 7.678 sin (B+1.374) Page 16/91

Part 2: Influence of Earth’s inclination • •

Earth is inclined, therefore the Sun’s projection on the Ecliptic is not linear This creates a difference between the real time and the apparent Sun

∆Tr (d) = -9.87 sin (2B)

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9. Motion of the Moon 9.1. The hidden side of the Moon The Moon rotates in 27.323 days. This is its sidereal period, also called its sidereal month. Which means RA of the moon increases by 48’ per day (vs. 4’ for the Sun). Because the Earth is moving around the Sun, the Moon’s full rotation relative to the Sun is actually longer. This is called the synodic month = 29.53 days. The synodic period drives the full moon cycle. The sidereal period is also the time that the Moon takes to rotate around the Earth  from Earth, we always see the same side of the Moon. The other side will always remain hidden from Earth. We say that the Moon’s rotation period and orbital period are the same. This is due to the tidal forces (“forces de marée”) applied for the Earth to the Moon, as indicated in the figure below (where the Moon is called the satellite). Let’s assume that the Moon is rotating faster than it is orbiting around the Earth. The Earth’s tidal forces create 2 small deformations (one on each side of the Moon), as indicated in (1). When the Moon is rotating, those deformations are rotating as well, and are now “in advance” of the Earth. The Earth’s tidal forces apply to those deformations, which tends to slow the rotation of the Moon by forcing them to go “backward” as indicated in (2). With time, the satellite will be rotating at the same speed as it is orbiting around the Earth. The same reasoning is valid also if the satellite was rotating slower than it is orbiting around the Earth. The tidal forces: the force on the left is stronger than on the right, making it seem the planet was exposed to two opposed forces.

as if

If the satellite was rotating faster than its orbiting speed, the tidal forces will also make it come closer to the planet. If the satellite was rotating slower than its orbiting speed, the tidal forces will make it go farther from the planet. In the case of the EarthMoon system, the Moon goes farther from Earth by about 3.8cm per year.

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9.2. Eclipses Eclipses happen when the Sun, the Earth and the Moon are more or less aligned. We speak about solar eclipse when the Moon masks (partly) the Sun, and about lunar eclipse when the Moon passes directly behind the Earth into its umbra (shadow). Eclipses do not happen every 27 days, though. This is because the plan of the Moon rotation around the Earth is inclined vs. the plan of the Earth rotation around the Sun: The Moon’s orbit is tilted by 5° with respect to the ecliptic (it varies between 5° and 5°18’ in 173 days). Like the Sun, the Moon is higher in the Summer. And both have almost the same angular size. Full moon

Nothing Nothing

Solar eclipse Lunar eclipse Represented as a drawing:

Hence, a lunar eclipse happens roughly 2 times a year. Since the Moon orbits in 27 days, it has done half a resolution around Earth in about 2 weeks. Hence, solar eclipse (most of the time, partial) and lunar eclipse happen roughly 2 weeks at interval. In total, there are therefore about 4 eclipses (lunar and solar) per year. This is why when perfect alignment, for a particular region on Earth (250km shadow), we can have total eclipses. Practically, solar eclipse happens 2 weeks before the lunar eclipse. In fact, using the notion of saros (18.6 years interval, where the Earth, Sun and Moon have exact same position), we can compute that there are about 4.6 eclipses per year. This is because lunar and Earth orbits are not multiple of each others. Page 19/91

In the North emisphere, when eclipse, the Moon goes from East to West (appear on the right) because it enters Earth shadow from West. Thinking in terms of Nodes :

9.3. Partial, Total and Penumbral eclipses Penumbral eclipse happen when the Moon enters the Penumbra only. Total or Partial eclipses happen when the Moon enter the Umbra.

Penumbral eclipse

When total lunar eclipse, the Moon does not totally disappear! It gets some light from the reflection of the light by the Earth.

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10. Planetary Motions 10.1.

Synodic period

The synodic period is the temporal interval that it takes for an object to reappear at the same point in relation to two or more other objects, e.g., when the Moon relative to the Sun as observed from Earth returns to the same illumination phase.

The Synodic period of two planets can be easily found by solving:

where S is Synodic period (i.e. when both align again), P1 the planet’s period of the faster planet, P2 the planets’ period of the slower planet, Remember that

where V is the planet’s speed.

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10.2.

