The Scale of the Universe The Measurement of distance in our Univers!
Chapters 12.1.1 Allday; Chapter 3 Silk
Measurement of Distance in the Universe
Two IMPORTANT concepts that you should know well from this topic:
1. the cosmological distance ladder: what it is, and the various “rungs”.
2. a standard candle: what it is, how you use one, what objects make good standard candles.
Overview: the Cosmological Distance Ladder
Geometric-based Standard candle-based Hubble’s law
1. “Spiral Nebulae” (1600-1929): Are they nebulae in our Milky Way, or are they galaxies like the MW?
Three Major Distance Controversies in the Recent History of Astronomy
2. Quasars: (1963-~1980) Are they strange objects shot out by our MW / nearby galaxies, or are they very luminous nucleus of distant galaxies?
3. Gamma Ray Bursters: (1967-1995): Are they exploding star remnants in our MW, or are they super explosive events out at the edge our observable universe?
The Expanding Universe: The Discovery of Galaxies
Our nearest large galaxy is Andromeda (M31) first mentioned in 1612 (soon after the invention of telescopes).
These objects were called “spiral nebulae”, and there were ~15000 cataloged by 1908. But it was not clear what these objects are (because their distances were unknown).
In 1912, Slipher found all “spiral nebulae”, with the exception of M31, are receding from us.
1923, Edwin Hubble determined the distance to Andromeda, using cepheid variables, settling the debate that these nebulae are galaxies.
1929: Hubble found that the speed of rescession is proportional to the distance of the galaxy from us.
Ca II doublet absorption lines (~3960Å)
spectrum of galaxy
The Hubble expansion: the spectrum shifts towards the red for more distant galaxies.
reference spectrum
Redshift and Hubble’s Law
- Define redshift z=(λ-λ0)/λ0=Δλ/λ0, where λ0=rest wavelength, and λ=observed wavelength (Note that λ/λ0=1+z) - if we interpret the redshift as a Doppler shift, then the recession velocity is: v=cΔλ/λ0=cz (note: this only works for
small z, the relativistically correct formula is (1+z)2=(c+v)/(c-v) )
The Hubble Law: cz=v=H0d, where H0=the Hubble constant, d=distance to the galaxy (note, again, this is true only for small z)
- We are able to measure distance to galaxies if we can measure redshift (easy), and know what H0 is.
The Hubble law implies that that the Universe is expanding (all galaxies are rushing away from each other), we will look into this in more details later in the next topic.
The Hubble constant has a unit of km s-1 Mpc-1, current best value: 71±3 km/s/Mpc.
Note that H0 has the equivalent unit of 1/time, hence it is also a time scale of the age of the Universe.
The Hubble constant provides a scale (both in time and space) for the universe, and is one of the most important/ fundamental constants for our Universe.
To determine the Hubble constant H0=v/d=cz/d , we need to measure the redshifts and distances of a set of galaxies. The difficulty lies in measuring the actual distances to (even) nearby galaxies.
Once we know what H0 is, we can then apply it to compute the distances to other (fainter and farther away) galaxies by simply measuring the redshift.
(Again, note that this simple relation between distance and redshift only applies to near-by galaxies (z«1). One needs to use a much more complicated full cosmological/relativistic formula for distant galaxies.)
How do we measure distance? 1. Geometric: triangulation (parallax)
2. Standard ruler (i.e., the apparent size of an object with known physical size)
3. Standard candle (i.e., the apparent brightness of an object with know luminosity)
In our everyday life, we use exactly the same three methods to estimate distances that we cannot measure directly.
The Method of Parallax
Small angle approximation: sin x ~ x (in radian)
For distance to stars, we use the baseline of the earth’s orbit. Our nearest star is α Centauri, which has a parallax of 0.75 arcsec, so that d=1.3pc=270000AU=4.3ly (reminder: 1 ly=9.46x1015m)
A scale model to give an idea of the relative distances and sizes: represent the sun by a marble of 1cm then: Earth=a grain of sand 1 m away Jupiter=a pinhead 5m away αCen= a marble 300km away!
