Review: Harvard sequence of stellar spectra

Review: Harvard sequence of stellar spectra
K K K 7 000 K 6 000 K 4 000 K 3 000 K 650 nm 400 nm Hydrogen Helium Calcium Iron Oxygen Magnesium Many molecules Credit: 2005 Pearson Prentice Hall, Inc. O B A F G K M He II (strong) He I, H H (very strong), CaII (weak) H (strong), CaII, Fe (weak) CaII, neutral metals (strong), H (moderately), neutral metals CaII, neutral metals (maximum) Strong molecular bands (TiO)

Review: Harvard sequence of stellar spectra
The spectral sequence is a sequence of decreasing effective tempereture The strength of the characteristic spectral lines varies systematically along sequence spectral type effective temperature [1000 K] Relative width of absorption lines Note on the terminology: He I = neutral helium He II = singly ionized helium

Review: Liminosity classification
Ia Bright supergiants Iab, Ib Supergiants II Bright giants III Giants IV Subgiants V Main sequense stars (dwarfs) VI Subdwarfs D (*) White dwarfs (*) DA, DB, DC

3.2.3 Spectral Classification of Stars
(j) Complete spectral classification luminosity class plus spectral type = two-dimensonal spectral classification e.g. G2V for the Sun, A0V for Vega, ...

3.2.4 Hertzsprung-Russell Diagram (HRD)
(a) General remarks Diagram showing the relationship between (a) absolute magnitude or luminosity and (b) spectral type or effective temperature (sometimes also color index) That means there exist different forms of this diagram, usually log L versus spectral type log L versus log T M versus B-V Right: HRD for stars for which accurate distances (relative error <10%) are known (mostly from the HIPPARCOS satellite). Ejnar Hertzsprung, Henry Norris Russell (~ 1910)

(a) General remarks The HRD displays the relationship between fundamental properties of the stars (→ key diagram for stellar structure and evolution) Stars tend to fall only in certain regions of the HRD: - main sequence (most prominent) - giants - supergiants - white dwarfs Right: Schematic HRD showing the densely populated areas as well as many of the well known stars. The diagonal lines illustrate the L-T relation for constant stellar radii R (i.e., the L-R-T relation from the Stefan-Boltzmann law; see Sect )

(b) The brightest stars 100 brightest stars in the sky. This HRD is biased in favor of the most luminous stars, which appear toward the upper left.

(b) The brightest stars … and the nearby stars 100 brightest stars in the sky. This HRD is biased in favor of the most luminous stars, which appear toward the upper left. HRD of the stars within about 5 pc of the Sun. The diagonal lines correspond to constant stellar radius.

(c) Star clusters Open clusters Globular clusters Pleiades Messier 3 Usually ~10 stars Loosely bound Concentrated toward the Galactic plane 2 Usually ~10 stars Compact Not concentrated to the Galactic plane 5

(c) Star clusters „Long“ MS (towards blue stars) MS turnoff point (if any) at small B-V Usually a few red giants only Apparent magnitude V Apparent magnitude V „Short“ MS (no blue MS stars) MS turnoff point (TO) redder (larger B-V) Many red giants, complex structure

(c) Star clusters MS: main sequence TO: main sequence turnoff point SGB: sub-giant branch RGB: red giant branch AGB: asymptotic giant branch HB: horizontal branch BS: blue stragglers P-AGB: post asymptotic giant branch Apparent magnitude V „Short“ MS (no blue MS stars) MS turnoff point (TO) redder (larger B-V) Many red giants, complex structure

3.3 Velocities and Distances of Stars
The nearby stars

Kinematik 3.3.1 Velocities v v r v r μ Radial velocity v
(a) Space velocity v r Kinematik v r t μ v Radial velocity v r Transversal velocity v t Space velocity v = √ v + v 2 2 r t

3.3.1 Velocities (b) Radial velocity Doppler effect
The frequency ν of a wave is changed by Δν when the observer moves with the velocity v relative to the source. This relative velocity corresponds to the radial velocity component. Δν v Δλ v Δλ = r = r v = c ν c λ c r λ 1 2 3 4 Fig.: Waves emitted by a source moving from right to left. The observer 1 (2) detects a higher (lower) frequency than the observers 3 and 4. (Credits: Wikimedia/Tkarcher/Tatoute) Fig.: Example for measuring the radial velocity component of the star Arcturur relative to the Sun.