Three laws of Kepler

a) The orbit of every planet is an ellipse with the Sun at one of the two foci

• • • •

Any point on orbit is at equal distance from each focus. r1 + r2 = 2a eccentricity e = dfoci / 2a (e=0 → circle) The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is

b)A line joining a planet & the Sun sweeps out equal areas during equal intervals of time The planet moves faster near perihelion, slower near aphelion. Therefore, the closest the planet to the Sun, the fastest.

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c) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. P2 = K a3 where K is constant.

Or more precisely:

whereM1+M2 is the total mass of the system

Where P is the orbital period of the planet orbiting and M2 its mass, M1 the other planet (Sun)’s mass, and a the semi-major axis of the orbit. G is the gravitational constant. This relation is due to the centripetal forces. Because M1 >> M2, K is constant for every planet of the Solar system. From this, we can get the planet’s speed for eclipses closed to circles (a≈R) using P=2πR/v:

P2 = a3

if units expressed in AU and years, since for Earth, using those units we find out K=1. a = R is circle instead of ellipse. VALID FOR SUN ONLY For other Stars, express Motherstar = β. Msun, then P2 = K. a3 where K=1/ β using AU and years.

Example: period of the ISS

ISS orbits at an altitude h = 370 km, Earth has radius of 6471 km and mass of 5.9272 x 1024. Hence P2 = 2 π2 R3 / GM → P = 5510s = 91.8 m.

10.3.

Centripetal force

In simple terms, centripetal force is a force which keeps a body moving with a uniform speed along a circular path and is directed along the radius towards the centre. For a satellite in orbit around a planet, the centripetal force is supplied by gravity. Centripetal force (N) F= m.a where m is the mass in kg and a the centripetal acceleration.

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Attraction force On Earth, F= m.g attraction force, where g= 9.82 ms-2 constant for Earth. Notice that the smaller r, the highest the F and the v.

10.4.

Newton's law of universal gravitation

Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them: where: • • • • •

F is the force between the masses G is the gravitational constant = 6.67 x 10-11 N.m2/kg2 m1 is the first mass m2 is the second mass, and d is the distance between the centers of the masses.

It can be shown that the Earth is completely equivalent to a point of same mass, concentrated in the middle. Using the above formula, we can compute the effect of gravity on someone at the surface of Earth of 59kg: 579N.

10.5.

Law of conservation of momentum

We define momentum as: p = m.v The momentum represents how easy/hard it is to modify an object’s course. In a closed system (one that does not exchange any matter with the outside and is not acted on by outside forces) the total momentum is constant. pA + pB = constant

This is particularly interested in the case of collision, when an object stops and the other starts moving.

Notice that since F=m.a and p=m.v, F is the rate of change of p. The angular momentum is also conserved : L = m.v.R if moving in a circle. The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation: R↓, so v↑. If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

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10.6.

Potential energy

Potential energy is the energy of an object or a system due to the position of the body or the arrangement of the particles of the system. For an object subject to gravity:

where ME and Mo are the masses of Earth

and of the object. Note that if object is located on earth, the Potential energy (in this case, energy is I release the object) becomes:

g So to sum-up: where

10.7.

constant for Earth

Law of conservation of energy

Energy is constant, if no other forces than gravity is applied to an object.

Exercise: finding the thermal velocity

This is the speed of atoms excited by a certain temperature: Ec = ½ mv2 = 3/2 kB T

Exercise: finding the escape velocity

Conservation of energy → Ec = ½ mv2 = -G. ME m/(RE+h) → v Comparing Thermal and Escape velocity, we can conclude whether those atoms remain in the atmosphere, or are expulsed by temperature.

10.8. Using

Tidal forces gives:

therefore

is the

acceleration on Earth due to the Sun, or Tidal acceleration, which varies depending on where we are on Earth:

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For there we get aT = a+ - a- → This can be also expressed as:

= 5.14 x 10-8g This is a rather small value. What about the tidal forces due to the Moon? Applying the formula to the moon, we get aTmoon = 2.2 aTsun. Tides repeat every 24h44 min. There’s a 12 min lag. This effect is even increased during full moon, when Sun and Moon are aligned with Earth. At quarter moon, the tide is the smallest.

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11. Waves 11.1. • • •

General definitions

f is the frequency, in Hz λ is the wavelength, in m Energy flux transported by wave is proportional to Amplitude2 , in (J/s)/m2, or W

λ.f = c where c is the speed at which the wavelength is traveling.

11.2.

