Cosmic Distances, Stellar Brightness, and The Hubble Constant

FAS Astronomers Blog, Volume 30, Number 3.

The Hubble Constant

The Universe is expanding. The farther a galaxy is from us, the faster it is moving away from us. We measure this expansion using something called the Hubble constant, which is the rate with which galaxies are receding from us as a function of their distance. More precisely, the Hubble constant is H₀ from the Hubble-Lemaitre Law, where velocity (v) = H₀ * distance (d). Therefore, H₀ = v/d and it is usually stated in units of (km/s)/Mpc, where Mpc is a megaparsec and is defined below.

So, let’s explore how distances and galactic recession velocities are determined. These are the two important measurements that allow us to determine the Hubble constant and the expansion of the universe.

Distance Measurements

When we look up at the night sky, we see stars. We can tell the apparent brightness of these stars. However, finding their distance isn’t all that easy. Stars appear to be laid out on a two-dimensional sphere. The constellations we observe are two dimensional renderings of star patterns that don’t really exist in three-dimensional space. Over the years, astronomers have used “steppingstones” (aka “standard candles” and “cosmic distant ladders”) that take them further and further out into the cosmos, providing what we hope are reasonable measurements of the distance to stars, galaxies, and beyond.

To start with, cosmic distances are so vast that they aren’t measured in miles or kilometers. Planets in our Solar System are millions or billions of miles away, but it is more convenient to use something called an Astronomical Unit (au). This is the average distance of the Earth to the Sun, which is just under 93 million miles. As an example, Mars is on the average 141.6 miles from the Sun (around 1.5 au). Neptune, the farthest official planet from the Sun, is around 2.8 billion miles out in the Solar System, which is 30 au. As an aside, the IAU says that the appropriate abbreviation for astronomical unit is au, not AU or A.U.

As we move out to stellar distances, a light year is a more useful unit. A light year is the distance light travels in a year (5.88 trillion miles). This is based on the speed of light in a vacuum, which is 300,000 km/sec or 186,000 miles/sec. Proxima Centauri, the closest star to the Sun, is 4.25 light years away. The Andromeda galaxy is around 2 ½ million light years. Just to make sure, a light year is a DISTANCE measurement and NOT a time measurement.

Parallax

Astronomers use parallax to find the distance to close by stars. As the Earth moves from one side of the Sun to the other, nearby stars appear to shift their position in a way your finger will appear to move when you hold it at arm’s length and alternate opening and closing one eye.

We can use parallax to find the distance to a star (D) using the distance from the Earth to the Sun (d), and the angle (2a) with which the star appears to shift over a six-month period. From the diagram below, tangent (a) = d/D, therefore the distance to a star is, D = d / tangent (a). If the angle, a, is small, then tangent (a) = a when measured in radians. Therefore, D = d / a_radians.

A parsec (PARalax SECond) is the distance at which a star’s parallax, using the Earth-Sun distance (1 au) as a base, is 1 arcsecond. In other words, the star’s apparent position will shift by 2 arcseconds over a six-month period. A parsec turns out to be 3.262 light years. A Megaparsec (Mpc) is one million parsecs or 3.262 x 10⁶ light years.

Cepheid Variables

Beyond a certain distance, parallax is no longer useful. Stars are too far away for their apparent position to shift using the Earth’s orbit as a base. Fortunately, astronomer Henrietta Leavitt discovered an important feature for a class of stars known as Cepheid variables. These stars pulsate with a specific regularity. Leavitt found that their brightness was related to their period of pulsation. So, just by observing how a Cepheid varies, its intrinsic brightness and then distance can be determined. For more on Henrietta Leavitt and her work, see the Harvard Computers.

RR Lyrae and TRGB

RR Lyrae stars are older population II stars currently in their red giant phase. These stars are found on the Hertzsprung-Russell (H-R) diagram at a point called the Tip of the Red Giant Branch (TRGB). They are fainter than Cepheid variables but exhibit a similar relationship between their periodicity and brightness, and therefore can be used to gauge distances.

Type 1a Supernovae

Farther out, type 1a supernovae can be used. A type 1a supernova is the explosion of a white dwarf star after it has absorbed material from a companion star and reached a critical mass called the Chandrasekhar limit. Because the star explodes at this known mass, its intrinsic brightness is known, and its distance can be determined. For more on type 1a supernovae, see Dark Energy.

Type 1a supernovae are not the same as type II supernovae, which result when a massive star exhausts its source of fuel and collapses in an instant. This collapse produces a shock wave that we see as a type II supernova. The two types of supernovae can be distinguished by their spectra. Type II supernovae contain hydrogen, while type 1a do not.

Other Methods

There are other methods of determining cosmic distances.

