Appunti completi Fundamentals of astronomy and astrophysics

Fundamentals of astronomy and astrophysics

Lesson 1: Spherical astronomy

Angle measurements

• Radians: 0<α<2π most used
• Decimals: 0<β<360°
• Degrees:
  • 0<θ<360°
  • 1 degree=60 minutes (‘)
  • 1 arcminute=60 arcseconds (’’)
• Hours:
  • 0<h<24
  • 1 h=60 minutes (=15 degrees)
  • 1 minute=60 arcseconds

Spherical triangle: Three points on the sphere connected by great circle

• The sum of the angles is always >180°
• The spherical excess is defined as E=A+B+C-180°
• The area of the spherical triangle is E*r² steradians
• The area of the sphere is 4π steradians

A coordinate system is defined by

• The direction of the perpendicular of a plane passing through the centre of the sphere
• The direction of one axis on the plane

Geographic coordinate system

Reference plane: Equatorial plane, perpendicular to the rotation axis of the Earth

• Meridians: Semicircle from pole to pole
• Parallels: Small circle parallel to the equator
• Longitude (λ): East (positive) or west (negative)
• Latitude (Φ): North (positive) or south (negative)

Since the Earth is rotating, its shape is slightly flattened.

Geodetic coordinate system

It uses as reference the normal (or plumb line) to the plane tangent to the surface of the ellipsoid.

• Geodetic longitude (λ): same as geocentric longitude
• Φ > φ Geodetic latitude (Φ): the plumb line does not point to the centre of the Earth
• The Earth is a geoid (sort of ellipsoid of revolution)

The World Geodetic System is composed of

• Standard coordinate system
• Standard spheroidal reference surface
• Gravitational equipotential surface
• NOT a reference plane normal to the surface

Celestial coordinate system

Astronomical horizon: Interaction between the reference plane and the celestial sphere

• Zenith and Nadir: Poles corresponding to the horizon
• Vertical: Any great circle passing through the zenith
• Local meridian: Vertical passing through the north

Coordinates

• Azimuth: Measured clockwise either from north or south
• Altitude (or elevation): Measured from the horizon along the vertical

Celestial equator: Projection of Earth's equator into the celestial sphere

North/South celestial poles: Projection of Earth's north/south poles into the celestial sphere

Ecliptic: The plane of Earth's orbit around the sun. Apparent path of the sun on the celestial sphere throughout the course of the year.

Vernal point: The intersection between the equator and the ecliptic when the sun goes from north to south.

Horizontal system

It is topocentric. During summer, the Sun DOESN’T rise/set closer to south than during winter.

• Coordinationations:
  • Declination: Angular separation from the equator ψ, -90° < ψ < 90°
  • Right ascension: Measured counterclockwise from the vernal point, 0° < α < 360° or 0h < h < 24h

The location of the centre of the reference system (e.g. surface or centre of Earth) is not required to define a celestial coordinate system.

The radius of the sphere where the celestial bodies are projected is NOT required to define a celestial coordinate system.

Parsec: Distance at which one astronomical unit subtends an angle of one arcsecond.

Lesson 2: Transformation of coordinates

The coordinates (ex, ey, ez) of P in x, y, z can be transformed in (fx, fy, fz) in X, Y, Z through a rigid rotation in x0:

We can also use polar coordinates. In general, every rotation can be considered as the result of three different successive rotations along one axis each time.

Equatorial and ecliptic share the same origin and x-axis. Three equations are needed to determine two angles and their signs.

Equatorial and horizontal have the East/West axis in common between the two reference planes, although they have different reference axes on those planes.

Right ascension (α) of a star: Distance of the star from the vernal point along the equator, measured anti-clockwise

Hour angle of a star: Distance between its projection on the equator and the south meridian, measured clockwise (HA)

Sidereal time

• The sidereal time is equal to the hour angle of the vernal point.
• When the vernal point is crossing the local meridian, the sidereal time is equal to 0:00.
• The sidereal time is equal to the sum of the hour angle and the right ascension of a star.
• WRONG: When a star is crossing the local meridian its hour angle is equal to sidereal time.