Doppler effect

Where vrec and vem are the speeds of the receptor or emitter. Or, if receptor is not moving: λ = λ0 (1-v/c) where c is the wavelength’s speed, and v the emitter’s speed. Notice that the frequency does not change with time: either it’s less than f (if emitter gets away from receptor), or it more than λ (if emitter gets closer to receptor). The sound effect we see when a car passes by comes from the amplitude difference with time. The frequency is higher, but does not change with time.

11.3.

Light

Light carries energy at a speed of c = 2.998 x 10^8 m/s Color is the frequency of light, of the order of 10^12 Hz for visible colors. Our eyes are only sensitive to the intensity of Red Green Blue (RGB) colors. Because each atom absorbs particular frequencies, by looking at the spectrum of light emitted by stars, we can find which atoms are present.

11.4.

Heat transfer and radiation

Heat transfer An object hotter than environment will lose energy until temperatures equilibrate. It can happen by: - conduction, i.e. through continuous contact - convection, i.e. through physical motion - radiation, i.e. hot objects glow losing energy to light. If energy is radiated at a rate L in J/s, at distance R

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Radiation

Radiation flow is distributed uniformly on surface on a sphere F=L/(4 π R2) • •

A hot object radiates The hotter the object, the smaller wavelength (gets blue): λmax . T= b constant b=2.9 x 10-3 m.K T in Kelvins, b in meter x Kelvins

where b is Wien’s law

Hotter objects radiate more: F = σ.T4 where F is the flux (i.e. power/m2) radiated, and σ = 5.67x10-8 W/m-2K-4 Stefan-Boltzmann constant Sunlight heat on earth is the solar constant b0 = 1361 W/m2 From this we can compute the luminosity of sunlight (L is fixed) L = 4 π d2 b0 = 3.83 x 1026 W

Flow is in W/m-2 Luminosity is in W (=J/s) Energy captured on Earth (in W/m2) = L/(4πD2) Energy radiated by Sun (in W/m2) = F.(4πRsun2)

Exercise: Flow produced by Sun at its surface

Flow from Sun at surface is: F = L/(4π R2) and Luminosity L=4 π d2 b0 Therefore F= (d/R)2 . b0 = 6.29 x 107 W/m2

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Exercise: Temperature of the Sun

Temperature of the part of the Sun that we see, at its surface: F = σ.T4 → T=5770 K = 5500 °C

Exercise: color of the Sun Using Wien’s law: λmax = b/T = 0.0029/5770 = 503 nm. This corresponds to the green color. Why not yellow?

The light emitted by the Sun if refracted by the Earth’s atmosphere. First, the blue is refracted (which is why the Earth looks blue from space). The Sun therefore looks yellow. Then, at sunset, the distance that lights covers in the atmosphere increases, and more light gets reflected – the sun looks red.

The highest frequencies get reflected first. 0K = -273.5°C

Exercise: finding the speed of an helium particle, knowing temperature

Use Ec = ½ mv2 = 3/2 kB T

12. Electromagnetic force (in Coulomb) to be compared with

gravity force.

Force can be attractive or repulsive. Opposite charges attract to most objects are neutral. Charge is conserved A charge creates and is affected by electric field A changing magnetic field creates electric field (Faraday 1831) A changing electric field creates magnetic field (Maxwell 1861) This leads to propagating waves with velocity c speed of light → light is an electromagnetic wave! Many waves’ frequencies are blocked by atmosphere, so we need to observe from space. • • • • • • •

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13. Solar system 13.1.

General

Our solar system is composed of: • the Sun (99.86% of total mass of the solar system) • 8 planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn (90% of the remaining solar system mass after the Sun), Uranus and Neptune. Distances from the Sun ranges from 0.39 – 30 AU • Their 175 natural satellites (or “moons”), most of them orbiting Jupiter and Saturn • 5 Dwarf planets: Pluto, Ceres, Eris, Makemake, Haumea • Billions of small bodies • All orbits are in the same plane, the Sun’s axis

All planets, as well as most other objects (except the Halley comet), orbit around the Sun in the same direction as the Sun’s rotation: counter-clockwise for an observer located on the North pole. All objects orbiting around the Sun do so in an elliptic path, from which one focus is the Sun. Planets’ orbit is nearly circular, while the smallest the other object, the more elliptic the path.