3.3.1 Velocities r v μ (c) Transversal velocity, proper motion t
Proper motion (pm) μ: Angular change in position over time (arcsec/yr) caused by the transversial velocity component of the star (plus the contri-bution from the Earth and the Sun). Fig.: Position change of Barnard's star between 1985 and 2005. Credit: Wikimedia, Steve Quirk v = r tan μ t μ is very small. Only for a few stars, the pm is larger than 1 arcsec/yr. (Highest: Barnard's star with μ ≈ 10 arcsec/yr). v = r μ t Therefore with the pm μ in radian per time interval.

3.3.2 Distances (a) General remarks
Geometric methods: relationship between angular diameter and linear size Photometric methods: relationship between apparent and absolute magnitude Distance modulus: m – M = 5 log r [pc] – 5 + A Requirements for useful distance indicators The relationships must be calibrated (difficult). The physics behind the relationships must be plausible.

3.3.2 Distances (b) Annual trigonometric parallax
παράλλαξίς (parallaxis) = alteration = displacement in the apparent position of an object viewed along different lines of sight. Nearby objects have a larger parallax than more distant ones. Can be used as a distance indicator (geometric method)). Parallax: As the viewing point moves, objects closer to the observer appear to move fast while the distant objects appear to move slowly or not at all. Credit: Nathaniel Domek (Wikimedia)

Apparent position change of a (nearby) star relative to distant stars due to the annual motion of the Earth around the Sun In this context, a star is „distant“ if its parallax is too small to be measured star Sun Earth position 1 Ecliptic plane

Apparent position change of a (nearby) star relative to distant stars due to the annual motion of the Earth around the Sun In this context, a star is „distant“ if its parallax is too small to be measured The distant stars can be considered as being all at the same (infinite) distance (celestial sphere) star Celestial sphere Sun Earth position 1 Ecliptic plane

Apparent position change of a (nearby) star relative to distant stars due to the annual motion of the Earth around the Sun In this context, a star is „distant“ if its parallax is too small to be measured The distant stars can be considered as being all at the same (infinite) distance (celestial sphere) 2 1 A nearby star describes an ellipse at the sphere = Projection of the orbit of the Earth around the Sun star Celestial sphere Sun Earth position 1 Earth position 2 Ecliptic plane

3.3.2 Distances (b) Annual trigonometric parallax r a π b 1 AU
Definition of the parallactic angle: sin π ≈ π [rad] = 1 AU / r[AU] r [AU] = 1 / π [rad] AU = astronomical unit = mean distance of the Earth from the Sun The parallactic angle is used to define a new distance unit, the parsec (pc) r [pc] = 1 / π [''] i.e., 1 pc is the distance from where 1 AU corresponds to an angle of 1 arcsec. 1 pc = (π [''] / π [rad]) AU = (360∙60∙60 / 2∙3.14) AU = 2∙10 AU 5

3.3.2 Distances (b) Annual trigonometric parallax r a π b 1 AU
How to measure the parallactic angle from Earth?

3.3.2 Distances (b) Annual trigonometric parallax π 1 AU r a b
How to measure the parallactic angle from Earth? From the Fig. (left), one finds π θ 2 1 α β φ P To reference star α + β + 2π = 180° φ = β + θ φ = 180° - α - θ (1) and 2 1 β = φ - θ 2 α = 180° - φ - θ 1 Inserting these two equations in (1) yields π = (θ + θ ) / 2 1 2

Simulation of the annual parallax of a nearby stars
Distances (b) Annual trigonometric parallax Simulation of the annual parallax of a nearby stars Jan 2013 Jul 2013 Jan 2014 Jul 2014 0.1'' Ross p = 0.318'' Measurement principle: relative position to stars in the neighbourhood Difficulties: Small position changes → measurement of very small angles with high accuracy The reference stars may move as well (most likely due to their peculiar motion in space) Right: example of the measurement of the annual trigonometric parallax of the star Ross 248

Calibration of the method: measuring the AU Kepler's third law: Consider the orbits of two planets with the semi-major axes a and the orbital periods P: The ratio of the squares of the periods is equal to the ratio of the cubes of the semi-major axes . (P / P ) = (a / a ) 2 3 1 2 1 2 If a is known for one planet, the orbital axes for any other planet can be found from its period. Assume that body 1 is the Venus, body 2 the Earth: 3 2 (a / 1 AU) = (P /yr) V V 1 AU Sun With a = 1 AU – d (see Fig.) we get Earth V V d V d V 1 AU = 1 - (P [yr]) 2/3 V Ecliptic plane The distance d of the Venus can be measured by the radar echo method with high accuracy. V 1 AU = /- 0.1 km Venus

