|In this EBook we discuss how we measure distance to celestial objects.|
|Tags||Astronomy, distance, units of distance, Parsec, cosmic distance ladder, trigonometric parallax, spectroscopic parallax, heliocentric parallax, Hipparchus, Hipparcos, GAIA, proper motion, luminosity, flux, inverse square law, apparent magnitude, absolute magnitude, distance modulus, standard candles, spectral class, luminosity class, Hertzsprung-Russell diagram, main sequence fitting, cluster fitting, Cepheid variables, RR Lyrae variables, Tully-Fisher relation, Type IA supernovae, optical redshift, Doppler effect, radial velocity, Barnard's star, cosmological redshift, expanding universe, Hubble's Law, Hubble's constant, cosmological model, cosmological distance, light travel time, comoving distance, angular diameter distance, luminosity distance, gravitational lens time delay, Sunyaev-Zel'dovich Effect, gravitational wave astronomy, observable universe, visible universe.|
|Prerequisites||EBooks Stellar Radiation; Stellar Evolution|
|First Published||April 2008|
|This Edition - 3.0||January 2019|
“The Universe is a big place” is easily the biggest understatement of the century.
If you could travel seven and a half times around the Earth in one second and with that same speed travel in one direction for 13.6 billion years, you would have reached the place where the furthest objects that we can see today were when the light we receive now, left those objects.
Finding distance to stars and other celestial bodies has been a challenge throughout the history of astronomy. Some methods go back a long way in time, such as the ones developed by the Greek astronomer Hipparchus (around 150 BCE), who determined the distance to the Moon with a method we now call parallax.
Within our Solar system the most common technique to measure distance nowadays is radar in various frequencies in the Electro-Magnetic spectrum. We know the orbits of planets, moons and even asteroids so accurately that we do not have many problems in knowing how far away any object is at any given time. That is why we can fly spacecraft into the far corners of our Solar system without too much of a problem as far as navigation is concerned.
In this EBook we will concentrate on distances outside our Solar system. We will find out that measuring distance at various scales of the Universe is far from trivial. Actually, measuring distance in deep space is arguably the biggest technical problem in astronomy today.
Most methods that we use, have been developed over the last two centuries and are the result of countless nights of systematic observations by many astronomers and what we could call, old-fashioned detective work to find relationships and physical properties that somehow relate to distance. That research is on-going and we still cannot say that we have a consistent and accurate picture of the true scale of the Universe.
Contemporary methods to determine distance outside our Solar system can be separated in three main groups:
- geometrical methods based on parallax,
- optical methods based on the star’s luminosity,
- optical and cosmological redshift
In this module we will discuss the most commonly used methods in each of these groups.
UNITS OF DISTANCE IN ASTRONOMY
The standard unit of length (or distance) is the metre (symbol: m).
In Astronomy we work with very large distances as compared to distances on Earth. Therefore we commonly use other units for distance in Astronomy.
- Astronomical Unit (AU) - average distance Earth - Sun
- Light year (ly) – the distance light travels in one year
- Parsec (pc)
The Astronomical Unit is used to express distances across the Solar System.
1 astronomical unit (AU) = average distance from the Earth to the Sun;
1 AU = 149,597,870,691 m = about 1.5 x 1011 m.
The Light Year is commonly used to express distances to objects, e.g. to stars. Note that the light year is not a measure of time!
1 light year (ly) = distance that light travels in one year
1 ly = 9.461 x 1015 m = 9.461 x 1012 km
1 ly = 63,240 AU.
To get an idea about how much a light year is, remember that light travels at about 3x108 m/s; this is about 7½ times around the Earth in one second.
For relatively short distances we use e.g.:
Distance to the Moon is in average 384,402 km or 1.282 Light second;
Distance to the Sun is 1 AU or about 8.3 Light minutes.
This unit also indicates how long ago the light that we see departed from the source.
The Parsec is a unit used to express distances outside our Solar system to stars and galaxies, etc.
1 parsec = 1 pc = 206,265 AU = 3.086×1016 m = 3.262 ly
This unit originates from determining the distance to a star by measuring parallax. It is the distance for which a star has a parallax of one arc second when the Earth moves 1 AU. We discuss this below.
Example: the Sombrero galaxy (M104) is at a distance of 50 Mly or 50,000,000 ly. It is also at a distance of 15.3 Mpc or 15,300,000 pc.
Readers who are not familair with the powers of ten notation (e.g. 105, 10-7) or with prefixes (e.g. Gly, Mpc) are referred to the handouts Scientific Notation and Prefixes & Symbols in the Physics Study Notes on this website.
COSMIC DISTANCE LADDER
The Cosmic Distance Ladder is the succession of methods by which astronomers determine the distances to celestial objects. The ladder analogy is used because no one technique can measure distances over the entire range required in astronomy. Instead, one method can be used to measure nearby distances, a second (the next rung of the ladder) can be used to measure nearby to intermediate distances, and so on.
Each rung of the ladder provides information that can be used to determine the scale for distances at the next higher rung. Because the more distant steps of the cosmic distance ladder depend upon the nearer ones, the more distant steps include the effects of errors in the nearer steps, both systematic and random ones.
Therefore the overall distance scale used in astronomy, is prone to systematic effects in any of these individual measurement techniques and these affect our knowledge about the scale of the Universe.
Measuring reliable distances in astronomy is one of its biggest ongoing challenges, and ultimately defines what we know about the size of the Universe.
|A simplified overview of various techniques on the Cosmic Distance Ladder, most of which will be described in more detail in this EBook, is given in this series of 9 videos by PhysicistMichael.|
When you hold one finger upright in front of you, and close one eye at the time, you will see that the background shifts with respect to the finger when you look with a different eye. This effect is called parallax.
The shift in background happens because the distance from your eyes to the finger is shorter than the distance to the background. When you stretch your arm to hold your finger as far as you can, you will notice that the background shift is smaller.
As babies we learn through trial and error, to estimate distance to objects in this way. After some practice, the brain will automatically interpret the difference between the image seen by each eye, as a distance or a difference in distance to objects. Because we have two eyes, we can see in 3 D.