Synodic day: Time between two consecutive passages of the sun across the meridian.

Sidereal day (ST): Time between two consecutive passages of a reference star across the meridian. It is faster than solar time.

• Star moves parallel to the celestial equator.
• Star CULMINATES when crossing the meridian (on south side).
• If the DECLINATION of the star is higher than the zenith angle of pole north, the star is always visible (circumpolar star).
• The altitude of the north celestial pole depends on the geographical altitude of the observer.

During the year, the sun moves around the ecliptic. The Mean Sun moves at constant angular velocity at the equator with a period of a tropical year. The Mean solar time is NOT equal to the right Ascension of the sun minus 12h.

Stars have the same angular velocity.

An annual aberration for a star with ecliptic latitude equal to 45° causes the star to describe an ellipsis with one axis perpendicular to the Ecliptic.

An annual aberration for a star with ecliptic latitude equal to 0° causes the star to move along a line parallel to the ecliptic.

Lesson 3: Telescopes mounts

Meridian transit telescopes move only in declination. The focal plane is equipped with a high-precision reticulum to accurately determine the time the star transits on the meridian.

They are NOT fixed in declination or moved only in right ascension.

There are also meridian radio telescopes.

Mount types

• Alt-Azimuth mounts
• Dobsonian mount: Big, cheap, portable, home-made, no-good for astrophotography
• Equatorial-German mount
• Equatorial-Open fork mount: Asiago
• Equatorial English or Yoke mount: Cannot point the celestial pole very big
• Equatorial Horseshoe mount
• Equatorial Cross-Axis mount: Asiago. Right ascension axis is supported at both sides.
• Alt azimuthal mount: Telescopio Nazionale Galileo
  • Equatorial Conservation of coordinates at every pointing
  • Pointing is easy directly from coordinates
  • Stars move both in azimuth and altitude with small adjustments simultaneously, at variable acceleration
  • Stars move along the declination axis at constant speed during the night- “simple” engine
  • Cannot point at zenith-degeneracy in the azimuth when approaching the zenith
  • NO problems in point-except the north pole
  • Field is fixed
  • Dome is just slightly larger than the telescope itself
  • Limited to 2.5m +
  • As big as you can imagine

Lesson 4: Perturbation of coordinates

Refraction:

• Light is refracted when passing through the Earth's atmosphere
• n = indice di rifrazione, is velocity of light in vacuum and is phase velocity of light in the medium
• It is not related to the movements of the Earth

Aberration: The finite speed of light causes a shift of the object in the direction of the observer's motion. The amplitude of aberration depends on the TANGENTIAL velocity with the three directions in the same plane.

• Annual Aberration: Velocity vector due to each orbital motion
• Diurnal Aberration: Topocentric velocity vector due to each rotation

Precession: Of Earth's axis due to the Moon and Sun pulling on the oblateness of the Earth. P=circa 26000 years

Nutation: Precession of the orbital plane of the Moon caused by the Moon’s orbit being inclined with respect to the ecliptic.

Stars have their own velocity with respect to the Sun:

• Radial velocity: Velocity in the direction of the observer
• Proper motion: Tangential velocity i.e. the projection of velocity onto the sky plane

Parallax: Is due to the finite distance between the object and the observer

• Baseline: Determine the amplitude of parallax
• Diurnal parallax: Change of direction due to each rotation. Depends on latitude
• Annual parallax: Baseline is the radius of Earth’s orbit
• Secular parallax: Maximum baseline = displacement of the Sun between two observations
• Trigonometric parallax: Is the only direct method to measure the distance of a star. BASELINE: 1u=radius of Earth orbit

Parsec: 206265 AU = circa 3.086×10¹⁶ m. At a distance of one parsec, one astronomical unit subtends an angle of one arcsecond. Arcsec = r[pc]=1/π. There are only 50 stars within 15 parsec.

