1 Introduction

• Various methods to measure distances from Solar System to cosmological scales
• Geometrical distances: the trigonometic parallax
  • Below ∼1kpc, the most accurate, simplest, and with least assumptions, method to measure distance, is trigonometric parallax
• For distances larger than ∼ 1kpc:
  • Photometric distances: using stars as reference candles
  • Galactic rotation curve
  • Light echoes
  • Supernova
  • Empirical scaling laws (e.g. Tully-Fisher)
  • Hubble's law

2 Trigonometric parallax

See here

3 Photometric distance

• General idea and main issue
  • What is measured is an apparent magnitude; if we knew the absolute magnitude, we would readily obtain the distance modulus DM=m-M, hence the distance d_pc in pc from DM=5log(d_pc)-5.
  • How to measure the absolute magnitude?
• Photometric methods
  • Color magnitudes of clusters
  • Standard candles: there are stars for which we know the luminosity from first principles (stellar evolution models) calibrated observationally. These stars are called standard candles: their luminosity is universal. Comparison of m and M gives DM. provided A is known.
  • Somehow, these methods rely on models to understand how a star can behave as standard candle, and to ensure that the assumptions are valid

3.1 Color-color diagram of a star cluster

• For example, U-B against B-V, for a star cluster
  • In a cluster: ∼ co-eval stars at the same distance (the size of the cluster must be negligible compared to ist distance)
• Various photometric HRD-like diagrams: Color-Color, Absolute Magnitude-Color

• General idea:
  • Comparing Color-Color diagrams of clusters serve to compare their distance; if the distance to one of these is known, then the distance of the others is known
  • Alternatively, one may compare the main sequence of C-CD to a calibrated main-sequence and make them match
• Problem: reddening.
  • Magnitude: m = M + DM + A, DM=5log10(d/pc)-5
  • As we have seen: colors correspond to flux ratios hence are distance-independent; U-B=(M_U-M_B)+(A_U-A_V)=(M_U-M_B)+E(U-B)
• Two-step method:
  • determine the reddening E(B-V): shift the observed main-sequence along the reddening vector in the CCD until it matches an 'unreddened' m-s (determined from neighbouring stars); the direction of the reddening vector depends upon dust properties, its amplitude depends on the extinction coefficient (A_V);
  • determine DM: shift vertically (i.e. along M-axis)

3.2 Standard candles and variables stars

• Variable stars (δ Cepheid, RR Lyrae): these stars have a periodic brightness modulation and the physics of their interior makes the period to depend uniquely on their luminosity, also known as Levitt-relation, P∼ L^α ⇒ measure the period, find the luminosity, compare to the measured flux and derive the distance
• Type Ia Supernova: the key feature of these exploding stars is that the luminosity they reach at maximum and the decrease of this luminosity with time – the light curve – is universal: measure the magnitude on a portion of the curve, compare with the universal light curve, find the absolute magnitude, hence the distance modulus
• Standard candles are the best tools to measure distances on galactic and intergalactic scales up to z∼1.5.
• Calibration of P-L relation or SNIa light curve is critical, but is extremely difficult in practice: metallicity effects, reddening.

Life cycle of stars

From the main sequence to red giant phase

• Plots: evolutionary track for a 5M_sun (intermediate mass) star
• Main sequence (A to C): core H-burning lasts ≈ 80 Myr
• C to D: core reaches the Schonberg-Chandrasekhar limit before He-core becomes degenerate (for M>2-2.5 M_sun)
  • core contraction, envelope expansion (R increases) to conserve Ω
  • H-burning in a shell surrounding the core
  • when T in the core reaches 10⁸ K, core He-burning; core contraction stops
  • new (thermal and hydrostatic) equilibrium:
  • very fast evolution (Kelvin-Helmoltz timescale, ∼2 Myr)
• Reg Giant Branch (RGB) (D to E)
  • extremely fast !
• red giant (at point E)
  • Helium burning in the core: red giant phase (path from D to E)
  • close the Hayashi line: deep convective zone
  • strong T-dependence of He-burning: convective core
• Similar evolution for M=2.5-10 M_sun

… to pulsating variable stars

Maeder 2009

3.3 Cepheids

3.3.1 Leavitt law

Period-luminosity curve of δ Cep

25 variable stars in the SMC, Leavitt & Pickering 1912

δ Cephei, a 4th magnitude F5 supergiant; P=5.37 d (from Stebbins 1908)