As a young lad I enjoyed drawing simple stereo images myself such as these pyramids.
If you bring them together you will see 3D. When viewing cross-eyed you will see an inverted pyramid. You can draw such pictures yourself and try both parallel and cross-eyed.
Watch an explanation of the difference between parallel and cross-eyed viewing here.
Seeing images in 3D is made easier by the historical stereoscope which forces each eye to see only one image.
With anaglyph images, the two images are overlapping and in two contrasting colours, e.g. red and blue.
With an anaglyph glasses you can then see the image in 3D.
The principle of parallax can be applied in astronomy as a powerful geometric tool to measure distances to nearby stars.
Parallax in Astronomy
An isosceles triangle is determined by the length of the base and the opposite angle.
If the base is fixed, the height (or distance) can be found by measuring the angle. In this geometric sense, the top angle is called the parallax.
In the examples on the previous page the basis is the distance between our eyes. In astronomy we can use that same principle, but the distance between our eyes is far too small to let our brains “see” distances to stars. We must use other, much larger baselines.
Distance to a comet
As an introduction to the use of parallax in astronomy we first discuss an example where the base of the parallactic triangle is the distance between two observers on Earth.
These photographs from the 1995 passage of comet Hyakutake were taken at the same time by two amateur astronomers at different places: one in Portugal and the other in Denmark.
It is clear that the comet's position appears shifted with respect to the background star.
This is an example of a parallactic shift; the comet is much closer to the Earth than the star, so that its position in the sky depends on the observer's location.
Bring the two images together to see the stereo image.
The apparent shift in position of the comet in the photographs arises from the fact that the observers were situated about 3000 kilometres apart, and the comet is much closer than the stars beyond. The baseline in this case is the distance of 3000 km.
With this information, together with the measurement of the apparent shift in the photograph, the distance to the comet can be calculated.
In the example above, the base was the distance between two different observers and the photographs were taken at the same time.
In most cases when using parallax in astronomy however, there will be one observer and the photographs will be taken at different moments.
Geocentric and Heliocentric parallax
We now use the Earth itself as a base. Take a photo two times in one night.
The distance between two positions of the same observer during one night is used as the base. Nearby stars appear to move more than stars further away. The distance between the two positions can be calculated from the time difference between the two observations, accounting for the rotation of the Earth.
Note that in this diagram the base could never be the complete diameter of the Earth because it would require the observer to see below the horizon.
Here we use the distance from Earth to the Sun as the base. Take a photo of the night sky in two different seasons, about half a year apart. The orbit of the earth around the sun is used as the base.
Nearby stars appear to move more than stars further away. The baseline can be up to 2 AU. This allows to measure distance traditionally to about 500 light year (or about 150 pc).
The green star is closest because it has the largest parallactic shift.
The blue star is further away, but still a lot closer than the red background stars, who do not appear to have any shift at all.
The animation below shows how parallax affects our perception of the apparent position of a star near to us in the course of a year.
It's position relative to other, farther away stars, seems to change because our viewing position shifts as the earth moves around the sun.
|Drag the slider in the animation to learn how we can deduce
the distance to a star by measuring the stellar parallax value.
(Origin of this great Flash movie unknown)
Calculating astronomical parallax
The angles involved are very small, typically less than 1 arc second. (Remember that 1 arc second = 1/3600 of a degree). To determine the distance to a star we can write (for small angles):
d = b / 2p = ½ b / p
where d is the distance to the star, p is the parallax angle expressed in radians (see diagram), and b is the baseline. In the case of heliocentric parallax, the base b is equal to 2 Astronomical Unit (AU) -- the diameter of the Earth's orbit.
Since there are 206,265 arc seconds per radian, the formula can be re-written as:
d (in AU) = 206,265 / p
with p now measured in arc seconds.
Or, if we write the distance of one parsec as 206,265 AU, we get:
d (in parsecs) = 1 / p (in arc seconds)
Thus one parsec is the distance for which the heliocentric parallax is 1 arcsecond.
This is the distance unit astronomers use most frequently, and it is equivalent to 3.26 light years.
This unit was chosen because it makes the parallactic formula very simple.
How can the measurement of parallax be improved?
- Make the base larger. For this reason heliocentric parallax is more accurate than geocentric parallax. With heliocentric parallax you can measure distance to stars that are further away. The practical limit is a distance of about 500 ly (150 pc).
- Measure the angle more accurately. This is most of all limited by the angular resolution of the telescope, which is related to atmospheric conditions and lack of all-sky visibility. To vastly increase this accuracy, measurements have been carried out in space. Examples are the Hipparcos and GAIA spacecraft.
Hipparcos (for High Precision Parallax Collecting Satellite) was an astrometry mission of the European Space Agency (ESA) dedicated to the measurement of stellar parallax and the proper motions of stars. The satellite was launched on 8 August 1989.
Accurate distance determinations have been obtained to about some 500 light-years distance.
ESA’s Gaia spacecraft was launched in December 2013 and is located in the L2 Lagrangian Point on the shadow side of the Earth. Gaia performs three main types of observations: astrometry (stellar position, parallax, and proper motion), photometry (magnitude) and spectroscopy (for radial velocity and astrophysics). So Gaia does much more than measuring distance from parallax, but that is one of the key objectives.
The GAIA distance catalogue now already contains 1.33 billion stars throughout our Milky Way galaxy.
In the first part of this movie we can see the effect of the annual parallax as observed by the Gaia spacecraft.
Up to now we have assumed that all objects we look at are stationary, and that while the Earth moves around the Sun, the stars we observe stay where they are. That is far from correct; stars themselves move like everything else in the Universe. All these motions are typically in the order of tens to hundreds of km per second. Because of the vast distances to us, these motions are hardly, if at all, noticeable within a human life span.
We need accurate and systematic measurements to find the actual motions of celestial objects. Any star can move in any direction with respect to us on Earth. When measuring that relative motion we must distinguish between two components
The transverse velocity of stars is the component of their motion perpendicular to our line of sight. We call that proper motion. The other component is the radial velocity or the velocity with which the star moves away from or towards us. That latter component does not affect the position of the star in our sky.