1 parsec (i.e., 1 pc) = 206265AU = 3.09x1013km =3.2 ly is the distance when the parallax is 1 arcsecond. (1 radian =206265 arc seconds) Clearly, the parallax decreases with distance;
The distance in parsec = inverse of the parallax angle in arcseconds E.g., parallax angle = 0.5 arcsec → the distance is 2 pc
The farther away the star, the smaller the parallax angle
Definition of parsec
How far can we measure distance via parallax?
Not very far: until recently only to about 100 light years when observations were made from the ground. The problem is atmospheric turbulence (’’seeing”)
From space, we can do much better, able to measure positions to an accuracy of more than 10 times better: e.g, the Hipparcos satellite Future missions: GAIA, SIM: to be launched ~2010, able to measure parallax of stars clear across the Milky way, i.e., a range of 20kpc!
But: How do we measure distances many Mpc away?? (We need to measure distance to at least some galaxies to derive the value of the Hubble constant!)
Standard Rulers: Angular size ϑ of an object size S at distance d: ϑ=S/d, (angle measured in radian) → d1/d2=ϑ2/ϑ1
Standard Candles: Relationship between flux f and luminosity L at distance d:
For two objects of the same L, at distance d1 and d2:
Astronomy Definitions and Concepts needed for the distance ladder
The Magnitude System - Astronomers use a somewhat unwieldy system to measure brightness and luminosity: it is based on logarithm and ratio. (The human eye interprets light logarithmically.)
m=-2.5 log(f/f0), where f=flux, fo=fiducial flux reference.
Note: 1) it is a negative log system → larger number means fainter! 2) 5 magnitudes = a factor of 100 3) the star Vega has a mag of 0 by definition, and everything else is expressed relative to it.
Ground-based 4m telescope 10m telescope (CCD detectors)
Just for fun, here is a scale of what various magnitudes mean.
Absolute Magnitude M
The Absolute magnitude is the magnitude of an object if it is placed at a distance of 10 pc from the observer.
This is effectively a measure of the luminosity of objects, since they are effective all at the “same” distance.
M=m-5 log(d/10pc) (see derivation in class) (m-M=5 log(d/10pc) is called the distance modulus) Our Sun has MV=+4.8, A typical bright galaxy (like our Milky Way) has M~-22 (i.e., the Galaxy is about 50 billion times brighter than the Sun
Photometric Systems
We measure flux through sets of filters which allow different wavelength bands (i.e., colors) to pass through to the detector.
An example of an often-used system: Johnson-Cousin UVBRI system: filter name central wavelength width U: 3650Å, 680Å B: 4400 , 900 V: 5500 , 900 R: 6500 , 1200 I: 8200 , 1500
There are many filter systems in use:
We define Colour as the difference of the magnitudes between two different filters (i.e., the ratio between the fluxes in two filters), e.g. B-V, V-R etc:
e.g.: B-V=-2.5 log (fB/fV), (this assumes the same f0 for both B & V)
Note that larger B-V means redder in colour (i.e., fewer photons in the B filter than the V filter -- recall the magnitude system is negative log.)
Example: a B-V color of 2.5 means there are 10 times more luminosity in the V filter than the B filter. (102.5/2.5=10)
More massive, more luminous, bigger, bluer, hotter, shorter lived
Less massive, less luminous, smaller, redder, cooler, longer lived
Spectral classification of stars:
Our Sun is G2
Example spectra of stars of different types
The Hertzsprung-Russell (HR) diagram: a plot of color vs magnitude for stars. Color implies: spectral type and temperature; magnitude: luminosity
The main sequence is where stars spend their nominal life time, until they use up the Hydrogen in the core. They then evolve off the main sequence to become giants etc
HR-diagram of a star cluster (all stars are of the same age). Note that all the bright (and blue) stars (types O,B, A, etc) have evolved off the main sequence. The distance of the cluster can be determined by the location of the main sequence on the HR-diagram.