The Cepheid period-luminosity relation

• Leavitt law: Period-Luminosity relation discovered by Henrietta Leavitt early 20th century
• Period: 2 to 100 d and intrinsic brightness -2 < M_V < -6 mag
• Measure P and m; P-M gives M, and comparison with apparent magnitude gives the distance modulus DM=5log(d)-5
• Calibration of the P-M (or P-L) relation: trigonometric parallax

The Period-Luminosity-Color relation

• A fundamental relation between the period and the density can be obtained easily:
  • Period ∼ sound crossing-time = R/c_s ∝ R/T^1/2
  • Virial equilibrium: E_pot = 2E_kin or kT=GMm_p/R ⇒ P ∼ ρ^-1/2
  • In practice: P = Q \(( \left<\rho\right> / \left<\rho_\odot\right> )^{-1/2}\), w/ Q = 0.035 - 0.050 days for Cepheids
• Period-Luminosity-Color (PLC) relation:
  • Stefan-Boltzmann law: L = 4π R² σ T_eff⁴ rewritten as M_bol = -5logR -10logT_eff + C
  • Map log T_eff as an intrinsic colour (B-V)₀
  • Use the Mass-Luminosity scaling law for nuclear burning stars in equilibrium to obtain a Period-Luminosity relation: L∝ M^α with M the mass, which depends on the average density
  • We obtain a general expression: M_V = α log P + β (B-V)₀ + γ
  • Numerically: log P = (3/4-1/2α)log(L/L_sun) - 3log T_eff + log Q + cst
  • Adopting α=3.3 for Cepheids, this gives: log (L/L_sun) = 1.67 log P + 5 log T_eff - 1.67 log Q + cst'

  • Translated in terms of absolute magnitude

    M = M₀ - 2.5log L = M₀ - 4.2 log₁₀ P + 12.5 log T_eff

Dealing with extinction: Observed Magnitude-Period

• Classical Cepheids (prototype δ Cephei):
• Giants to supergiants, young intermediate-mass stars, found in the disk population and in young clusters;
• Period 1 to 100 d
• Disk midplane implies that reddening is important: observe at longer wavelength
• Note that at longer wavelength (J, H, K bands), the Magnitude-Period relation is steeper and with less scatter, hence more accurate

Tammann et al A&A 2003

Freedman & Madore ARAA 2010

3.3.2 Advantages and limitations

• δCepheids:
  • Young stars, located close to the Galactic Plane: extinction is important
  • Bright, hence can be used over intergalactic distances: M_V ∈ [-8:-2]
    • Can be seen with HST in host galaxies of SNe Ia at d up to 50 Mpc
  • But only long-period (P>10 d) are bright enough
  • In the MW, ≈ all long-period Cepheids live at d>1 kpc ⇒ parallax precision better than 100\(\mu\)as
• RR Lyrae (subgiants to giants)
  • Population II stars, metal-poor
  • Found in globular clusters in the halo and bulge
  • Lower mass than Cepheids and not as bright as Cepheids: M_V≈ 0.6
  • Period 0.2 to 1 d
  • But M_V spreads over a narrow interval: M_V ∈ [0.5:1.0] (depends on metallicity): nearly constant M_V
    • Note: metallicity is defined as [X/H] = log10[n(X)/n(H)]_* - log[n(X)/n(H)]_Sun
    • [X/H]=-1 means that element X the star has 1/10th the Solar abundance

The variable stars zoo

Gaia DR2

Pulsating stars, Catelan & Smith 2015

The Gaia view of variable stars

Gaia DR2 and here for a movie

4 Light echoes

• Light echoes: interaction of light with ambient material
  • light of a transient event scattered by a dust cloud in the vicinity of a mass loss star (e.g. RS Pup)
  • SN explosions
• Measurement:
  • a time series showing different parts shining progressively
  • difference in time gives the distance (assumptions on the geometry, light emission mechanism)

4.1 Geometry of light echoes

4.2 Light echo from SNIa and the distance to the LMC

Light echo from SNIa 1987A in the LMC

• Interaction of light emitted by the explosion reaches a ring of gas left by the star before the explosion
• Why it is a ring and not a sphere is actually not known

• Light curves emitted by atoms ionized by photons from the explosion:
  • Finite speed of light: different arrival times from different parts of the ring
  • No light until t₀; then, closest part shines first; max. intensity at t_max, when entire ring is illuminated
• Recover the ring inclination, t₀, and t_max: actual size of the ring
• Actual size / angular size = distance; derived distance to the LMC: 52±3 kpc