The annual parallax we can observe from Earth cannot be separated from the proper motion of a star
We will always observe the combination of the two. The difference however is that annual parallax is periodic, it repeats with a cycle of one year, whereas proper motion is generally secular, it shows the movement of the star in one general direction.
Barnard’s star (also dubbed “Greyhound of the Skies”) is the star with the largest proper motion. It is one of the closest stars at almost 6 ly and it has a proper motion of just over 10 arcseconds per year. Its annual parallax is about 0.55 arcseconds.
The diagram shows observations of Barnard's star over a period of more than one and a half year. The periodic motion due to annual parallax is clearly visible. The fitted straight line shows the proper motion after the parallax has been removed from the data.
If we want to calculate the distance to Barnard’s star from annual parallax, we must first remove this proper motion from the observations. Its annual parallax of 0.55 arcseconds gives a distance of
1 / 0.55 = 1.8 pc or 5.9 ly.
The star's proper motion is 10.55 arcseconds per year.
A detailed account of measuring Barnard’s Star is given in this publication by R.J. Vanderbei.
The movie we linked to above shows in the latter part the proper motion of stars. Both the parallax and the proper motions shown are based on data from the Gaia spacecraft.
See the ESA page and download option of the movie here.
LUMINOSITY AND DISTANCE
This chapter is a summary of our EBook Magnitude & Distance. Go there for more details on the Inverse Square Law, the magnitude scale and Distance Modulus.
Luminosity (as we also explained in our EBook Stellar Radiation) is a measure of the amount of energy that is emitted per unit of time. The physical quantity is Power which is expressed in the unit Watt or Joule (energy) per second.
We are all familiar with the power of light bulbs that is expressed in Watt.
In astronomy power is called luminosity and it is expressed in units of Lsun (the luminosity of our Sun as the reference).
Lsun = 3.83x1026 Watt or Joule/sec.
Flux and the Inverse square law
Flux is the amount of power going through a unit area. This diminishes with the square of the distance.
This is simply a geometric effect expressed as:
where F is the flux, L is the Luminosity and r is the distance to the star. This is called the Inverse Square Law because flux decreases with the square of the distance.
Stars appear in the sky to have a particular brightness but the actual energy output of a star may be vastly different to what we see.
As a practical example, if someone takes two torches, one bright and the other one not so bright and walks to a distance of say 50 m, then we will see the bright torch still brighter than the less bright one, but both are less bright than at a closer distance.
If you now leave the less bright one at that distance and walk further say to 100 m with the bright one, that bright one may now look fainter than the one that was left at a distance of 50 m.
If we don’t know the distance to the torch (or a star), the brightness we see (we call that apparent or relative brightness) may give a wrong indication of the actual brightness.
The brightness of the torch or of a star diminishes with distance according to how the flux diminishes with the Inverse Square Law.
This means that if the distance increases by a factor of two, the brightness diminishes by a factor of two-squared or four. If the distance is tripled, the brightness decreases with a factor of three-squared or nine, etc.
So how can we find the distance to a star with the brightness that we observe?
The method of finding distance we are going to discuss is based on assumptions about the luminosity of the star.
Suppose we know the luminosity (L in the image), then with the Flux (brightness that we observe) we are able to calculate the distance with the Inverse Square Law.
In astronomy it is common to express brightness not in Flux but in Magnitude. So let us discuss that first.
In astronomy it is common to use the concept of Apparent Magnitude, which is the measure that expresses brightness as we see it.
The magnitude scale has a historical origin but has been redefined in the modern era. The magnitude Scale is explained in detail in our EBook Magnitude & Distance.
In Summary apparent magnitude (symbol m):
- indicates the brightness of a star as we see it
- is a scale that runs opposite to brightness (the smaller the magnitude, the brighter the star)
- is centred at the apparent magnitude of Vega (m = 0)
- allows negative values for bright stars
- is a logarithmic scale in which five units in magnitude correspond to a factor of 100 in brightness
Absolute magnitude uses the same scale as apparent magnitude. It has a value which is defined as the apparent magnitude a star would have if it were located at a standard distance of 10 pc from Earth.
As an example the Sun has an apparent magnitude m = -26.73 but it has an absolute magnitude of +4.75. That is the magnitude we should see if the Sun was at a distance of 10 pc.
- we have a measure of the absolute magnitude (M) of a star and
- we compare that with the apparent magnitude (m),
then we can use the Inverse Square Law to find the distance.
The method used for this in astronomy is related to (m - M) which is called the Distance Modulus.
For this purpose astronomers use the Distance Modulus which is derived from the inverse square law and the definition of the magnitude scale.
This is described in more detail, together with numerical examples in our EBook Magnitude & Distance.
The expression that relates distance D with apparent magnitude (m) and absolute magnitude (M) is a logarithmic one:
This formula gives the distance to the celestial object in pc (parsec).
Because absolute magnitude is defined as the magnitude at a distance of 10 pc, the difference (m - M) indicates how far a celestial object is inside or outside a circle with radius of 10 pc.
For a calculated example see our EBook Magnitude & Distance.
The reason that we discussed luminosity, flux, the inverse square law, magnitude and the distance modulus in this chapter is that
- IF we have a way to know the luminosity or absolute magnitude of a star,
- and measure its apparent magnitude,
we have found a way to calculate the distance.
Celestial objects of which we (think we) know the luminosity or absolute magnitude are termed Standard Candles.
We will now look at different examples in astronomy of measuring distance to celestial objects using knowledge of the luminosity or absolute magnitude, objects that are termed Standard Candles.
Any object that can be considered as a Standard Candle in astronomy needs to have the following properties:
- it must be easy to identify and not being confused with a different type of object
- it must have a know luminosity related to some physical property we can measure
- it should preferably be very bright so we can use it to large distances.
Fortunately there are several ways in which we can find estimates for a star’s luminosity. These methods are generally deducted from particular physical processes in a star (or entire galaxy) that are indicative for the object’s luminosity. These techniques allow us to estimate distances, even at inter-galactic scale.