Cepheid stars
blue red
bright
faint
main sequence HR diagram of fields stars, contains stars of different ages evolved stars (giants)
Using HR diagram to measure distance:
By comparing the relative brightness of two clusters, we can tell their relative distances from us.
For the same stellar temperature (i.e., colour), the stars in Pleiades are ~7.5 times dimmer than those in the Hyades cluster .
Therefore: the Pleiades is 7.5½ =2.7 times further away.
We know the Hyades is 151 lyrs away → the Pleiades is 408 lyrs away
Distance Ladder: What is the length of the Golden Gate Bridge?
1. Friend: 2m 2. car ~ 2xfriend ~4m 3. big truck ~ 5xcar ~ 20m 4. container ship ~ 12x container ~240m 5. length between towers on Golden Gate ~ 5x length of ship ~1200m
Actual length of Golden Gate between towers: 1280m What do you have to worry about? Your “standard rulers” may not be identical at each rung.
The Concept of Distance Ladder
Use successively more luminous (for standard candles) or larger (for standard rulers) distance indicators as we go out further in distance, with each more distant indicator calibrated via the closer one(s).
e.g., we measure the distances to a set of closeby objects, from this set (now with known distances) we pick out a subset of (luminous, or large) objects that can be seen at much greater distances. This subset (with known luminosity or size) then serves as the next set of distance indicators (i.e., the next “rung” on the “ladder”), allowing us to estimate objects at greater distances.
( example of car, truck with cargo container, cargo ship, bridge...)
Rung 1: main sequence in the HR diagram
With stars that we are able to measure the distances using parallax, we can derive the luminosity (i.e., the absolute magnitude) of the stars, and hence calibrate the HR diagram absolutely. → we are able to measure distances to star clusters out to as far as the Large Magellanic Cloud (~50kpc) - this method depends essentially on the fact that stars of a given color (i.e., spectral type) on the main sequence have the same luminoisty.
This works primarily with star clusters, with many stars at the same distance.
(Note: before Hipparcos, we are not able to go far enough to include enough stars to do that. A method called “moving cluster” was used to measure distances to the couple nearest star clusters to calibrate the main sequence (see page 33 of Silk’s book.)
The Distance Ladder
Rung 2: “Special Stars”: in particular: Cepheid Variables
Cepheids are pulsating stars in the late stage of evolution. They are bright (hence can be seen to large distances), and their period of pulsation is proportional to the luminosity. By identifying these stars and measuring their periods, we know how luminous they should be, and so measuring their flux gives us the distance. The period-luminosity relation can be calibrated using Cephieds in star clusters of known distances.
The Period-Luminosity Relation of Cepheid Variables
The P-L relation was discovered in the 1920’s by Henrietta Leavitt and her co-workers at Harvard, and it was the key for Edwin Hubble to determine the distance to Andromeda (the closest external galaxy).
Characteristic light curve (magnitude vs time) with sawtooth shape variation.
Schematic of (averaged) mag vs period
Light curve (real data) of a very bright cepheid
Period-Luminosity relation of Cepheids
Cepheids can be detected from ground-based telescopes out to about 7 to 10 Mpc.
Using Cepheids: identify Cepheids by its variability and light curve shape. Measure its period, and hence get its luminosity from the period-luminosity relation. The difference between the absolute mag and the apparent mag gives the distance to the star.
The HST Hubble Constant Key Project
The HST can resolve stars out to galaxies in the Virgo Cluster (about 15 Mpc away). The Key Project’s aim is to find Cepheids in nearby galaxies and galaxies in Virgo to measure H0 to 10% accuracy.
Example images and light curve from the Key Project.
20 galaxies were observed in total; the project determined the average distance to the Virgo cluster to be 17.8±1.8 Mpc. (The recession velocity of the cluster can be easily measured.)
Is this sufficient to determine the Hubble constant? The answer is no!
