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Massimo Capaccioli » 11.Cosmic distance scale


Introduction

Distance D is a fundamental parameter to qualify observations of celestial objects and related phenomena, and to understand the physics that rules all of them. Usually, we measure directly angular sizes, \alpha, and fluxes, F_\lambda, only. These observables are related to actual sizes, R, luminosities, L, then masses, M, mass-to-light ratios, M/L, etc. as follows:

R=\alpha D \propto D;

L=4\pi D^2 F \propto D^2 F (neglecting extinction and all cosmological effects);

M=(\sigma_0^2/G)\alpha D \propto D, where \sigma_0 is a distance independent estimate kinematical parameter (a circular velocity);

M/L\propto D^{-1}.

Astronomical distances must be related to the standard unit (meter) used in terrestrial physics. In cosmology, attention must be paid to understand the difference between luminosity distance, D_L, and angular diameter distance, D_s = D_L/\left(1+z\right)^2.Classically, there are only two direct methods to measure the distances of celestial bodies.

  • One is based on the trigonometric parallax: it is linked to the standard metric unit and applies only to objects relatively close to us.
  • The other is based on the so called Hubble law, i.e. to the dependence of distance D on redshift z: it works better and better as the distance of the sources increases but has no direct link to the metric unit and in fact must be calibrated externally.

Trigonometric distance

Trigonometric parallax is based on a purely geometric effect and is therefore independent of any physical assumptions. Due to the motion of the Earth around the Sun, the positions of nearby stars on the sphere change relative to those of very distant sources (e.g., extragalactic objects such as quasars). The latter therefore define a fixed reference frame on the sphere. In the course of a year the apparent position of a nearby star follows an ellipse on the sphere, the semimajor axis of which is called the parallax \theta. The parallax depends on the radius r of the Earth’s orbit, hence on the Earth-Sun distance which is, by definition, one astronomical unit 1 AU. Furthermore, the parallax depends on the distance D of the star,   \frac{1}{D} = \tan{\theta} \approx \theta. The trigonometric parallax is also used to define the common unit of distance in astronomy: one parsec (pc) is the distance of a hypothetical source for which the parallax is exactly \theta = 1''. With the conversion of arcseconds to radians, 1 pc = 206265 AU = 3.086 \times 10^{18} cm.

Geometry of the trigonometric annual parallax. Proxima Cen has the largest stellar parallax, θ = 0”.7723 ± 0”.0024

Geometry of the trigonometric annual parallax. Proxima Cen has the largest stellar parallax, θ = 0''.7723 ± 0''.0024


Trigonometric distance: Hipparcos

As distance increases, trigonometric parallax becomes more and more difficult to measure. At D=100\ pc, the parallax is so small that cannot be no longer measured from Earth because of atmospheric blurring [How small? Why not measurable?]. The limit is extended by a factor up to 5 using ad hoc space observatories such as the European Space Agency satellite Hipparcos (HIgh-Precision PARallax COllecting Satellite). Its catalogue Ticho-2 contains positions, proper motions, and 2 color photometry of over 2.5 million Milky Way stars.

Distance within the Solar System

The corrected third Kepler law: \frac{P^2}{4 \pi^2}= \frac{a^3}{G(M_{\odot}+m)}, provides the relative distance from the Sun of each planet of mass m. For instance, writing P^2=ka^3 (possible as M_\odot\gg m), then k=1 if ais in Astronomical Units and P in sidereal years. In order to know the value of 1 AU, we need to find the distance of one body from the Sun or, better, from us at a given epoch. This may be done by measuring the flight-time of a radar signal reflected by the body:2t=2D/c. In order to appreciate the complexity of this method, consider that the power P_r that we get back for a given transmitted power P_t is proportional to the cross-section \sigma of the target and to the aperture A of the receiver, and it dims with D^4: P_r=P_t\left(\sigma AF/D^4 \right). Note also that that the resolution depends on the time-length of the wave-packet and that the accuracy depends of the ability to model the pattern propagation factor F (that is, the value of c in the various environments crossed by the signal).[Search in the web how Eratosthenes and Aristarchus measured the radius of the Earth and the distances to the Moon and the Sun.]

Laser ranging technique to measure distances of Solar System bodies close enough to the Earth.

Laser ranging technique to measure distances of Solar System bodies close enough to the Earth.


Pros and cons of parallaxes

Pros:

  1. Fundamental geometric measurement of distance.
  2. It can be measured directly.
  3. It is used as a primary calibrator for other less direct distance indicators [see ahead], thus constructing a cosmic ladder.

Cons:

  1. It is limited to nearby stars (D \ll 1\ kpc).
  2. Ultimately, even our estimates of distances to the most remote galaxies rests on a reliable measure of parallax to the nearest stars.
Flux conservation. [Always? Write the condition mathematically.]

Flux conservation. [Always? Write the condition mathematically.]