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Kepler's laws of planetary motion are three scientific laws describing orbital motion, each giving a description of the motion of planets around the Sun. Most of the largest natural satellites are in synchronous rotation, with one face permanently turned toward their parent. All the 4 giants have rings. A planetary ring is assumed to be quite instable, and disappear after a few thousand or millions years. Hence today’s planetary rings are quite recent. One object from the ring is either attracted back to the ring, and therefore stays in the ring, or attracted by the planet (and therefore disappears within the planet). Hence rings have distinctive edges. •

70.5% is Hydrogen, 27.5% is Helium, 2% is Metal

13.2.

The Sun

The Sun is a yellow dwarf, as 20-40 other billions yellow dwarf in the Milky way (for a total of 200-400 billions starts). Each second, the Sun merges 564MT of Hydrogen and produces 560MT of Helium. The difference, for a weight of 4MT, produces energy and is radiated as light and solar wind. Every 150M years, the sun looses the equivalent of 1 mass of Earth. The Sun is in its mid-life. In 5 bn years, it will become bigger, more bright, colder, more red: a red giant. It will then be several thousand times more bright than today. The Sun is a star Population I: it is born from supernovae’s explosions, which created heavier metal. It is widely assumed that the presence of heavier metal in the Sun is required to form planets, grouping metals together.

13.3.

Interplanetary medium

In addition to light, the Sun also radiates a continued flow of charged particles (a “plasma”) called “solar wind”. This flows extends at a speed of 1.5M km per hour, creating an atmosphere called Heliosphere up to 100AU far from the Sun. These particles are called Interplanetary medium. Page 31/91

The solar wind explains why the second tail of comets (the plasma tail) always points away from the Sun. On Earth, solar winds can create continuous current on the high-voltage power lines, creating overload of the power transformers. The Earth’s magnetic field (magnetosphere) protects us more or less against the solar wind. When they penetrate near the poles, they create the Aurora Borealis. The plasma is ejected from the Sun at a speed of on average 450 km/s (between 400 – 800 km/s) and is composed of 73% of hydrogen and 25% helium, roughly 10^6 T per second. Because the Sun is rotating, the magnetic field lines form a spiral, called Parker’s spiral.

13.4. • • •

Age of the Solar system

Oldest rocks on Earth are 4.4bn years (“Gy”) Oldest rocks on the Moon are 4.4 - 4.5bn years. Oldest meteorite is 4.54bn years

Our best estimate is 4.55-4.58bn years, using radioactive dating. Atom of atomic number N and of atomic mass A has its nucleus which contains N protons (positive) and A-N neutrons (neutral). A indicates the number of nucleons Nuclei can have same N but different A. They are then called isotopes.

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13.5.

Nuclear/radioactivity decay

Most combinations are unstable and decay via: • α decay: emission of Helium nucleus, i.e. 2 protons and 2 neutrons •

β decay: emission of electron with conversion of neutron → proton, or emission of a positron with conversion of proton → neutron: or

• Fission: breakup into two smaller nuclei All these decays are usually accompanied by creation of γ rays, and produce heat. Using Ec = ½ mv2 = 3/2 kB T (thermal velocity), we find that vhelium ≈vescape and therefore the Earth is loosing Helium. But it is producing Helium also, through α decay.

13.6.

Radiometric dating

We use decay process, for example using Carbon 14: If for an atom, we know the half-life period t½ (i.e. the period in which half of the remaining atoms have decayed), then the number of atoms remaining (i.e. that have not yet decayed) is: where N(t) is the number of atoms remaining, and t½ the half-life period. Which means that every half-time period, half the atoms have decayed. Since atoms have no memory, after the half-time period the process starts again. Use ln to solve. Carbon 14 dating can estimate a date from a few hundreds year to 50,000 years.

13.7.

Radiometric dating (Uranium-lead)

This method can estimate a date from 1M years to 4.5bn years, with 0.1-1% precision, and uses the fact that Uranium decays into lead through 2 routes instead of 1. Hence the 2 routes should give the same dating, which in practice is not the case. Hence we reduce uncertainty. 1. 238U to 206Pb, and 2. 235U to 207Pb Under conditions where the system has remained closed, and therefore no lead loss has occurred, the age of the zircon can be calculated independently from the two equations:

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And

In practice, the results of both equations differ slightly, because Fission tracks and micro-cracks within the crystal will create conduits deep within the crystal, thereby providing a method of transport to facilitate the leaching of Pb isotopes from the zircon crystal. We use the upper intercept of the Concordia method to evaluate the age of the sample.

13.8.