But being mindful of the principle of the Distance Ladder (see page 4), we must have several objects in any particular class of standard candles to which we can also measure distance with another technique, so that we can mutually calibrate the distance scale.
Below we will describe a few of these techniques in detail and summarise several others.
We will start with a couple of methods that are based on the magnitude and the distance modulus we discussed above.
These are Spectroscopic Parallax and Main Sequence Fitting.
Spectroscopic parallax has nothing to do with parallax. But since this expression is customary among astronomers, the parallax method we discussed before is often referred to as trigonometric parallax.
The spectroscopic parallax technique requires that a star's apparent magnitude and its spectrum have been observed. Information obtained from the spectrum is used to find the star's position on the Hertzsprung-Russell Diagram (HR-diagram).
We discussed the HR-diagram in our EBooks Stellar Radiation and Stellar Evolution.
In the HR-diagram Absolute Magnitude is sometimes listed along the vertical axis as an alternative to Luminosity. (see diagram).
Astronomers have classified stars according to their spectra into spectral classes (also called spectral type). The major spectral classes are type O, B, A, F, G, K, M where the O-class is the most luminous (and hottest) and the M-class is the faintest (and coolest). Each type is subdivided into 10 finer divisions (0-9), as A8 or F0.
This picture shows standard spectra for each class along the horizontal axis.
Therefore the Spectral type as determined from the observed spectrum, defines the horizontal position of the celstial object on the HR-diagram.
Luminosity class is a classification of the luminosity of a star. In our EBook Stellar Radiation we explained that as the Morgan-Keenan luminosity and it can also be obtained from the observed stellar spectrum.
The luminosity class of the celestial object defines its the vertical position on the HR-diagram.
Once the star's position on the HR Diagram is identified by the intersection of its spectral class and luminosity class, one can read off its absolute magnitude.
The absolute magnitude combined with the already observed apparent magnitude and the distance modulus (see above) then provides the distance to the star.
|Use this calculator of the University of Nebraska-Lincoln that very well illustrates the method of spectroscopic parallax.
In the animation chose Spectral type (top left), Luminosity Class (bottom right) and Apparent Magnitude (bottom right).
This produces the distance (in pc) from the Distance Modulus.
This method is generally not very accurate for individual stars. In particular the luminosity class is often not sharply defined on the HR-diagram.
But if we can process a large number of stars that are about at the same distance and we limit these to stars of luminosity class V - Main Sequence stars - we have the generally more accurate method of Main Sequence Fitting to determine distance.
Main sequence fitting
Main sequence fitting also uses the HR-diagram to determine distance but is always applied to clusters of stars.
Star clusters formed at the same time from the same cloud of gas and dust and are in a group where they are gravitationally bound. This conveniently means that they are all located at about the same distance.
As the name of the method suggests, it is limited to star clusters that are still on the Main Sequence in the HR-diagram. This means that they are fusing hydrogen to helium and are in Luminosity Class V (this has been explained in our EBook Stellar Radiation). Whether a cluster is on the Main Sequence depends on the age of the cluster. If the cluster is too old, stars will have left the Main Sequence. This is a limitation of the method.
Main sequence fitting places the stars of the cluster on the HR-diagram with the vertical position initially derived from their apparent magnitude.
On the diagram nearby main sequence stars are plotted whose distances are well-known from trigonometric parallax where absolute magnitude is used as the y-axis variable (idialised by the red line in the diagram). The vertical position of the cluster is now systematically different from the reference Main Sequence stars due to the unknown absolute magnitude. By shifting the cluster's stars to fit the actual main sequence, the vertical displacement is now used as a measure of distance, that can be calculated with the distance modulus.
Potential difficulties with main sequence fitting are related to whether all stars plotted are actually on the main sequence. Some stars may have already left the Main Sequence and will appear more to the upper right in the diagram. Foreground main sequence stars that not belong to the cluster will appear brighter and above the cluster main sequence stars. Background main sequence stars appear fainter and below the cluster main sequence stars.
Cepheids, also called Cepheid Variables, are stars which brighten and dim periodically. This behaviour allows them to be used as cosmic yardsticks out to distances of a few tens of millions of light-years.
In 1912, Henrietta Swan Leavitt noted that variable stars, called Cepheid stars, in the Small Magellanic Cloud (SMC) galaxy brighten and dim periodically.
What she determined was that the brighter the Cepheid, the longer its period. Observing these stars in the SMC she could reasonably assume that they were all at about the same distance. This means that their apparent magnitude is a relative measure for their luminosity. And the luminosity appeared to be a function of the period of brightness fluctuations of these stars. This discovery means that once the period is measured, the luminosity can be calculated. And this allows to calculate the distance to the Cepheids with the inverse square law.
|Plot from a paper by Leavitt in 1912.
The horizontal axis is the logarithm of the period
of the corresponding Cepheid,
and the vertical axis is its magnitude.
The lines connect points corresponding to the stars' minimum and maximum brightness.
Credit: Leavitt and Pickering; via Wikipedia.
Cepheids are reasonably abundant and very bright. Astronomers can identify them not only in our Galaxy and the neighbouring Magellanic Clouds, but in more distant galaxies as well. This method works up to some ten million light-years when Earth-bound telescopes are used. But in the present day, space-based telescopes such as the Hubble Telescope, have used Cepheids to much further distances. One of the design goals of the Hubble Telescope was actually to observe distant Cepheid variables.
Observing a galaxy in the Virgo cluster called M100, astronomers used the Cepheid variables observed there to determine its distance - 56 million light-years.
Distance scale from Cepheids
Leavitt determined the period-luminosity relation of Cepheids mainly with stars in the relatively nearby Magellanic Clouds, assuming they were all at about the same distance. But what is the distance to the Large and Small Magellanic Clouds? These small galaxies are too far for the traditional parallax method from ground based observations. So the calibration of the Cepheid method with an independent distance method has been of great interest to astronomers ever since Leavitt's days.
The importance of Leavitt's discovery at the time was that the method gave accurate relative distances.The famous Great Debate could finally be settled because it paved the way for Edwin Hubble's ground breaking discovery that there is a Universe outside the Milky Way.