Hipparcos (ESA): High precision Parallax Collecting satellite

Gaia: Formerly global interferometer for astrophysics

Lagrangian point: L2 is 1.5 million km from Earth on a Lissajous orbit

Lesson 5: Movements of Earth

Solar day: Interval between two consecutive passages of the sun through the local meridian

Sidereal year: Interval between two consecutive passages of the sun over a fixed direction with respect to the background stars

Tropical year: Interval between two consecutive passages of the sun on the vernal point

The stars ideally provide a fixed reference frame to determine the complex movements of the observer. The ecliptic plane is much more stable than the equator.

The instantaneous rotation axis is influenced by the presence of the Moon and the Sun (precession and nutation).

The instantaneous rotation axis may not coincide with the minor axis of the geometrically best-fitting ellipsoid (free or Eulerian nutation).

The speed of the vernal point: G=50’’290966+0’’02222T, T = Julian date

The Moon’s orbit is inclined by 5’’9’ to the ecliptic plane. The distance of the Moon and the Sun change during the lunar month and tropical year.

The only physical effect that is NOT concurring to luni-solar precession and nutation is: the Moon shows always the same side to Earth.

Nutation: Collection of the short period movements of the equator. The main contributor is the Moon. It changes the origin of the ecliptic longitude and the obliquity λ and ξ.

Free or Eulerian nutation: Variation of the rotation axis due to forces internal to the body.

Polar motion

• 1 year with meteorological causes affecting the momentum inertia
• 14 months due to ocean atmosphere circulation and regional sea surface temperature circle.

Lesson 6: Measuring the time

Seconds: Duration of 9192631770 oscillation between two hyperfine levels (F=4, F=3) of fundamental energy levels of cesium 133 when an atom is far from magnetic fields and at sea levels.

Mean solar day: 86400s

Solar/sidereal day: Time between two consecutive passages of the sun/star across the meridian. Sidereal time is faster than solar time.

Apparent sidereal time: Determined directly from observations

Mean equinox: Point where the vernal equinox would be if there was no nutation

Mean sidereal time: Hour angle of mean equinox. WRONG: Obtained by averaging the Apparent Sidereal time over one year

Equation of equinoxes: Δλ cos(ε) − Δε, with nutation in longitude and instantaneous obliquity of the ecliptic.

Sidereal day = synodic day + 1

The motion of the sun in the sky is not uniform due to:

• The Sun moves on the ecliptic, not the equator
• Earth eccentricity

Mean sun: Fictitious sun moving at constant angular velocity on the equator, complete revolution in one tropical year.

Mean (solar) time: Hour angle of the mean sun + 12h.

Equation of time: E = M − Θ = (λ − ⨀) − ⨀. The mean solar time at Greenwich is called universal time (UT).

UT0: Apparent one measured from a particular observer determined by local sidereal time and longitude. Includes polar motions and shift of the rotational pole with respect to the geographic pole

UT1: Referred to the Greenwich longitude. Obtained by tracking extragalactic radio sources. Proportional to the rotation angle of Earth with respect to distant quasars

UT2: UT1 with seasonal variation smoothed out

3 types of irregularities:

• Secular slowing down of rotation
• Seasonal variations due to meteorological causes
• Irregular fluctuations of geographical origin

TAI: International Atomic Time, quantum mechanics definition. Ideal time proceeding at a perfectly constant rate proper time.

TT (or TDT): Terrestrial Dynamical Time. Affected by relativistic time dilation. TT = TAI + 32.184

Proper time: Clock attached to an observer

Coordinate time: Theoretical time scale freed by relativity effects

TDB: Dynamical time of the barycenter. For dynamical studies, TT must be transformed to coordinate time, according to the position and velocity of the observer with respect to the barycenter of the solar system.