The creation of the solar system

Conditions for a molecular cloud to stay a cloud Recalling that Ec = ½ mv2 = 3/2 kB T → v=f1(T) And that v2esc =G.M.m/R → vesc =f2(R) The cloud will remain stable if f1(T) = f2(R), i.e. R=[G.M] / [3 kB T]. This can be also expressed as: Ec < EG → 3/2 kB T< G.M.m/R

Jeans instability

Where M is the mass of the cloud, and m the mass of each particle. Hence, if clusters form, temperature gets higher and creates stars. In practice, if there’s a singularity, or a non-conformity, the cloud will split. The solar system started as a molecular cloud. Then fragmentation and gravitational collapse created by a nearby supernova creates a fragment of around 3000 MassSun, and 2000-20000 AU in size, from which the Sun is a member of an open cluster now dispersed. If the mass had a small rotation movement, when it collapses the mass concentrates in one point, and therefore the rotation speed increases (conservation of the angular Page 34/91

momentum). That is why our galaxy, and all stars and planet, orbit together in the same direction. The closer the planet to the axis, the faster is moves. Matter orbiting around the center flattens to a ring towards the center of the rotation, exactly as what happens we turn around quickly, holding strings, which move towards our center of orbit.

13.9. Kelvin–Helmholtz contraction: gravity contraction→ heat As the universe concentrates, heat increases. This is also true for a star or planets.

This is due to the law of conservation of energy, where

. If d ↓, then

energy must be radiated through temperature increase. At the center of our galaxy, temperature ≈2000 K, with highest planet density. Then the farther the planet from the center, the colder it gets, and the more solid we can find on the planet.

13.10. Terrestrial formation 1. 2. 3. 4. 5. 6. 7. 8.

Grains of dust collide and adhere As soon as they reach 1km in size, they are bound by gravitation The larger the object, the faster it grows. The growth rate is proportional to R4 Objects grow for 100,000 years, at which point they are called protoplanets (R ≈ 1000km) Because of the Kelvin–Helmholtz contraction effects, the gravitational force heats the elements until the planet melts. The planet becomes spherical because 1) it melts and 2) of gravity Chemical differentiation occurs: heavier materials sink to core Gravity is opposed by pressure force: pressure increases when closer to the center Compression heats the core

9. Then, protoplanets accrete into larger planets, called planetesimals. It ends up with 100 Moon-Mars sized planetesimals. Page 35/91

10. Gravitational interactions distorts orbits 11. Collisions lead to merger or ejection, leaving large Venus or Mars. Other get stripped to core 12. Orbits settle to near-circular orbits in 10-100 Myears, around the Sun

13.11. Beyond the Snow line The snow line is the distance from the Sun where it is cool enough for hydrogen compounds such as water, ammonia and methane to condense into solid. The temperature is estimated around 150K. The snow line of the solar system is around 5AU, hence Jupiter is right on the outside of this line. 1. Outside of this line, the gravity of the Sun is lower, and gas giants like Jupiter acted like our solar system: it aggregated nearly all the remaining H and He gas, and grew very rapidly until gas in orbit exhausted. 2. Jupiter rotates, exactly like our solar system, but faster (in 10h) 3. Jupiter creates a flatten ring, exactly as the Sun creates its planets orbiting Saturn is further from Sun, it started later so captured less gas, but repeated the same process.

13.12. Orbital resonance Orbital resonance occurs when planets’ orbital periods the Sun are related by a ratio of small integers: nP1 = mP2. • Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits, regardless of the Sun’s attraction. • Often, resonance occurring at 2-4 AU disrupts planet formation, which creates asteroid belts around the planet. This is the case in particular for Jupiter and Saturn. • In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists.

Kirkwood gap Resonance can also occur between asteroids gravitating around the Sun at a distance coming at resonance with Jupiter’s orbit. At those distances, no asteroids can be found. This is called the Kirkwood gap.

The Nice model The Nice model, developed at the Nice observatory, explains why the Gas giants are where they are. It proposes the migration of the giant planets from an initial compact configuration into their present positions. The four model proposes that after the dissipation of the gas and dust of the primordial Solar System disk, the four giant planets (Jupiter, Saturn, Uranus and Neptune) were originally found on near-circular orbits between ~5.5 and ~17 astronomical units (AU), much more closely spaced and more compact than in the present.