4.3 Light echo from supernovae

Yang et al ApJ 2017

4.4 Light echo from Cepheids

Kervella et al 2008

• Goal: calibration of long-period Cepheids; long-period Cepheids (the brightest) are used to measure extragalactic distances
• Idea: see the modulation of the reflection nebula by the Cepheid light curve + different arrival time of light from the Cepheid to various locations in the surrounding gas
• RS Pup: a 41.4 d period Cepheid is located ≈ 2 kpc; trigonometric parallax is uncertain
  • Observations with 3.6 m ESO New Technology Telescope (NTT), La Silla; ESO Multi-Mode Instrument (EMMI): multipurpose imager and spectrograph
  • Phase shift: propagation time due to light speed: projected distance hence distance to 1.4% accuracy (1992±28 pc)

5 Eclipsing binaries

• Roughly 50% of stars are found in multiple systems; some are eclipsing ones;
• Several types of binary systems, classified according to the detection technique: optical double, visual binary, astrometric binary, spectroscopic binary, eclipsing binary
• Eclipsing binaries are classified according to their separation compared to the size of their equipotential surfaces called Roche surfaces
• Detached eclipsing binaries (DEB) are the key to a 1% determination of H₀ by Riess et al 2019
• DEB systems provide geometrical mean to measure the distance

5.1 Types of eclipsing binaries

Equipotential surfaces

5.2 Detached eclipsing binaries

• DEBs present several features that make them well suited for distance determination
  • radii smaller than Roche lobes
  • light curves show well separated absorption dips: this allows accurate determinations of the radius and mass of each star
  • when radii much smaller than Roche lobes, stars are little deformed and ∼ spherical: spectral analysis is simpler (T_eff and log(g) are ∼ uniform)
• Mass and radii: from Kepler's 1st law, both objects in a binary orbit move about the center of mass in ellipses, with the center of mass occupying one focus of each ellipse. For simplicity, let us assume circular orbits seen with an inclination angle i≈90°. More general expressions can be found in general textbooks.

Mass and orbits derivation

• Kepler's 3rd law gives the sum of the masses: \[ P^2 = \frac{4\pi^2}{G(m_1 + m_2)} a^3 \]
• Since a=a₁ + a₂ = P(v₁ + v₂)/(2π), we obtain \[ m_1 + m_2 = \frac{P}{2\pi G}\frac{(v_{1,r} + v_{2,r})^3}{\sin^3 i} \]
• Finally, if we know the inclination \(i\), the measurement of P, v_1,r and v_2,r lead to a₁, a₂, m₁, and m₂

Stellar radii and effective temperature ratio from the light curve

Flux measured in the three phases 1, 2, and 3:

• Total luminosity of a star: L=4π R² σ T_eff⁴, and the measured flux is \(S=f\times L/4\pi d^2\), where \(f\) is a reduction factor (distance, reddening, atmosphere, detector, etc)
  • \(S_1 = S_s + S_l = f \sigma (T_s^4 R_s^2 + T_l^4 R_l^2)\)
  • \(S_2 = S_l = f \sigma (T_l^4 R_l^2 - T_s^4 R_s^2)\)
  • \(S_3 = f \sigma [T_l^4 (R_l^2 - R_s^2) + T_s^4 R_s^2) = S_l(1-R_s^2/R_l^2) + S_s\)

• We thus have: \[ \frac{S_1-S_2}{S_1-S_3} = \frac{F_s}{F_l} = \left(\frac{T_s}{T_l}\right)^4 \]
• Note also that \(\rho^2 = R_s^2/R_l^2 = (S_1-S_3)/S_2\)

Why using late-type DEBs ?

• Distance to the DEB can be obtained from: \[ d^2 = \frac{R_l^2 F_l}{S_3/f} [(1-\rho^2) + \frac{F_s}{F_l}\rho^2] \] where we note F=L/(4π R²) = σ T_eff⁴ the surface flux.
• The above expression is useful because it involves, as far as possible, ratios instead of absolute measurements;
• However, we need to know \(f\), and the intrinsic surface flux F_l:
  • For F_l: either T_l and F_l(T_l) from stellar atmosphere models, or from an empirical relation (V-K) vs F(V-K)
  • For \(f\): requires the extinction curve and the total-to-selective absorption R_V

    • Exercise: Show that f=10^[-0.4 E(B-V) (k(λ-V) + R_V)], with k(λ-V) the normalized extinction curve E(λ-V)/E(B-V)

  • Atmospheric lines (e.g. Balmer lines, HeI or HeII) can strongly constrain the atmosphere model (log g)
  • SED of the star constrains E(B-V) and R_λ
  • However, for early-type stars (meaning O, B), accurate flux calibration is difficult (atmospheric models are NLTE; metallicity); but most of all, the extinction curve for these stars is poorly constrained (see this article for a detailed discussion on the limitations of early-type stars). Final precision is ∼5-10%.