While Cepheids are great Standard Candles (they give an accurate relationship between luminosity and the observable period), their calibration with other distance methods is still at the forefront of research. A key issue is the distance to the Cepheids in the Magellanic Clouds. The current GAIA space mission is expected to make a major contribution to the distance scale by improving the parallax derived distance to the Magellanic Clouds.
Then we will finally have an accurate calibration between the geometric and luminosity based distance scales. Because Cepheids can now be observed to great distances this calibration means a more accurate distance scale in the Universe up to more than 100 million ly.
RR Lyrae are variable stars that are in the Red Giant phase, late in the star’s evolution. Due to instability in these Red Giant stars, their size and therefore luminosity changes periodically. Similar to Cepheids they exhibit a period-luminosity relationship which allows their luminosity to be determined from their period. Unlike Cepheids, this does not work well in the visible spectrum, but it does in Infrared. Alternatively the colour-period relationship is being used for RR Lyrae.
RR Lyrae are low metallicity stars, meaning that they are original old stars and typically found in globular clusters above and below the galactic plane of our Milky Way galaxy. Therefore these stars are used to determine the distance to those clusters. Because these stars are not very luminous they can also be observed in the most nearby galaxies, but not beyond.
The Tully-Fisher relation describes the link between the luminosity of a spiral galaxy and its rotation. The rotation of a galaxy causes a widening in the spectral emission lines that can be observed.
Basically the assumption is that the bigger the galaxy is (and thus more luminous), the faster it rotates. The detailed physics of this relationship is beyond the scope of this EBook, but it is important to mention that the Tully-Fisher relation is being used as a standard candle to determine inter-galactic distances since 1977.
A similar method is the Faber-Jackson relation published in 1976.
Type Ia Supernovae
In our EBook Stellar Evolution we discussed the Type Ia Supernova, which is the runaway nucleosynthesis of a White Dwarf in a binary system. The importance of this type of event as a Standard Candle is the fact that in principle the Supernova produces a consistent peak luminosity which is related to the Chandrasekhar limit, above which the White Dwarf re-ignites. The observed light curve of the supernova can give an accurate measure of its peak luminosity, which makes these events very important for measuring distance. An obvious limitation of supernovae is that you have to catch them when they go off; they don't hang around.
A big advantage is the fact that these supernovae, similar to the previously discussed Tully-Fisher relation, can be observed up to extremely large distances of up to ten billion ly (1010 ly).
There is more
We discussed the most commonly mentioned Standard Candles with which distances are being measured in astronomy. There is a lot more to this and sizing up the Universe is one of the biggest ongoing technological challenges in astronomy.
A good overview can be found here
On page 4 we already mentioned the playlist on YouTube by PhysicistMichael.
Limitations of Standard Candles
Several types of challenges exist for any class of standard candle, limiting the ability of astronomers to measure accurate distances.
The principal one is calibration, determining exactly what the absolute magnitude of the candle is. It is essential to be able to find enough members of the class with distances obtained from another Standard Candle method, so that their luminosity can be calibrated accurately.
Another problem lies in recognizing members of the class, and not mistakenly using the standard candle calibration for an object which does not belong to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.
Different stellar clusters or galaxies do not generally have all types of stars in them, so standard candle techniques cannot always be used.
In some cases the use of a particular class of objects is known to be of limited accuracy, but when no other class is available, the class is useful simply because there is no alternative.
These issues cause major limitations and uncertainties with measuring distance in the Universe and are subject to continuing research and discussion.
|``It is much easier to make measurements
than to know exactly what you are measuring.''
J. W. N. Sullivan
Before we can go into the use of Redshift to find distance, let us explain the term Redshift as a measure of velocity. This relates to the Doppler effect.
It is a daily experience for all of us to hear the effect of the changing of wavelength and frequency of sound waves when the source is moving with respect to the observer. This is called the Doppler effect, named after Christian Doppler, who formulated this principle in 1842.
Imagine for instance that you are standing on the side of a race track and you hear a race car coming towards you and then passing by. The sound you hear distinctly changes its frequency when the car passes you. When the car approaches, it has a relatively high frequency and when it passes, the pitch becomes lower. Only at the precise moment that the car is right in front of you (no relative velocity between the car and yourself) you hear the true frequency of the sound of the car.
The Doppler effect is caused by the fact that, while waves travel at a certain speed, due to the relative motion the number of waves that reach the observer per second changes, hence the apparent frequency changes.
The doppler effect allows us to calculate radial velocity, i.e. velocity along the line of sight, of a source with respect to the observer, from the change in received frequency. When the source moves towards us we measure a higher frequency (shorter wavelength) and when the source moves away we hear a lower frequency or (longer wavelength).
Check out this multimedia tutorial of the School of Physics, UNSW.
Doppler effect in astronomy
Because EM-waves have wavelike properties, they behave in principle in the same way as sound waves as regards the Doppler effect.
We all know the risk of getting caught speeding by a camera on the side of the road. Our speed is measured with radio waves or infrared light using the Doppler effect.
In astronomy we can observe line spectra from all celestial objects that are radiating, even in wavelengths outside the visible spectrum. We discussed this extensively in our EBook Stellar Radiation.
These emission or absorption lines are ideal objects to use for the Doppler effect. Astronomers know the line spectrum for a given chemical element from the laboratory. When the celestial object is moving away from us, the frequency becomes lower (compare with lower frequency when the race car is receding), thus longer wavelength. The lines are horizontally shifted towards the longer wavelength. In the visible spectrum that is towards the red side of the spectrum. Therefore this shift is called a redshift. Alternatively, when the celestial object is moving towards us, the lines will be blue shifted.
Because these line spectra can be observed with great detail using modern spectrographs, the Doppler shift can be measured very accurately, revealing the relative (line-of-sight) speed (or radial velocity) between the celestial object and us here on Earth as observer.
|Ignoring relativistic effects (which must be taken into account when the relative speed becomes very large), the radial velocity can be determined from the observed Doppler shift with:|
|This ratio is called redshift, denoted by z|
Example Barnard's star
When we discussed proper motion in the Parallax chapter, we saw Barnard’s star had a large proper motion (perpendicular to our line of sight.