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After several hundreds of millions of years of slow, gradual migration, Jupiter and Saturn, the two inmost giant planets, cross their mutual 1:2 mean-motion resonance. This resonance increases their orbital eccentricities, destabilizing the entire planetary system. The arrangement of the giant planets alters quickly and dramatically. Jupiter shifts Saturn out towards its present position, and this relocation causes mutual gravitational encounters between Saturn and the two ice giants, which propel Neptune and Uranus onto much more eccentric orbits. These ice giants then plough into the planetesimal disk, scattering tens of thousands of planetesimals from their formerly stable orbits in the outer Solar System. This disruption almost entirely scatters the primordial disk, removing 99% of its mass, a scenario which explains the modern-day absence of a dense trans-Neptunian population. Some of the planetesimals are thrown into the inner Solar System, producing a sudden influx of impacts on the terrestrial planets: the Late Heavy Bombardment.

13.13. Timeline of the creation of the Solar system

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14. The Earth 14.1. • • • • • • •

General

71% of its surface is water Above this is atmosphere of mostly N2 and O2 Surface is rocky: Si The core is rich in metals: Fe, Ni. Inner core is solid, outer core is liquid Mantle is made of rocks Average density 5500 kg/m3 (rock is 3000 kg/m3) Pressure, density, temperature increase with depth

14.2.

Internal Heat

Heat is generated in the core by : • Radioactive decay • Kelvin-Helmholtz The mantle drives the convection: fluids go up, then down. Because of that, the crust - broken into plates - is dragged by mantle. This creates mountains at the surface. New crust arises from volcanic processes. Heat loss: 87W/m2 at the surface. However, most heat on Earth comes from the Sun’s radiation. It loses energy as radiation into space.

14.3.

Finding temperature of Earth

Ignoring Earth’s atmosphere Energy captured on Earth (assuming Earth is black): Iin and also

(L is in W)

therefore

(T is of Sun)

Energy radiated by Earth (assuming Earth is black): Iout

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At equilibrium: Iin = Iout →

gives 279K, i.e. 6°C which is too cold!

In fact, Earth is blue and therefore reflects about 0.367 of the radiation, therefore Iabs = (1-a) Iin

where a = Earth’s albedo = 0.367 →

gives 248K which is even

colder (and below zero)! TEARTH is the temperature of the planet based only on the light received by the Star.

Greenhouse model Greenhouse effect explains why temperature is higher: • The atmosphere is transparent to the incoming sunlight (visible) • The atmosphere partially (g) absorbs the infrared light radiated by Earth, through it’s molecules, reradiating part of this energy towards Earth. At equilibrium, each medium is such as Fin=Fout. • Atmosphere: g.σ Te4 = 2.σ Ta4 • Earth’s Surface : σ Te4 = σ Ta4 + Fin Solving the equations gives Fin = (1-g/2). σ Te4 Te = (1-g/2)-1/4 Tno greenhouse

This gives:

a=0.367 and g=0.21 → Te = 292K

a depends on clouds, g depends on molecules present in the atmosphere. Notice that changes in a & g can alter climate drastically.

14.4.

The Atmosphere

N2 and CO2 were released when minerals were cooked at high temperature. H2O was imported from outer system as ice, during heaving bombardment period (3.5bn years ago) Rain creates oceans which dissolve CO2 and fix it in sediments Plants released O2 initially taken up by Fe and S

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14.5.

Earth magnetism

Earth is a magnet, roughly aligned with rotation axis. 1. The core – metal-rich - is being heated by the inner code. This drives convection. 2. Along with the rotation of the Earth, this creates a magnetic field which aligns itself with the rotation axis 3. Every 500 years, the N/S change polarity unpredictably

Charged particles of Solar wind are trapped by field lines into radiation belts. This prevents this intense flux of charged particle to arrive on Earth. The solar winds deforms the Earth’s field, in particular on the North pole where they penetrate Earth’s atmosphere. This gives the auroras in the poles.

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15. The Moon

Those linear marks indicate that the Moon is shrinking.

• • • • • • •

Temperature 370K day, 100K night (nights/days are 2 weeks long) No liquid water because water requires atmospheric pressure to retain it. Ice in crater shadows 35K No atmosphere because the Moon is not big enough to attract the molecules Lunar surface is a museum of history, because on Earth those marks are removed by tectonic activities Moonquakes are caused by Earth’s tidal forces No big magnetic field Mineral composition indicates it's a peace of Earth. Probably from a giant impact by a Mars-sized meteorite in early Earth history. This explains why Moon is poor in metals, since metals are in the Earth's core,

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Combining crater dating with radiometric dating of lunar samples and meteorite leads to history of bombardment rates, and 3.9bn years ago the period of heavy bombardment.