How far does it reach? The limit of the distance mesurement by the annual trigonometric parallax 1 r = π relative distance error r π = Δr Δπ r = Δr π 1 Δπ max = ( ) Δr r min The maximum distance r is set by the accuracy of the parallactic angle Δπ (i.e. the smallest measurable angle) for the constraint of a relative error better than 10% (Δr /r < 0.1) we get modern ground-based observations: Δπ = 0.01'' Hipparcos: Δπ = 0.002'' modern ground-based observations: r = 10 pc Hipparcos: r = 50 pc min max

3.3.2 Distances (b) Annual trigonometric parallax Accuracy
Improvement of the position accuracy and the number of stars with measured positions and parallaxes large angles small angles between ~1600 (Tycho Brahe) and ~1990 steady improvement of the accuracy by a factor ~2.5 per century jump ~1990 by the astrometry satellite Hipparcos another jump and a revolution of the data base (number of stars!) with astrometry satellite Gaia in next years (>2016) Gaia Credit: ESA

3.3.2 Distances (c) Star stream parallax
Star stream: Group of stars with common space motion

Star stream: Group of stars with common space motion Hyades : nearest open cluster, connected with a stellar stream

K: direction of motion (convergence point)
Distances (c) Stellar stream parallax (c) Star stream parallax d r tan ω = v / v t r v = μ r μ: proper motion (pm) t tan ω = μ r / v r Therewith observer v [km/s] tan ω r [pc] = r 4.74 μ[''/ yr] The measurement of the angle ω requires the knowledge of the convergence point K that can be found from the extrapolation of the proper motion vectors (Fig.). Distance to the Hyades: pc K K: direction of motion (convergence point) α δ

The knowledge of the Hyades' distance enables us to calibrate the main sequence of this star cluster (i.e. transition from apparent magnitudes to absolute magnitudes). Colour-absolute magnitude diagram for the Hyades 2 M V 4 6 8 10 0.0 0.4 0.8 1.2 1.6 2.0 B-V

3.3.2 Distances (d) Star cluster parallax
Comparison of the Hyades and the Pleiades in the CMD Compared with the Hyades main sequence (MS), the Pleiades MS appears to be parallel shifted towards fainter magnitudes. This is (mainly) a distance effect: the Pleiades are in greater distance than the Hyades. V B-V 0.0 0.4 0.8 1.2 1.6 2.0 2 4 6 8 10 12 M If r and the extinctions A and A are known, r can be found from the CMD m – M = 5 log r -5 + A When we assume that all stars of a cluster are at the same r, MS stars of the same B-V have the same M , … we get from m – m = 5 log (r /r )+ (A - A ) P H

Colour-Magnitude Diagram (CMD)
Distances (d) Star cluster parallax Colour-Magnitude Diagram (CMD) Principle: Given are absolute magnitudes of MS stars that define a „standard MS“ M = M (B-V) (e.g. the MS of the Hyades) Absolute magnitude M calibrated „standard MS“ (absolute magnitude M) B-V

Distances (d) Star cluster parallax Colour-Magnitude Diagram (CMD) Principle: Given are absolute magnitudes of MS stars that define a „standard MS“ M = M (B-V) (e.g. the MS of the Hyades) For a cluster of unknown distance d the apparent magnitudes m = m (B-V) of MS stars are known. MS of a cluster of unknown distance (apparent magnitude m) Magnitude (M or m) calibrated „standard MS“ (absolute magnitude M) B-V

Distances (d) Star cluster parallax Colour-Magnitude Diagram (CMD) Principle: Given are absolute magnitudes of MS stars that define a „standard MS“ M = M (B-V) (e.g. the MS of the Hyades) For a cluster of unknown distance d the apparent magnitudes m = m (B-V) of MS stars are known. The comparison of m with M at the same B- V yields the distance from the distance modulus m - M = 5 log r – 5 +A MS of a cluster of unknown distance (apparent magnitude m) m - M Magnitude (M or m) calibrated „standard MS“ (absolute magnitude M) B-V

Distances (d) Star cluster parallax Colour-Magnitude Diagram (CMD) Principle: Given are absolute magnitudes of MS stars that define a „standard MS“ M = M (B-V) (e.g. the MS of the Hyades) For a cluster of unknown distance d the apparent magnitudes m = m (B-V) of MS stars are known. The comparison of m with M at the same B- V yields the distance from the distance modulus m - M = 5 log r – 5 +A In that way, the calibrated part of the MS (i.e.the standard MS) can be expanded MS of a cluster of unknown distance (apparent magnitude m) m - M Magnitude (M or m) calibrated „standard MS“ (absolute magnitude M) B-V