• The Tully-Fisher relationship shows the correlation between the intrinsic brightness and rotation rate of spiral galaxies. Once a galaxy’s intrinsic brightness is known, its distance can be determined.
• Baryonic Acoustic Oscillations (BAO) result in a lumpiness in the way galaxies are spread. This lumpiness can be used to gauge the distance between galaxies, which if known, can be used to determine the distance in way similar to parallax.

Stellar Brightness

For many of the above techniques, the intrinsic brightness of a star is determined. The distance to the star is then found by comparing the star’s intrinsic (actual) brightness to its apparent (observed) brightness. The farther away a star is, the dimmer it appears. In general, apparent brightness = intrinsic brightness / distance², and distance = √(intrinsic brightness / apparent brightness).

A star’s brightness is usually measured by its luminosity. Luminosity (L) is a measure of the light emitted by the star. Technically, lt is the energy per second produced by the star, and it is proportional to the star’s radius² and the temperature⁴. However, it is most often stated as a ratio to the luminosity of the Sun (L_o). Huge type O stars can have a luminosity of 10⁶ times that of the Sun. Small type M red dwarfs can be as faint as 10^-5 that of the Sun.

Stellar Magnitude

A star’s magnitude is another measurement and uses a scale ranging from negative numbers (brightest) to positive numbers (dimmest). For example, when we look up into the night sky, going from brighter to dimmer, Sirius has an apparent magnitude of -1.46, Vega is +0.03, Rigel +0.18, and Polaris +1.96.

There are two ways in which a star’s magnitude is measured.

• Apparent magnitude (m) is the brightness of the star as seen from the Earth.
• Absolute magnitude (M) is the brightness of the star from a fixed distance of 10 parsecs (32.6 light years).

The difference in one magnitude corresponds to a 2.512 ratio in brightness. The number 2.512 is known as Pogson’s ratio after astronomer Norman R. Pogson. He noticed that in the ancient stellar magnitude system of Hipparchus, which was later adapted by Claudius Ptolemy and used by astronomers into the nineteenth century, 1^st magnitude stars were around 100 times brighter than 6^th magnitude stars. So, Pogson formalized the difference and made 2.512 the basis for the modern stellar magnitude system. With this, the difference in magnitude of x corresponds to a ratio in brightness of 2.512^x, and 2.512⁵ = 100. For example, a star of magnitude 1 is 2.512 times brighter than one of magnitude 2, 2.512 x 2.512 times brighter than a star of magnitude 3, 2.512 x 2.512 x 2.512 times brighter than a star of magnitude 4, and so on.

The Doppler Effect

The rate with which galaxies are receding from us is determined by the doppler shift of light coming from stars within these distant galaxies.

The Doppler effect for light is similar to what we experience for sound. When a fire truck or ambulance rushes by, we notice the frequency of the siren shifts from high (moving toward us) to low (moving away from us). The frequency of light shows this same characteristic depending on whether the source is moving toward us (frequency is higher) or away from us (frequency is lower). The greater the speed, the more the frequency changes.

Light Spectra

The electromagnetic spectrum of stars shows dark bands at specific frequencies depending on the elements found in a star’s atmosphere. Each element absorbs light only at specific frequencies, which show up as dark bands unique to that element. The farther a galaxy is from us, the faster it is moving away, and the more these bands are shifted toward the red end of the spectrum. We say these galaxies exhibit a “redshift.” A few local galaxies, such as Andromeda, are “blue shifted” because they are moving toward us. However, most galaxies are “red shifted” indicating that the universe is expanding.

Galactic Recession Velocity

The Doppler shift in frequency is associated with the velocity of the source object, f_o = f_e [1 / (1 + v/c)], where f_o is the observed frequency, f_e the emitted frequency, and c the speed of light in a vacuum.

Because the frequency (f) and wavelength (λ) of light are related (f = λ/c), we can also measure this shift in wavelengths. In this case, the change in wavelength is also associated with the velocity of the source object, λ_o = λ_e (1 + v/c), where λ_o is the observed wavelength and λ_e the emitted wavelength. The Doppler shift and corresponding cosmic distance is often expressed as z = (λ_o – λ_e)/λ_e.

In any event, it is easy to observe the spectra frequency shift and determine the rate with which a star is receding from us or moving toward us.

The Hubble Tension

That’s it! If you know an object’s brightness, you can figure out its distance. Combine that with its red shift, which gives the velocity of recession, and you can determine Hubble’s constant! … Almost.

Today, there are some problems with the measurement of the Hubble constant. This is called the “Hubble Tension.” When distance measurements are used, the Hubble constant is around 73 (km/s)/Mpc. When the Hubble constant is calculated from the Cosmic Microwave Background (CMB) using the standard model of the universe called the Lambda Cold Dark Matter (LCDM) model, it is around 67 (km/s)/Mpc. So, there is either something wrong with our distance measurements or with the standard model of the universe. No one quite knows which, so more on this in a future article.