•

Lunar density is not much higher than that of rocks, therefore we deduce its core is very small.

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16. Planets detection methods 16.1.

Astrometry

First other extra solar planets discovered in 1988. There are now 853 planets found, around 672 stars. Only 32 planets, in 28 systems have been detected by imaging. In a system with 1 planet orbiting a star, the star is slightly orbiting around the center of gravity of the system. We have: Mstar.Rstar = Mplanet.Rplanet Rsystem = Rstar + Rplanet is fixed By combining both equations, we get: Rstar = (Mplanet /Msystem).Rsystem By carefully analyzing the complex path of a Star, we can get information on orbiting planets.

16.2.

Radical velocity

We can measure the Star’s speed using Doppler effect: λ = λ0 (1-v/c) This is valid of course if the planet is in our plan, because to measure the Doppler effect, the planet must go away/closer from us. Forcegravity, system = Forcecentripetal, star Which gives using Rstar = (Mplanet /Msystem).Rsystem : The farther the planet from the Star, the more effect it has on its velocity. Jupiter’s effect on the Sun is 12.5 m/s. Vplanet 498 planets in 386 systems have been detected by radial velocity measurements.

16.3.

Transit method

If a planet crosses (transits) in front of its parent star's disk, then the observed visual brightness of the star drops a small amount. The amount the star dims depends on the relative sizes of the star and the planet. For example, in the case of HD 209458, the star dims 1.7%. This method has two major disadvantages. Page 43/91

•

•

First of all, planetary transits are only observable for planets whose orbits happen to be perfectly aligned from the astronomers' vantage point. The probability of a planetary orbital plane being directly on the line-of-sight to a star is the ratio of the diameter of the star to the diameter of the orbit. About 10% of planets with small orbits have such alignment, and the fraction decreases for planets with larger orbits. For a planet orbiting a sun-sized star at 1 AU, the probability of a random alignment producing a transit is 0.47%. Therefore the method cannot answer the question of whether any particular star is a host to planets. Secondly, the method suffers from a high rate of false detections. A transit detection requires additional confirmation, typically from the radial-velocity method.

However, by scanning large areas of the sky containing thousands or even hundreds of thousands of stars at once, transit surveys can in principle find extrasolar planets at a rate that could potentially exceed that of the radial-velocity method. The main advantage of the transit method is that the size of the planet can be determined from the lightcurve. When combined with the radial-velocity method (which determines the planet's mass) one can determine the density of the planet, and hence learn something about the planet's physical structure. The transit method also makes it possible to study the atmosphere of the transiting planet. When the planet transits the star, light from the star passes through the upper atmosphere of the planet. By studying the high-resolution stellar spectrum carefully, one can detect elements present in the planet's atmosphere. 290 planets in 235 systems have been detected via transit. The Kepler telescope has found 2321 candidate planets in 1290 systems.

16.4. • • •

What have we found? Between 1-40% of (Sunlike) stars have planets Only a tiny zone has been explored. These methods are sensitive to Hot Jupiter: big planets orbiting close to stars.

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17. Analyzing Stars 17.1.

Laws of conservations of charge and of electrons

1. Law of conservation of charges: Whatever the reaction, the total charge remains unchanged before/after the reaction has occurred. 2. Law of conservation of electrons: Whatever the reaction, the number of electrons remains unchanged before/after the reaction has occurred.

17.2.

Chemical reactions to create heat

The Sun gets part of its heat through chemical reactions. It burns 10-19 J per atom, or 6.107 J per kg of H. The Sun produces in this fashion 6.4.1018 kg/s, hence the Sun would live around 10000 years if using this process only.

17.3.

Nuclear Fission to create heat

Reminder: 1 atom = 1 nucleus + electrons orbiting = a bunch of nucleons + electrons orbiting Recall that nucleon = neutron or proton. Atoms are not charged, because the charges of the nucleons and the electrons cancel each others. Why don’t nuclei break up under electric repulsion? A strong, short-range (10-15m) attractive force binds the nucleons. This gravity force is called the nuclear force. If we can break the nucleus, then the nucleons get away from each other, and liberate electrostatic energy that was used to bind them together. Practically, the binding energy per nucleon peaks around iron (Fe). Same process happens for decay.

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17.4.

