Appendix D: Cosmic Distance Scales
The scale of space
We use rulers to measure the sizes of objects on our desks and miles or kilometers to measure distances to other cities. Clearly, it would not make sense to use inches to describe the distance between New York and San Francisco. Likewise, when we think about objects like planets and stars, the units of miles or kilometers are too small. We need a different measuring stick. The ideal metric will match our intuition or experience with the physical world. Some of the common “measuring sticks” that astronomers use are:
- Astronomical units (AU) – the distance between the Earth and the Sun – this is handy for comparing distances to planets within our solar system. One AU is 1.49 × 108 km, or 93 million miles. It takes light about 8 minutes to travel 1 AU.
- Light years (ly) – the distance that light travels in one year (note that this is not a measure of time, it is a measure of distance). This unit is handy for describing distances to stars within our galaxy. When we see the light from a star that is 40 light years away, we are seeing light that left that star 40 years ago. One light year is 9.5 × 1012 km or 63,240 AU.
- Parsecs (pc) – formally, this is the distance at which the orbital radius of the Earth (1 AU) subtends one arcsecond (or 1/3600 of a degree). Using parsecs as a metric for distance greatly simplifies our calculations for distances to stars that are measured with trigonometric parallaxes. One parsec also equals 3.26 light years or 206265 AU. We use parsecs for describing distances to stars in the Milky Way galaxy, and kiloparsecs (kpc = 1000 pc) or megaparsecs (Mpc = 1 million pc) for describing distances to other galaxies.
However, astronomers have the flexibility to use any handy, scale-appropriate metric. The mass or radius of a planet can be arbitrarily described in Earth units. The mass, radius, and luminosity of other stars are usually given in solar units (i.e., relative to the Sun).
Consider the scale of the solar system. How does the size of the Earth compare to the size of Jupiter? Or to the size of the Sun? How many Astronomical Units (AUs) from the Sun to Jupiter? What are these distances like in terms of light travel times? How long does it take a photon to travel from the Sun to the Earth?
Memorizing numbers?
There is no need to memorize charts of numbers, but it is worth remembering the “round number” distances to a few planets in our solar system in terms of Astronomical Units.
- Mercury is 0.4 AU from the Sun (you will learn that many of the exoplanets that have been discovered orbit much closer than 0.4 AU to their host stars).
- Jupiter is another solar system landmark at 5 AU.
- The light travel time from the Sun to the Earth is 8 minutes and the light travel time to other planets just scales with their distances. So, it takes about 40 minutes for a photon of light to travel from the Sun to Jupiter.
- The edge of our solar system? It’s about 120 AU from the Sun (about 3 times farther than the distance to Pluto). Amazingly, the Voyager 1 and 2 space probes, which were launched in 1977, have left the boundary of the solar system.
Other handy numbers for the solar system:
- Jupiter is about 300 times the mass of the Earth
- The Sun is 1000 times the mass of Jupiter
- The radius of Jupiter is about 11 times the radius of Earth, but 1/10th the radius of the Sun.
NASA provides a handy fact sheet listing information about solar system planets in metric or Earth ratio units.
Calculating sizes
One of the main confusions that students have when considering how to measure the sizes of various objects is how to convert a qualitative understanding of how big or small an object is into a quantitative measurement. The difficulty starts with a realization that while distances are a fairly well-understood concept, the amount of space an object takes up can also be measured in terms of areas and volumes. The space we inhabit is a three-dimensional reality, and so there are three different but related ways to describe the amount of space an object takes up. Depending on how many dimensions of space is used to describe an object, we will look at its length, area, or volume.
Areas are calculated by multiplying two perpendicular length measurements together. Depending on the context, sometimes these length measurements are the same, but sometimes they are different. In many mathematics and engineering classes, precise formulae for calculating the areas of different shapes and surfaces are derived and presented, but it will often suffice for us to estimate the areas of objects based on simply multiplying the length of the object by itself (a mathematical operation called squaring because this is the area of a square). Occasionally, we may find it useful to know the area of a circle or the surface area of a sphere (a lot of objects in space are approximately circles or spheres). In this case there are two formulas of interest
Area of a circle = π × radius × radius = π r2 ≈ 3r2
Surface area of a sphere = 4 × π × radius × radius = 4π r2 ≈ 12r2
where the radius (r) is the distance from the center of the sphere or circle to its edge.
Volumes are calculated by multiplying three perpendicular length measurements together. Similar to areas, the choices of which lengths to choose to multiply are dependent on the particular forms that are of interest. The simplest volume calculation just relies on multiplying the length of the object by itself three times (a mathematical operation called cubing because this is the volume of a cube). Occasionally, we may find it useful to know the volume of a sphere
Volume of a sphere =
4/3 × π × radius × radius × radius = (4/3)π r3 ≈ 4r3
When calculating areas and volumes, the length units are also multiplied together two or three times often resulting in very different numerical scales. It comes as a great surprise to many to learn that while there are one hundred centimeters in a meter there are one million cubic centimeters in a cubic meter.
To illustrate the different scales associated with the lengths, areas, and volumes of objects, some select examples are given in Table 1. Notice how the areas are roughly (but not exactly) the square of the lengths and the volumes are approximately the cube of the lengths for each example.
| Concept | Dimension | Standard Unit | Size of Proton | Size of Human | Size of Earth | Size of Milky Way |
| length | 1 | meter (m) | 8 × 10-16 m | 1 m | 6 × 106 m | 5 × 1020 m |
| area | 2 | square meter (m2) | 2 × 10-30 m2 | 2 m2 | 5 × 1014 m2 | 5 × 1041 m2 |
| volume | 3 | cubic meter (m3) | 2.5 × 10-45 m3 | 6 × 10-2 m3 | 1 × 1021 m3 | 7 × 1060 m3 |