Dynamic or moving cluster parallax

Let’s assume that a cluster of stars of diameter d (invariable) moves (away from or towards us) with a velocity V, identical for all stars. The figures explains how to derive the distance from measurements of radial velocity and proper motion.

Let's assume that a cluster of stars of diameter d (invariable) moves (away from or towards us) with a velocity V, identical for all stars. The figures explains how to derive the distance from measurements of radial velocity and proper motion.


The Hubble law

In 1929 Hubble showed the existence of a correlation between distances of galaxies and their redshifts. This had been predicted by the dynamical solutions of Einstein’s equation derived by Friedman, Lemaitre, Robertson and Walker (FLRW), which describe a simply connected, homogeneous, isotropic, expanding or contracting universe. For any such cosmology the correlation is linear, cz=H_0 D, up to rather large distances (where higher order terms, such as the acceleration parameter q_0, come into play). The proportionality constant H_0, named Hubble constant, measures, at the present time, the ratio of the rate of change of the cosmological scale factor R to its current value, and is currently given in km/s/Mpc. Its reciprocal is the Hubble time, t_{H}=1/H_0, which is an indication of the age of the universe. The figure shows a modern Hubble diagram for the brightest cluster galaxies (BCGs) compared with Hubble’s discovery paper original plot.

Modern Hubble law for the brightest cluster galaxies (BCGs) compared with Hubble’s discovery paper original plot.

Modern Hubble law for the brightest cluster galaxies (BCGs) compared with Hubble's discovery paper original plot.


The realm of distance indicators

In principle, the Hubble diagram can be built by:

  1. identifying a set of objects (standard candles) having the same absolute magnitude M, even if the latter has an unknown value;
  2. measuring the apparent magnitudes of these objects, m_c, properly corrected (see later), so that the relation: m_c-M=5\log D - 5, provides distances D, although in an unknown scale;
  3. measuring cosmological redshifts cz_0 of these objects, i.e. the recession velocities cleaned by the peculiar motions of the observer and of the sources.

Since peculiar motions of galaxies are superiorly limited, their contribution to the measured redshifts decreases with 500\ {\it pc}<D<1~Mpc, where neither one of the two methods is utilizable (see figure in the next page). The gap must be filled up, as we want to calibrate the Hubble law (i.e. find the value of H_0) against the trigonometric parallax, which in turn is linked to the standard metric unit. To this purpose we need to use the so called distance indicators (see the figure in the next page).

The “gap” between direct distance ladders

The figure shows how the increasing errors with distance for parallaxes and decreasing for the Hubble law create a gap between the two direct methods, from  D= 500 pc to D 1 Mpc

The figure shows how the increasing errors with distance for parallaxes and decreasing for the Hubble law create a gap between the two direct methods, from D= 500 pc to D 1 Mpc


Gauging distance beyond parallaxes

One of the major astronomical challenges in the second half of the XX century has been the determination of the value of the Hubble constant, of paramount importance for its cosmological significance. In fact, H_0 provides an estimate of the age and the “size” (better, the scale) of the universe. Its current value is 67.15 \pm 1.2 \frac{\frac{km}{s}}{Mpc} (ESA’s Planck Mission 2013) and more will be illustrated in the coming lectures. Here we recap the motions of the observer.

The “evolution” of H0

How the value of the Hubble constant has changed since the discovery of the Hubble law, in 1930. Most of the progress is due to the late Allan Sandage and Gerard de Vaucouleurs.

How the value of the Hubble constant has changed since the discovery of the Hubble law, in 1930. Most of the progress is due to the late Allan Sandage and Gerard de Vaucouleurs.


Motions of the observer

Remember the correct definition of Astronomical Unit: radius of the circular orbit of a massless particle orbiting about the Sun in one sidereal year.

Remember the correct definition of Astronomical Unit: radius of the circular orbit of a massless particle orbiting about the Sun in one sidereal year.


Motion about the Sun


Reducing observed radial velocities to the Sun

Radial velocities spectroscopically measured need to be reduced to the Sun, so to remove the component due to the orbital velocity of the Earth projected unto the direction of the target at the time the observation is made. Consider an equatorial system centered on Earth, with the x-axis towards the Vernal Equinox (unit vectors: \boldsymbol{i},\boldsymbol{j},\boldsymbol{k}). Let {\bf V}_* be the radial velocity vector of the target seen by an observer in the direction of the unit vector {\bf r}_*, and {\bf V}_\odot the apparent orbital velocity of the Sun (opposite to that of the Earth {\bf V}_E) at the time of the observation.

It is: {\bf V}_{obs}=\left({\bf V}_*-{\bf V}_E\right)\cdot {\bf r}_*= \left({\bf V}_*+{\bf V}_\odot\right)\cdot {\bf r}_*,

where:

\boldsymbol{V_{\odot}} = (30 km / s) \times [\cos{(\lambda_{\odot} - \pi/2)}\boldsymbol{i} + \sin{(\lambda_{\odot}- \pi/2)} \boldsymbol{j}]  , with \lambda_\odot the ecliptic longitude of the Sun at the time of the observation (there are simple formulae to compute it from the observing time expressed in Julian days [what is it?]).