Let us now find out what the radial velocity is of Barnard's star.
The graph gives a small part of a high resolution spectrogram from the star compared with that of the Sun. The wavelength range covers part of the orange section of the visible spectrum.
Any distinct peaks (absorption lines) in the Barnard's spectrum can be used to measure the wavelength shift, when we find corresponding peaks in the Sun's spectrum. Knowing that the Sun does not have a significant radial velocity with respect to Earth, any shift we see must be due to the radial velocity of Barnard's star.
We can use the red peak just to the right of the 610 nm that has a corresponding green line to the right of it or the red peak at 612 nm that has a corresponding green peak to the right of it.
The first thing we note is that the Barnard spectrum (red line) is shifted to the left, meaning towards the smaller wavelength. This is a "blue shift" and it means that Barnard's star is moving towards us.
|One of the green peaks is at about 612.218 nm. The red peak is at 612.000 nm.
The measured shift gives thus a change in wavelength of -0.218 nm.
When we now apply the Doppler formula we get:
change in wavelength / original wavelength = -0.218 / 612.218 = -3.561x10-4
With the speed of light being 3x105 km/s we get:
radial velocity = -3.561x10-4 x 3x105 = -106.8 km/s.
The negative sign indicates that it is moving towards us (blue shift).
Hence Barnard's star is moving towards us with a radial velocity of -106.8 km/s.
With modern spectrographs an amazing radial velocity accuracy of about 3 m/s is possible, provided we have very sharp spectral lines that can be carefully calibrated with the stationary equivalent.
Locally, meaning within our own galaxy and towards the nearest neighbour galaxies, redshift is a very important technique to measure relative velocity. Chief applications are in studying rotation of binary systems, finding exoplanets, and measuring galaxy rotation, including that of the Milky Way.
In this EBook we are concerned with measuring distance and as such the redshift discussed isn't very relevant. But we needed to explain the basics of redshift as a Doppler effect, to pave the way for the discussion how redshift does play an important role to measure distance at much larger scale. In that situation redshift is not caused by radial velocity, but by the expansion of space, and we talk about Cosmological Redshift.
Above we described Optical Redshift in astronomy that occurs when an object (e.g. Barnard’s star) has a radial velocity with respect to us as observers. In that case it actually is a blue shift because the star moves towards us. It will be a Redshift for any star that is moving away.
Another cause of Redshift is that space itself expands. Objects in it may move relatively or not, but the fact that space expands, means that far away objects will always show a Redshift in their spectra. Generally this is called Cosmological Redshift and is denoted by z. Objects need to be far away enough to exceed any optical redshift due to their relative motion and make the expansion observable.
Before we go into detail, let us review the history of the discovery of the expanding Universe and what this has meant for the development of our understanding of the Universe during only the last hundred years.
|It is less than one hundred years ago that astronomers were uncertain whether our galaxy the Milky Way was all there is or that the actual Universe contains many more galaxies and must be “inconceivably large”.
Of historic significance is the “Great Debate” between astronomers Heber Curtis (left)
and Harlow Shapley in 1920.
Discovery of the Expanding Universe
The astronomer Vesto M. Slipher was the first to measure the spectrum of the Andromeda galaxy in 1912. This was no simple feat in those days. He continued these measurements and by around 1925, he had measured 45 of these spectra. At that time galaxies were called “spiral nebulae”, and their nature was actually unknown. Were they galaxies like our own Milky Way and outside our galaxy or were they nebulae with a spiral shape, inside our Milky Way? At that time it was not even known what our Milky Way galaxy looked like.
Slipher discovered something very intriguing in these spectra, when he measured the red shift to find radial velocities. With the exception of the Andromeda Galaxy they were all moving away from us. This was a puzzle, also because there were not yet accurate distances to these “spiral nebulae”, but this red shift discovery looked significant for the astronomers of the day.
Edwin Hubble made observations of Cepheid variable stars in various spiral nebulae in 1922 and 1923 and proved that these objects were outside the Milky Way galaxy, using Henrietta Swan Leavitt's period-luminosity relationship for Cepheids. This technique, described earlier in this EBook, was decisive to end the discussion among astronomers about whether there was more to the Universe than just the Milky Way galaxy.
Combining these distance measurements with Slipher’s red shift measurements, Hubble and Milton L. Humason discovered a rough proportionality of the objects' distances with their redshifts. The larger the distance, the larger the receding velocity. In 1929 Hubble and Humason formulated the empirical Red shift Distance Law of galaxies, nowadays simply termed Hubble's law.
This law dictates that an object twice as far away as another object, has a velocity that is also twice as much as the velocity of the closer object. This suggestion was astounding because the general consensus in those days was that the Universe was static and not changing in size at all.
Earlier, in 1917, Albert Einstein had discovered that his newly developed General Theory of Relativity indicated that the universe must be either expanding or contracting. Einstein himself believed, like everybody else, that the Universe was static, and he therefore introduced a cosmological constant to the equations to avoid this "problem". When Einstein heard of Hubble's discovery, he was quoted as saying: "this is the biggest blunder of my life". He visited Mt. Wilson observatory to discuss the discovery with Hubble and others.
Hubble and Humason were able to plot a trend line from the 46 galaxies they studied. Though there was considerable scatter in the data, the conclusion was made that recession velocity and distance are proportional, and they obtained a value for the Hubble constant of 500 km/s/Mpc (this constant determines the slope of the line in the graph). Hubble estimated a value of about 500 km/s/Mpc (red line).
This original value is far from the modern value, primarily because of the difficulty of acquiring accurate large distances in space, and the fact that they used nearby galaxies, that also have their own peculiar velocity with respect to the Milky Way. As an example, the Andromeda galaxy is now known to have a velocity towards us (blue shift) of 300 km/s. The expansion of space can only clearly be demonstrated with objects that are much further away.
|History of Hubble's constant, until 2000 (left) and recently (right).
Error bars indicate uncertainty.