Distances (d) Star cluster parallax Colour-Magnitude Diagram (CMD) Principle: Given are absolute magnitudes of MS stars that define a „standard MS“ M = M (B-V) (e.g. the MS of the Hyades) For a cluster of unknown distance d the apparent magnitudes m = m (B-V) of MS stars are known. The comparison of m with M at the same B- V yields the distance from the distance modulus m - M = 5 log r – 5 +A In that way, the calibrated part of the MS (i.e.the standard MS) can be expanded Plejades Hyades Magnitude (M or m) Useful tool to calibrate the MS up to the brightest stars. B-V

3.3.2 Distances (d) Star cluster parallax -8 M v
B-V v 8 4 -4 -8 NGC2362 H+χ Per Plejades M41 M11 Coma Hyades Praesepe M67 Calibration of the MS by stepwise „continuation“ of the standard MS Once the MS is calibrated, the absolute magnitude M of a MS star can be derived from B-V … and if the apparent magnitude m is measured, the distance is given by the distance modulus

Distances (e) Pulsating stars … see later (Sect. Variable Stars)

3.3.3 The nearby stars Proxima Centauri – the closest star to our Sun
How can we efficiently select nearby star candidates from the huge number of background stars? Proxima Centauri, the small red star in the center was only discovered in 1915 and is only visible through a telescope. Credit & Copyright: David Malin, UK Schmidt Telescope, DSS, AAO

3.3.3 The nearby stars (a) Selecting nearby star candidates
Statistically, proper motion is a good distance indicator, because μ = v / r t A high proper motion can be a hint at a small distance. For example: Proxima Centauri on three photographic plates from different epochs (*) (*) SuperCosmos Sky Survey (SSS); image size 2' x 2' 1976 (blue) 1982 (infrared) 1993 (red) With μ = 3.9 arcsec/yr, Proxima Centauri is among the top 20 fastest stars in the sky! Reference objects

Expected to be a complete, volume-limited sample
The nearby stars (b) The 3 pc sample Expected to be a complete, volume-limited sample ⊙ Sp type = Spectral type V Name π [''] r [pc] m log L/L Sp type Proxima Centauri 0.770 1.3 11.05 -4.22 M5V α Cen A 0.750 -0.01 0.20 G2V α Cen B 1.33 -0.35 K5V Barnard's star 0.545 1.8 9.45 -4.35 Wolf 359 0.421 2.4 13.50 -4.70 M6V BD +36°2147 0.397 2.5 12.52 -2.26 M2V Luyten 726-8A 0.387 2.6 Luyten 726-8B 13.02 -4.40 Sirius A 0.377 -1.46 1.37 A1V Sirius B 8.30 -2.52 DA Ross 154 0.345 2.9 10.45 -3.32 M4V Majority (7/11 = 64%) of the stars in double or triple systems Majority (9/11 = 82%) have spectral type later than the Sun and/or are fainter Only one brighter main sequence star (Sirius A) One White Dwarf (Sirius B)

3.3.3 The nearby stars ? (c) The problem of incompleteness
The 5 pc sample is expected to be highly complete. However, the number of stars within 5 pc is too small for statistics. The sphere around the Sun up to 25 pc is usually considered the immediate solar neighbourhood. It contains ~ 2000 known stars. From the extrapolation of the local density at r < 5 pc it is expected that ~63% of the stars in the immediate solar neighbourhood are still missing in current catalogues. Distance r (pc) 300 200 100 Number of systems within r Expected number of systems for a constant space density Known systems Assumed to be completely known The (in)completenes of the stellar database for the nearby stars ? 5 pc sample 10 pc sample 25 pc sample 47 stars known complete (?) 229 known 130 missing (36%) ~2000 known ~3500 missing (63%)

3.3.3 The nearby stars (d) The stellar luminosity function
The solar neighbourhod is the only region in the Universe where we have a chance to measure statistical properties of a volume-limited sample of stars. incomplete An important statistical function is the luminosity function = distribution function of the luminosities, or the absolute magnitudes, of the stars (Fig.) Note the strong peak at faint absolute magnitudes (M ~ 10-15), which mainly represents M dwarfs (see Table 3pc sample). V

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