Nuclear fusion to create heat: PP chain

Two atoms of H can crash and merge if 1) they have enough kinetic energy to overcome electrostatic repulsion, and 2) right alignment. 1. When they merge, the 2 protons create a proton and a neutron. Because of the law on conservation of charges, one positive electron (positron) gets created. Because of the law of conservation of the quantity of electrons, a neutrino (ν) gets created. 1

1

2

+

H + H → H + e + νe + 0,42 MeV ( H is also called deuterium) The positron e+ positron annihilates 2

2.

itself with an electron of a nearby H atom, which creates energy through 2 photons +

-

e + e → 2γ + 1,02 MeV

3. The deuterium can then repeat the same process with another H atom 2

1

3

H + H → He + γ + 5,49 MeV

4. Two 3He will eventually merge 3

3

4

1

1

He + He → He + H + H + 12,86 MeV

PP1

This process creates 4.3 10-12J through 1 He atom, guaranteeing the Sun ~ 1011 years (100 bn years) However, this process happens very infrequently (1 in 5bn years), and only temperature in the core is high enough for this process to occur in the Sun, which represents 10% of the Sun. Hence this fusion process guarantees 10bn years. Which is also roughly the Sun’s life, by the way.

Exercise: quantity of He produced since the Sun’s birth Energy produced since birth: Luminosity (in J/s) x Sun’s life in s = 3.83e26 x 1.4e17s = 5.4e43 J Energy produced by 1 He atom in PP chain: 4.3e-12 J/atom of He Mass of 1 He atom ≈ 4 protons = 6.65e-27kg/atom Therefore; 8.41e28kg of He have been produced in the core (on top of the 27% already present uniformly).

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17.5.

Solar Structure

We know the structure of the Sun thanks to Helioseismology. The sun creates acoustic waves @3mHz. This method can also apply to other stars, in this case we talk about asteroseismology.

The core • • • •

R ≤ 25% Rsun 7 x 10^6 K ≤ T ≤1.57 x 107 K 2 x 104 kg/m3 ≤ ρ ≤ 1.5 x 105 kg/m3 M ≈ 40% Msun

The Luminosity of the stars are determined by their mass, because the heavier the star, the more it contracts the core.

Inner Mantle • • • •

25% Rsun ≤ R ≤ 70% Rsun 2 x 10^6 K ≤ T ≤7 x 106 K 103 kg/m3 ≤ ρ ≤ 2 x 104 kg/m3 Heat is transmitted through charged plasma, transit time 1.7 x 105 years takes a charged particle to transit from the core to the outer edge

Outer Mantle • • • •

70% Rsun ≤ R ≤ Rsun 5780 K ≤ T ≤ 2 x 106 K 2 x 10-4 kg/m3 ≤ ρ ≤ 103 kg/m3 The plasma becomes opaque, heat is transmitted through convection

Chromosphere • • • •

Density is low, but temperature increases with altitude h < 2000 km 5780 K ≤ T ≤ 50,000 K 10-10 kg/m3 ≤ ρ ≤ 2 x 10-4 kg/m3

Corona • • • • •

2000 km ≤ h < 1.3 Rsun T ≈ 2 x 106 K ρ ≈ 3 x 10-12 kg/m3 Can be observed though UV or X-Ray wavelength Temperature is very high, therefore particles are very fast, and can escape gravity of the Sun: solar wind of charged particles.

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17.6. • • • •

•

•

•

Solar weather

The Sun is covered by sporadic sunspots, varying in 1 year cycle. Looking at those sunspots, we can see that the Sun is orbiting in ~ 25 days. Sunspots pair (+/-) appear first at mid-latitudes, and later near equator. Spots are regions of increased magnetic fields, therefore they modify the density of atoms and therefore temperature. The charge varies from one cycle to the next, because the magnetic field reverses between cycles. The equator rotates faster than the poles! This deforms the convection zone: the field gets elongated in the equator vs. the poles. Reconnection releases energy, every 11 years reverses polarity. Sunspots are those regions where the magnetic field rotates when getting released Reconnection releases magnetic energy through charged particles: up to 6x1025J, in gas @107K. This is the solar wind, which takes ~3 days to arrive to Earth, if projected in our direction

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17.7.

Parallax, or Finding distance from the Star

To find the distance from a Star to Earth, we look at the angle at which we see the Star, from 2 distance points.

Dsun/D = tan(θ) → θ = Atan(Dsun/D) ≈ Dsun/D Atan(r) ≈ r if r