5.3 Late-type DEBs in the LMC

• Articles: Pietrzyński et al 2013 and Pietrzyński et al 2019
• Main advantage: accurate (2%) relation between the surface-brightness/colour relation (SBCR) between the surface brightness F_l and the V-K colour excess
• Method:
  • Use the Optical Gravitational Lensing Experiment (OGLE) which provides time-dependent photometry for 35 million stars in the LMC for >16 yr !
  • Main uncertainties: SBCR (value of R_V has little influence: changing from 3.1 to 2.7 translates to 0.3% uncertainty on the distance) and zero-point of photometry;
  • Total error budget (2013 paper) is 2.4% on the distance

Late type DEBS in the LMC

• Distance to LMC barycenter (2013):
  • Distance modulus: 18.493 ± 0.008 (stat) ± 0.047 (syst)
  • d=49.97 kpc ±0.19(stat)± 1.11 (syst)

The key to a 1% fundation of the H₀ constant: improved SBCR

• SBCR obtained by combining precise NIR and V-band photometry and angular diameter precise to 1% with ESO/VLTI/Pionier; Surface brightness S_V: S_V = α (V-K)₀ + β
• Increase early-type DEBs sample size: from 8 to 20; decreases statistical uncertainties

• Final error budget:

  The systematic uncertainty on the LMC distance includes the following contributions: calibration of the surface brightness–colour relation (0.018 mag, or 0.8%), photometric zero points (0.01 mag for both V and K bands, or 0.5%), and reddening absolute scale (0.013 mag, or 0.6%). Combining these quadratically we obtain 0.026 mag, or 1.1%.

• Actually, 0.026 mag is 1.2% uncertainty
• Additional sources of uncertainty: metallicity has a limited impact;
• Distance modulus: DM=18.476±0.004 (statistical) ± 0.026 (systematic) mag

The distance ladder

6 Cosmological distances

• Studies of external galaxies (star formation history, etc)
• Study of the large scale structures (galaxy clusters, Big Wall, etc)
• Models of the Universe

6.1 The first extragalactic object

• Using a Cepheid, E. Hubble (1927) was able to compute the distance to the Andromeda Galaxy (M31)
• His value, 300 kpc (actually a factor two lower than the modern determination) implies that M31 is outside the M-W.
• This was the first proof for the existence of structures outside the MW

Nowadays

• Total: 1566 Cepheids in 19 SNIa hosts monitored w/ WFC3 (Riess et al 2016)

6.2 Hubble's law

• Cosmic expansion:

\[ \boxed{v = cz = H_0 D} \]

• Redshift z is easily measured
• Current expansion rate: H₀ ≈ 70 km/s/Mpc (Planck 2018 value: 67.7±0.4 km/s/Mpc)
• Assuming H₀ is known, measure of z gives the distance. Yet, to apply this law:
  • Get rid of peculiar motions (galaxy velocities in clusters, etc) ≪ cz
  • Distances must be large enough for z to be dominated by cosmic expansion (random velocities due to gravitational interactions, e.g. w/ galaxies in the vicinity, remain small compared to the Hubble velocity)

6.2.1 Our peculiar motion

• Our galaxy is moving
  • the MW is part of the Virgo Cluster
  • gravitational attraction caused by the cluster mass
• CMB dipole anisotropy
  • CMB is isotropic
  • but appears anisotropic due to the motion of the Solar System: v_Sun/CMB = 369.82±0.11 km/s towards (l,b)=(264°,48°)
  • The amplitude of the dipole is 3362.08±0.99 μ K_CMB. Can you recover the value of v_Sun/CMB ?
• Local Group wrt CMB: v_LG=620±15 km/s

6.3 Measuring cosmology with Supernovae

• Supernovae are the brightest events: how to use them as distance indicators ?
• Supernovae: based on their optical spectra, four types
  • Type Ia: a white dwarf (degenerate electron core) in a binary system is brought above the Chandrasekhar limit (M_ch ≈ 1.44 M_sun) by accretion from a giant companion; collapse and rebound, leaving only a degenerate gas of neutrons (neutron stars, pulsars); one example is the Crab Nebula (explosed in 1054); SNIa are the most luminous and homogeneous;
  • Type Ib,c: massive star undergoing core collapse
  • Type II: mass > 8 M_sun; no degenerate core; complete explosion; used to measure distance with the expanding photosphere method;
• Supernovae:
  • intrinsic brightness (observable in the distant Universe)
  • ubiquity (both nearby and distant Universe)
  • type Ia provide accurate (8%) distance measurements
  • type II provide distance accuracy ≈ 10%
• acceleration of universe expansion
• Nobel Prize 2011: Perlmutter, Riess, and Schmidt