The Local Standard of Rest

We define as Local Standard of Rest (LSR) to a fictitious point S_{LSR} in Solar neighborhood (thus staying on the Galactic Plane) which moves on a circular orbit about the center of the Milky Way. Let’s now built a reference system centered at the Galactic Center, with R as the radial coordinate pointing to S_{LSR}, q as an azimuthal angle on the Galactic Plane, and z as the distance from the Plane. The corresponding velocities are:

P=dR/dt, Q=R(dq/dt), Z=dz/dt.

The rotational rate at the distance of the Sun gives for the LSR:

P_{LSR}=0,Q_{LSR}=Q_0, and Z_{LSR}=0,

where Q_0 is the circular velocity at the Sun’s distance.

Peculiar velocities are then defined as:

U=P-P_{LSR}=P, V=Q-Q_{LSR}=Q-Q_0, W=Z-Z_{LSR}=Z.

Let’s find out a way to measure the peculiar (residual) velocity of the Sun.


Peculiar motion of the Sun

Let’s consider a system of N particles with velocities {\bf v}_i relative to the baricenter. Let us sum up the velocities of the N-1 particles with respect to the N-th.Since the average of all velocities is zero:

\displystyle{{\bf v}_N=-\sum_1^{N-1} {\bf v}_i.}

Moreover:

\sum_1^{N-1}\left( {\bf v}_i- {\bf v}_N\right)=\sum_1^{N-1} {\bf v}_i- {\bf v}_ N\left(N-1\right).

Thus: {\bf v}_N=-\frac{\sum_1^{N-1}\left( {\bf v}_i- {\bf v}_N\right)}{N}.

The speed of the Sun relative to the LSR: U =-10 km/s, V = +5 km/s, W = +7 km/s.

The direction of the Solar Apex is SW of the star Vega (constellation of Hercules); its coordinates are: R.A. = 18h 28m 00s, Dec. = +30°, lG= 56.24°, bG=22.54°.

 

The Virgo cluster

The Virgo cluster of galaxies.

The Virgo cluster of galaxies.


Recap of the formulae for the redshift

The Doppler effect requires different formulae to link the spectral shift z to the velocity V of the source relative to the observer. For small velocities compared to the speed of the light, V\ll c, the classical, purely radial Doppler effect is simply: \displaystyle{z=\frac{\Delta\lambda}{\lambda_0}=\frac{V}{c}}. In Special Relativity the redshift depends on the radial component of the velocity as: \displaystyle{ 1+z=\sqrt{\frac{1+\beta}{1-\beta}}\simeq 1+\frac{V_r}{c} + \dots} where: \beta=V/c. There is also a transverse effect, where the frequency measured by the observer, \nu_0, is never larger than that of the source, \nu_s: \displaystyle{\nu_0=\frac{\nu_s}{\gamma\left(1+\beta\cos\theta_0\right)}}, where: \displaystyle{ \gamma=1/\sqrt{1-\beta^2}}, and \theta_0 is the angle between the direction of the emitter and the of the incoming light, in the frame of the observer. In General Relativity there is a redshift due to the gravitational effects of a mass M. It writes: z = \frac{1}{\sqrt{1 - r_{s}/r}}  where  \displaystyle{ r_S=\frac{2GM}{c^2}} is the Schwarzschild radius, and r is a distance in the Schwarzschild metric.

Exercise: compute the parallax ellipse

Let’s make the exercise of spherical trigonometry of computing the trajectory that a star makes in a year on the celestial sphere .owing to the parallx.

Let's make the exercise of spherical trigonometry of computing the trajectory that a star makes in a year on the celestial sphere .owing to the parallx.


Compute the parallax ellipse


Compute the parallax ellipse


Compute the parallax ellipse


Compute the parallax ellipse


Compute the parallax ellipse


Compute the parallax ellipse


I materiali di supporto della lezione

G. Bertin, Dynamics of Galaxies. Cambridge Univ. Press, 2000.

J. Binney & S. Tremaine, Galactic Dynamics, Capter 6, Cambridge Univ. Press, 1987.

J. Binney & S. Tremaine, Galactic Astronomy, Capter 10, Cambridge Univ. Press, 1998.

C. C. Lin & F. H. Shu 1964, Ap.J., 140, 646, 1964.

A. Toomre & J. Toomre, Ap.J., 178, 623, 1972.

W. Freedman, "The Measure of Cosmological Parameters", stro-ph/0202006.

R. Sanders, "Observational Cosmology" astro-ph/0402065.

J. Jensen, J. Tonry and J. Blakeslee, "The Extragalactic Distance Scale", astro-ph/0304427.

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