At present this relationship between distance and velocity is referred to as Hubble’s Law and it is the foundation for the cosmological model of an expanding Universe. There is no doubt about its validity, although the actual value of Hubble’s constant (the ratio between velocity and distance) has been under discussion ever since 1929. Finding the present value of the Hubble constant has been the result of decades of work by many astronomers, both in analysing the measurements of galaxy red shift and in calibrating the steps of the Cosmic Distance Ladder. During these decades, the value of Hubble’s constant has been the subject of many debates among astronomers. Hubble’s original estimate was 500 km/s/Mpc, but we now know that the value is much smaller.
Above we reviewed the history of the discovery of the expanding Universe which is one of the biggest breakthroughs in Cosmology.
Let us now go back to cosmological redshift as a consequence of this expansion.
The expansion of the Universe is not an expansion of space into another surrounding space, but it is the expansion of space itself. This is difficult to understand because we do not have any phenomenon in our daily life to compare this with.
|Ants on an inflating balloon or the Expanding Universe
One way to imagine what it means is to watch an inflating balloon.
The surface of the balloon is a 2-dimensional representation of our 3-dimensional space.
On it the distance between any two points increases when the balloon inflates and points that are twice as far apart move twice as fast.
Note that the ants do not move themselves on the balloon!
The expansion of space is now a fundamental part of the Cosmological Models that are used to describe the evolution of the Universe. As a consequence, the distance between any two objects in space (e.g. galaxies) is increasing. The further these objects are apart, the faster they will recede from each other. Hubble’s Law describes this expansion. The apparent motion due to the expansion is called the Hubble Flow. It does not mean that galaxies are actually moving with respect to each other.
But if we measure redshift of objects that are very far away, what do we actually measure?
Looking at the simple formula for Optical Redshift we used before (the actual Doppler effect)
we see that if the redshift is equal to 1 the velocity of the object is equal to the speed of light. But nowadys we routinely measure redshifts that are much larger than 1.
This means two things:
- When we apply the theory of general relativity (which we must for velocities that are a significant fraction of the speed of light) we see that the expansion at large distances actually is going faster than the speed of light. Since this is an expansion of space itself, it does not violate the speed limit c, dictated by special relativity. For these calculations we need a cosmological theory that also involves the Hubble constant. This is not a Doppler effect.
Cosmogical Redshift means that photons are "stretched" while they have been travelling towards us. This "stretching" is observed as a change towards longer wavelength. The larger the cosmological redshift we observe, the longer the photons have been travelling, i.e. the longer ago they left the source. The actual meaning of the cosmological redshift is that it tells us how much the universe has expanded since the light was emitted.
|The wavelength of the emitted radiation is lengthened due to the expansion of the Universe. In this animation, the galaxy on the left was formed a long time ago, while the galaxy on the right was formed more recently. Although each galaxy emits the same wavelength of light, the light from the left hand galaxy has spent longer travelling through the expanding Universe, and has therefore experienced a greater ‘stretching’ (redshift). Thus the more redshift is measured the longer these photons have been travelling.
Source: Cosmos, Swinburne Astronomy Online
But the key question for our discussion is: Can cosmological redshift be a measure for distance?
What about distance?
The reader may have wondered that ever since we started to discuss Redshift, we have only discussed how to find (radial) velocity, not distance. So why is redshift discussed in this EBook which is about measuring distance? The answer is that we needed this basic introduction to both optical and cosmological redshift to discuss the principal difficulties of measuring distance in really deep space and even to illustrate that the very concept of “distance” needs more scrutiny.
The naïve approach to derive distance from redshift is to use Hubble’s Law in reverse. From the measured redshift we can calculate velocity with the Doppler formula (corrected for relativistic effects) and with Hubble’s Law we can then work backwards to find the distance. This is quite inaccurate for various reasons.
Hubble’s law describes the expansion of space but it ignores any local relative motion between the source and us as observers. Such motion is often dubbed “peculiar” motion. If we are really “close” (let’s say nearby galaxies) we can largely ignore the expansion of space and only observe peculiar motion. An example is the blue shift of the Andromeda galaxy. In that situation we can only use standard candles we discussed before to find distance. Redshift will only give radial velocity.
At larger distances this becomes blurred by the expansion of space but we still may see the effect of peculiar motion. This is a particularly difficult distance range in which we must both model peculiar motion and cosmological expansion.
At very large distances we only see the expansion. In the latter case Hubble’s law describes this, but then we cannot ignore relativist effects. The simple Doppler formula is not good enough. Objects that are moving with respect to an observer at high velocity (in comparison to the speed of light) exhibit time dilation due to this relative motion. Time passes more slowly on those objects and their atoms emit radiation with lower frequency. This would hold for relative motion in any direction, not only radial velocity. Another red shift is predicted by general relativity. Strong gravitational fields can cause time dilation. This is called gravitational red shift. This effect is generally small but becomes significant e.g. near a black hole.
We need a Cosmological Model
Cosmological and gravitational redshift cannot be explained by the Doppler effect, they require a cosmological theory that formulates the expansion of the Universe. Developing such a theory is the prime focus in modern Cosmology.
In that context, if we want to measure distance, we need to raise the question what do we mean by distance in an expanding Universe?
Distance in an expanding Universe
We now reach a fundamental problem when we are describing techniques to measure distance in the Universe. What do we mean with distance?
The light we receive at the present day from objects that are very far away, has been travelling for a long time, up to many billions of years. During that time the Universe has been expanding, so what do we really mean if we talk about THE distance to those objects?
In cosmology there are different definitions for distance, each of which will give very different outcomes for an object that is far away (has a large redshift).
Here is a simplified list.
- The “naive” Hubble’s law distance is what we have discussed above, when we use the relativistic Doppler red shift formula and Hubble’s Law. Even if we correct for relativistic effects it gives very large “distances” for large red shift.
- Light travel time (or Lookback time) is simply how long ago the light left the object that we now see. This is more a time measure than a distance measure and it requires us to have an estimate of when the light left the source. This is generally found as a fraction of the age of the Universe, derived from Hubble’s Law and redshift. According to this measure the edge of the observable Universe is at 13.8 billion years of light travel time. We could say that light travel distance is light travel time multiplied by the speed of light. That would give us the distance to the source at the time the light we receive left the source.