SN Ia light curve

• Decay rate of luminosity correlates with absolute magnitude
• Applies to Branch Normal SNIa and also to peculiar type Ia

Phillips ApJ 1993

SN Ia light curve

• Decay rate of luminosity correlates with absolute magnitude
• Universal light curve in each band; and also for color index
• Light curve is strongly wavelength dependent
• However, time of maximum magnitude depends on photometric band (reddenning): taking B max. as reference, U-max is reached 2.8 days before, while V-max is reached 2.5 after.
• Correct for interstellar reddenning (multi-λ)

The B band light curve of 22 SNe Ia

⇒ Type Ia SNe can be used as standardized candles

Distinguishing cosmological models

• Need to find high-z SN Ia
• Problem: occurence rate of SN Ia is weak; few times per Myr in MW-type galaxy
• 4m-class telescopes: 1/3 degree² down to R=24 mag in less than 10min → 10⁶ galaxies to z<0.5 in one night
• It takes ∼ 20 days to reach maximum luminosity ≈ 14 rest frame days at z=0.5 → observe the same fields three weeks apart (before and after full moon)
• K-correction for distant SN Ia: photometric bands must be redshifted

6.3.1 A Hubble diagram for z<0.4

Riess et al 2016

6.3.2 The distance ladder

• use various distance tools from local to distant Universe
• use one tool to calibrate another to be used at a larger distance, etc

The Cepheid-SNIa distance ladder (Riess et al 2016)

6.4 Indirect distance estimator: the Tully-Fisher relation

• The Tully-Fisher (1977) relation links the maximum rotation speed of a disk galaxy with its luminosity, that is, the stellar mass \[L \propto v_{\rm max}^\alpha,\quad\alpha\approx4\]
• \(v_{\rm max}\) is a distance-independent quantity, while the stellar (or baryonic) mass does depend on the distance
• Proper calibration of the Tully-Fisher relation thus provides a distance measurement tool, in the local universe;
• TF relation applies to spiral galaxies (similar relation for elliptical is the Faber-Jackson relation)
• Explaining the TF scaling is a challenge because the rotation speed not only depends on the baryonic mass and size, but also on the radial distribution of the dark mass which itself depends on the dark matter halo on larger scales;
• TF relation may be fundamentally linked to the baryonic mass however (McGaugh et al 2000)
• Although simple arguments can be used to understand why the rotation speed may scale with the stellar mass, the TF (or FJ) relation are not entirely understood.

The Tully-Fisher method

• Left: Calibrated TF relation for galaxies from Local, Sculptor, and M81 groups; absolute magnitude against the H21cm linewidth; distances obtained from Cepheids; (Pierce & Tully 1992)
• Middle: determination of v_max from H21 cm (Macri et al 2000)
• Right: Absolute H-band magnitude M_H against H21cm linewidth (Aaronson et al 1982). Slope=10, hence L∝ Δv⁴

Tully-Fisher method in practice

• v_max is measured
  • spatially resolved rotation curve – v_rot(θ) – possible for nearby galaxies
  • determine the edge of an average H21cm spectrum: FWHM ∼ 2 v_max
• Less scatter in red bands (less extinction)
  • Steeper (hence more accurate) in H band
• Uncertainties: cosmic scatter, photometric (extinction correction) and velocity width measurements (inclination), distance assignments

Physical basis of the Tully-Fisher relation

• We can show that M∝ v_max⁴
  • Observations show that M/L_H (H-band luminosity) is ≈ constant for all spirals
  • Therefore L∝ v_max⁴
  • How to show that M∝ v_max⁴ ?
    • Virialized baryonic mass distribution: 2T+Ω=0 gives Mv_max² ∼ M²/R_max or M∼ R_maxv_max²
    • Mean surface brightness <I>∼ L/R_max²; observationally, <I> ∼ cst (Freeman's law)
    • Also, M/L_H ∼ cst for all spirals
    • Thus: L_H ∼ M and R_max ∼ L_H^1/2 hence L_H ∼ L_H^1/2 v_max² which leads to L_H ∼ v_max⁴