The Cosmic Distance Ladder Explained
Distance is something peculiar. It’s very hard to gauge if a celestial object is far away and bright or close and dim. We know that the closest galaxy, Andromeda, is ~2 537 000 light-years away and that Eta Carinae, one of the Milky Way’s brightest blue supergiants, is distanced at ~7 502 light-years. How do we know such distances as we can never actually measure them?
Let’s introduce the cosmic distance ladder. It’s used to approximate distances to objects which we could never actually measure. Each step of this ladder is based on extensive research and allows us to determine distances to very distant celestial objects. The further we go up the ladder, the less precise the distance calculation becomes.
Every distance is measured relative to the Earth.
Radar Ranging (10^-4 light-years)
distances up to the ø solar system
(0.0001 light-years ≈ 946 052 840 kilometers)
For very small distances simple radar will suffice. We shoot an electromagnetic beam in the form of radio waves towards the object of which we want to measure the distance, such as a nearby planet. It is possible to measure the time it takes before the electromagnetic waves reach us back and thus we know the distance. There are several downsides to this method. It can only be used for extremely close objects, such as our surrounding planets. This technique also suffers from redshift.
redshift All kinds of waves have a certain wavelength, it defines whether they are radio waves, microwaves or for example visible light. Visible light is perceived as colors to us. Colors are nothing more than waves with a certain wavelength. Redshift is the elongation of those waves, the wavelength increases and thus shifts more and more to the red side of the visible spectrum. Blue is the color that has the shortest wavelength of the visible spectrum and red the longest. This is also exactly why red is used for taillights. Longer wavelengths go through dust and air without scattering as much as shorter wavelengths, such as blue.
Radar ranging is not very practical to measure longer distances due to redshift. It’s also not very useful to measure the distance to other stars. It has to be possible to reflect a beam of light on the surface, which stars do not do very well. Let’s say it does, measuring the distance to a star with radar ranging will still pose some issues. Take Sirius for example, it’s the brightest white dwarf of our Milky Way and located at 8.61 light-years. Meaning that light will take 8.61 years to go there and another 8.61 years to get back. With this technique, we’ll have to wait more than 17 years before we’ll actually have measured the distance to Sirius. Instead of waiting for more than 17 years, measuring those distances can be done by using something called parallax.
Parallax (10^2 light-years)
distances up to the ø Milky Way
(100 light-years ≈ 9.46 * 10¹⁴ kilometers)
Distances up to 100 light-years can be measured accurately using parallax. Several stars are located within this distance region, such as Proxima Centauri, located at 4.3 light-years from Earth. Proxima Centauri is the closest star to our sun. Parallax is something we observe in our daily life. Distant objects go by slower than objects much closer to us. Think about a train ride and how the farthest objects are apparently stationary while closer objects rush by rather fast. Parallax is the apparent displacement of an object when an observer changes its point of view.
Before diving further into measuring these immense distances, we should know more about the parsec (parallax arcsecond), what it is and how it is used to define distances in space.
1 pc = 3.08567758 × 10¹⁶m = 3.26163344 light-year
parsec The parsec is used in astronomy to define distances. It is the distance measured when you observe 1 arcsecond at a length of 1 AU. An arcsecond is 1/360 of a degree.
AU The Astronomical Unit is defined as the average distance from the Earth to the sun. AU’s are mostly used to better comprehend relative distances. When talking in AU’s, it is easier to grasp the concept of distance relative to other objects than talking in kilometers.
With parallax, when knowing the distance to a nearby object, we can measure the distance to objects farther away. We do this by measuring the angle change when observing the distant object at 2 separate points with 6 months apart. The more it shifts from our perspective, the closer it is, hence the parallax phenomenon.
We can measure the distance to a star p by finding the diameter of the rotation of the Earth around the sun b (300 million kilometers) and the angle a which denotes the apparent displacement of the star against the background stars.
p = (b/2) / tan(a/2)If we want to measure the distance in parsecs, the equation becomes even more simple. With a the angle in arcseconds, the following equation p = 1/a will give you the distance p in parsecs.
Limitations arise when measuring angles, from telescopes on Earth, smaller than 0.01 arcsec (100 parsecs) due to atmospheric disturbance. Space telescopes can go up to 0.001 arcsec (1 000 parsecs). If you consider the size of the Milky Way to be 30 000 parsecs wide, this method is rather limited. That’s where main sequence fitting comes in.