UTC: Coordinated Universal Time. Pace of TAI, origin coincident with UT1 within 900ms. Discontinuous but practical and inexpensive.

International date line: Imaginary line of navigation on the surface of Earth that runs from the North Pole to the South Pole and demarcates the change of one day to the next.

Tropical year: Determines the succession of seasons. Time between two consecutive passages of the Sun on the vernal point.

WRONG: Sidereal year. Time between two consecutive passages on the vernal point on a given star.

Season starts when the Sun has longitude = 0 (spring)

Due to eccentricity:

• Warm seasons in the northern hemisphere last 7 days longer than southern
• At aphelion, Earth receives about 7% less energy from the Sun at perihelion

Aphelion happens during the northern summer (coincidence)

The Northern Hemisphere is dominated by land (low heat capacity)

The Southern hemisphere is dominated by water

In July, the land-dominated Northern hemisphere is tilted toward the Sun, overall temperature is higher.

Twilight

Illumination of Earth’s lower atmosphere when the Sun is just below the horizon. 3 types (according to the geometric solar elevation angle of center Sun GCS):

• Civil Twilight: When GCS is 6° below the horizon
• Nautical Twilight: When GCS is 12° below the horizon
• Astronomical Twilight: When GCS is 18° below the horizon

The calendar actually in use is the Gregorian Calendar.

Lesson 7: Radiation mechanisms

Quantum Theory of Light

• Light is both a particle and a wave: a photon has Energy and Momentum but no mass
• Energy of photon: E=h*v (h=Planck’s constant, v=frequency), increasing frequency increases energy
• 1 Ångström = 10^-10 m, wavelength=c/v (v=frequency)

Electromagnetic Spectrum of Light

• Visible light has a wavelength that goes from 400 to 750 nm

The Structure of an atom

We consider the Bohr Model: electrons orbit around a nucleus composed of protons and neutrons.

The orbit of an electron depends on its energy. These energy levels cannot take arbitrary values: they are quantized.

Transition between energy levels

• Absorption (bound-bound transition): A photon with energy E corresponding to the jump between two levels is absorbed, an electron jumps up between the corresponding levels
• Spontaneous emission: An excited electron jumps to a lower level, emitting a photon with random phase and direction
• Stimulated emission: A photon hits an excited electron, the electron goes to a lower state and a photon with the same direction and phase of the inducing photon is emitted: radiation is coherent (spontaneous: incoherent)
• Ionization (bound-free transition): The energy of the absorbed photon is higher (in absolute term) than the energy of the bound electron (Ep>|Ee |), the electron leaves the atom with momentum ΔE/c
• Recombination (free-bound transition): A free electron is captured by an ionized atom, the excess momentum is emitted as a photon
• Free-free radiation: An electron is scattered (change of momentum) by an atom, without being absorbed
• Scattering: Absorption of a photon followed by an instantaneous emission at the same wavelength, but with different direction. Goes by different names depending on the relative size and charge of the scattering particle

Emission and Absorption Spectra

• A continuous spectrum (uniform distribution of photons as a function of wavelength) can be produced by heated-up material (black body) or hot gas at high pressure
• Light from a continuous source travels through a material and it is absorbed by the atoms in the material at the wavelengths corresponding to the jumps between the energy levels
• Atoms in a (low-pressure) hot gas get excited by collisions, then they re-emit at the wavelengths corresponding to the energy levels of the elements in the gas

Each chemical element absorbs/emits at specific wavelengths, depending on the configuration of the energy levels.

Line Profiles

• Despite quantization of energy levels, line profiles are not infinitely narrow and sharp
• The natural width of the line γ is defined as:
• It can be demonstrated that γ is the full width at half maximum (FWHM) and the line profile is a Lorentzian
• Doppler broadening: Atoms move faster at higher temperatures. The natural profile of each atom is Doppler-shifted according to its velocity. The wavelength distribution profile is a Gaussian