Problem: recall Ho=v/d, where v is the expansion velocity, but the velocities (i.e., redshift) we measure for the galaxies in Virgo include the motion induced by the gravitation field of Virgo; also, as it turns out, our Galaxy is falling towards Virgo (from the gravitational pull of the cluster).
So, in general, the velocity we measure contains two components: the expansion velocity, and a “peculiar velocity” due to local gravitational field: v=vexp+vpec → Ho=(v-vpec)/d
Two ways to deal with peculiar velocity: I) correct for the peculiar velocity by: (1) using the average velocity and average distance of the Virgo cluster galaxies; (2) try to estimate how fast we are falling towards Virgo. (The measured velocity of Virgo is ~1200 km/s, the Virgo-centric infall is estimated to be ~200km/s; i.e., the true expansion velocity of Virgo from us is ~1400km/s).
II) Measure galaxies much further away so that the peculiar velocity (typically around 300 km/s) is much smaller than the expansion velocity: i.e., if vpec« vexp, then Ho=(v-vpec)/d → Ho=v/d
Need another rung in the distance ladder...
Secondary Distance Indicators
Cepheids are called “primary distance indicators” - we set up secondary distance indicators (much more easily seen than Cepheids, i.e. more luminous or larger) by choosing some standard candles (or rulers) that are found in galaxies already with distance calibrated by the Cepheids. - these can then be used to get the distances to galaxies much further away than Virgo.
There are many secondary indicators tried and used by astronomers over the last 80 years. We will focus on one of the most useful ones: Supernova.
Supernova Type Ia (SNIa)
Carbon fusion occurs when the mass of the white dwarf goes beyond 1.4 solar masses
SNIa is an excellent standard candle:
The peak luminosity (after correction for decline rate) is used as a standard candle; with M=-19.2 with a dispersion of <20%
Compared with Cepheids: M~-5, dispersion of ~20%. SNe are ~500000 times brighter, and can be seen across 75% of the observable universe! (We will see later in the course how SNeIa at high redshift measure the cosmology of the universe.)
Results on Hubble Constant:
Most modern results come in between 65 to 75 km/s/Mpc: HST Cepheid key project + SNe Ia: 71 ±2(random) ±6 (systematic) km s-1 Mpc-1
There are other direct methods of measuring Ho (i.e., without distance ladder) which we will discuss later: e.g., CMB fluctuation (a standard ruler method), gravitational lensing delay.
Mapping the Universe: Redshift Surveys
Redshift + the Hubble law allow us to map the third dimension in space, and produce a (almost) true map of the Universe.
In the past, galaxy redshifts were measured one at a time (using a slit spectrograph). Modern imaging spectrographs are “multi-object” spectrograph, which can target tens to hundreds of faint galaxies simultaneously. -- this allows us to carryout very large redshift surveys of 10s and 100s of thousands of galaxies.
Distribution of bright (<17 mag) galaxies on the sky
Galactic Plane “zone of avoidance” -- molecular clouds, dust grains block light from out side of our galaxy.
An early redshift survey: the CfA survey (1980’s)
Nearby bright galaxies, spectra taken one at a time
as on the sky
Modern large scale redshift surveys:
The 2DF (”2 squared Degree Field”) Survey - ~200,000 galaxy redshifts, in strips of 50 to 100 degrees on the sky.
The Sloan Digital Sky Survey (SDSS): 10000 sq deg in the northern sky, 750,000 galaxy redshifts
(Both of these are not very deep, going out to z~0.2, or under 1 Gpc)
Example of a 2dF slice
Deep “Pencil-Beam” Surveys:
Targeting fainter galaxies (hence higher redshift), over much smaller angles: E.g., the CNOC2 Survey (Yee et al, 2000).
Cone opening 1.5 degrees
Schematic Maps of the Universe
A mass density volume contour map of the nearby Universe (~100Mpc)
Summary of key points
- Redshift: z=(λ-λ0)/λ0=Δλ/λ0