Main Sequence Fitting (10^5 light-years)
objects no farther than neighboring galaxies
(10 000 light-years ≈ 9.46 * 10¹⁷ kilometers)
The main sequence is a specific region of a certain luminosity and temperature where most stars live the most part of their lives. It is also known as the Hertzsprung Russell diagram. We are going to need to know more about this diagram in order to explain main sequence fitting. This method is mostly used to determine the distance to star clusters, not individual stars. Those clusters are kept together by their collective gravity.
HR diagram Every life of each star starts somewhere on this diagram. It contains several regions.
- left lower corner are the white dwarfs
- right upper corner are the red (super)giants
- left upper corner are the blue (super)giants
- left upper to right lower diagonal are the main sequence stars
Our sun is a main-sequence star. It lives on the diagonal and will eventually (in about 5 billion years) go the right upper corner to form a red giant. This happens when the fusion of hydrogen in the core stops and fusion begins in the outer layers. After only the heavier elements remain, from which the star cannot obtain enough energy anymore by fusing them, it will become a white dwarf. It will stay this way for the rest of its life until all energy is gone and it will turn into a brown dwarf. The more massive the star, the more likely it will become a blue or red supergiant.
So how can we derive distance from the HR diagram and where stars are located on it? Following the HR diagram and the main sequence, we know that there’s a strict correlation between temperature and luminosity. If we know the apparent luminosity measured as is on Earth m and know the temperature of the star, we get a shift in absolute luminosity M and apparent luminosity m. Which in turn gives us the distance if we already know the distance of main sequence stars, by for example parallax. This is also called color-magnitude diagram (CMD) fitting. This can only be done with clusters of stars that consist of stars mostly on the main sequence. When stars get older they drift away from this diagonal and it makes it harder to calculate the distance to those star clusters.
Measuring even farther takes us to two types of standard candles. One type of candle are the Cepheid variables and another type Ia supernovae. A standard candle has a known luminosity. Emitted light diminishes in brightness the further it travels. If we know the luminosity of the emitting source L and how bright it reaches Earth F, it is possible to calculate the distance r.
F = L/4πr² (power_per_unit_area = luminosity / area_sphere)Cepheid Variables (10^7 light-years)
distances up to the ø Local Galactic Group
(10 000 000 light-years ≈ 9.46 * 10¹⁹ kilometers)
The first type of standard candles are the Cepheid variables. Henrietta Leavitt discovered in 1908 that the periodicity of the star was closely related to its luminosity. If we know its periodicity, we know its luminosity and thus also how far away the source is located. Cepheids with longer periods are more luminous than those with shorter periods. She correctly deduced a period-luminosity relationship.
We know distance d because of the apparent luminosity m how we see it on Earth can be measured and the absolute luminosity M can be deduced from the period-luminosity diagram.
m - M = 5 * log(d/10)Type Ia Supernovae (10^10 light-years)
distances up to the ~1/10 ø Observable Universe
(10 000 000 000 light-years ≈ 9.46 * 10²² kilometers)
Type Ia supernovae have known luminosity because they are always formed under the same conditions. Let’s explain what supernovae are.
supernova When a star is too heavy to support itself, it collapses violently. This triggers an explosion and is called a supernova. Stars are in constant equilibrium, they find themselves in hydrostatic balance.
hydrostatic balance Nuclear fusion in the core generates pressure. This pressure pushes the star apart. While pushing the star apart, particles will find themselves farther and farther removed from one another. The mass density reduces and collisions will diminish. Gravity takes over and pushes all of the mass back together, mass density increases, collisions are more frequent and more outward pressure is generated. This will decrease mass density again and fewer collisions will happen. This goes on and on for billions of years until all hydrogen and helium are gone and the star cannot support nuclear fusion in the core or its outer layers anymore. Gravity is too strong and crushes the star. It all depends on the weight of the star. Less massive stars will not explode violently, they will change into white dwarfs. More massive stars will end their lives with a bang and produce a supernova. Their remains will be either a neutron star or a black hole depending on their mass and whether or not they exceed the Chandrasekhar limit. This limit is the maximum stable mass of a white dwarf and is currently about 1.4 solar masses. Everything heavier than that will result in a supernova and a neutron star or black hole.
Type Ia supernovae are formed when a white dwarf finds itself in a binary star system with another red dwarf. The heavier white dwarf will siphon mass from the red dwarf until it reaches a critical mass and surpasses the Chandrasekhar limit and goes supernova. Because this type of supernova always happens under the same circumstances and thus also have a known luminosity, they can act as a standard candle.
The distance ladder starts with simple radio waves, goes into apparent displacement based on distance, redshifting according to the HR diagram and ends with a known luminosity of standard candles. All of these make it possible to measure extremely distant objects and help us to explore and understand the universe even more.