- Comoving distance tells where the object is now when we receive the light. This measure has a scale that grows with the expansion of the Universe. On this basis the edge of the visible Universe is now at about 46 billion light years. This is probably the most realistic measure of distance in an expanding universe.
- Angular diameter distance is more or less the opposite of comoving distance. It is a measure of where the object was with respect to us, when the light left the object.
- Luminosity distance is the distance derived from the luminosity. This is not a realistic distance measure, because due to the expanding Universe, the travelling light photons are stretched (cosmological red shift), which causes the object to appear much dimmer than its actual luminosity would suggest. Therefore the luminosity distance will always give a (much) larger value than any of the other distance measures.
A more detailed list can be found here
All distance measures above but the first, are precisely defined in cosmology and depend on assumptions about the cosmological model that is used for the expanding Universe.
A comparison of cosmological distance measures for different ranges of redshift z.
Let us summarise our discussion about Cosmological Redshift.
It is important to note that the only thing in the context of this discussion that we can measure,
and fortunately very accurately, is Redshift z.
The conclusions we attach to those observations depend on the particular cosmological interest we have,
and if that pertains to distance, it depends on the type of distance indicator we use.
Cosmology is a very complex field of astronomy, and it is no wonder that the idea of “distance” in the context of large redshifts is often misquoted and misunderstood in the popular media.
Cosmologist Ned Wright has set up an online calculator for various distance related concepts,
within a particular choice of cosmological model, as:
The largest redshift object ever measured to date is the galaxy GN-z11 that has a redshift of 11.09.
Using the calculators linked above, find out the value of the various distance indicators for this galaxy.
Note: First enter the redshift in the first calculator, and then enter the obtained value for light travel time in the second calculator.
Going back to measuring distances at the largest scale, there are various new ideas being pursued for physical indicators of distance. Importantly these are independent of redshift and thus give an opportunity for calibrating methods based on redshift.
We briefly describe three of these techniques, with links for further reading.
Gravitational Lens Time Delay
One promising technique is the use of the effect of gravitational lensing of images of quasars. When we see a quasar through two different paths because of a gravitational lens, the two paths differ in length and therefore also the travel time of the light. When the quasar varies in intensity abruptly (as they do), we see that variation with a time delay in one path as compared to the other.
This time delay is a measure for the distance. This method does not yet give accurate results, primarily because the lensing effect depends on the unknown mass distribution of the lensing cluster of galaxies. When this improves this method will be an important indicator for large distances, because it is independent of any other method and thus does not require calibration like other methods in the Cosmic Distance ladder.
The large mass of a galaxy cluster including its dark matter bends passing light rays, forming an "Einstein Ring".The HST image shows parts of this ring from the blue, distant galaxy. The lensing cluster is at a distance of about 4 Gly.
The Cosmic Microwave Background radiation (CMB) is affected in certain parts of the sky by hot gas in galaxy clusters if the CMB happens to pass through. With this very difficult technique, the observed angular diameter of the galaxy cluster could be a measure for its distance. Importantly this distance is independent from redshift.
Gravitational Wave Astronomy
Since 2017 astronomers have been able to detect gravitational waves caused by merging neutron stars in a binary system. In some cases it has been possible to also measure the same event in the EM spectrum as a gamma-ray burst. The amplitude of the gravitational waves enables scientists to determine distance to this event, independently from any other technique in the distance ladder.
The Observable Universe
The age of the Universe is estimated at 13.8 Gyr which is the same as the light travel time to the edge of the Universe. It is not very useful to say that the size of the Universe therefore is 13.8 Gly. We saw before that light travel time means the time that the photons we receive now have been travelling through an expanding universe. If we multiply that with the speed of light what does that tell us more about the size of the Universe?
The observable Universe is that part of the Universe that is within our present light travel time range. This means that the oldest light we can see is that which left the source at “first light” of the Universe. This is a physical limit caused by the finite speed of light and is independent of the technology we use for observations. We will never be able to look any further.
There is a small difference between “Edge of the Observable Universe” located at the Big Bang and “Edge of the Visible Universe” which is from the time of “recombination” , the time that the Universe became transparent for EM-radiation.
A proper distance indicator for the size of the Universe is the comoving distance we discussed above.
The comoving distance to the edge of the Observable Universe is 46.5 Gly and to the edge of the Visible Universe is 45.7 Gly.
An interesting fact is that we on Earth are at the centre of the observable Universe. We should say “Our” observable Universe. An alien who looks around from a distant galaxy will see a different observable Universe that is centred around that galaxy. If we can see that galaxy there is an overlap between the two observable Universes. If not, we can have no meaningful discussion about it.
Every year that passes the Universe will have expanded a bit further. So the edge will be further away from us over time. This also means that we will be able to see new light that finally has a chance to reach us as regards the age of the Universe. Of course it can only be light that was emitted after the Big Bang (more precise after the epoch of recombination). But that won’t help very much, because at those extreme distances the Universe is already expanding faster than the speed of light. This is not a violation of Special Relativity as it is space itself that is expanding; it is not a movement of mass. So this means that there is a “cosmic event horizon” or Hubble Sphere beyond which we have no “causal connection” to the Universe. In cosmology there are theories about the Universe beyond our observable Universe, but we can never test those theories with observations.
Read more here
Image: Earth (top) at the centre of our Observable Universe.
See full size here
WOBBLY DISTANCE LADDER
Undoubtedly we will continue to see improvements in many techniques involved in the Cosmic Distance Ladder. And new techniques are being investigated.
In the close range, measurements by the GAIA spacecraft are making major contributions to the accuracy and range of the method of parallax. In the medium range better observation techniques improve the standard candle of Cepheid variable stars and further afield the value and accuracy of Hubble’s parameter is likely to improve.
However some “standard” candles may turn out not to be so standard after all. The problems of calibration between the various distance measurement techniques will continue to form a large obstacle, because of the lack of distance indicators that work on overlapping distances.
Measuring accurate distances remains one of the biggest technical problems in astronomy.