The Cosmic Distance Ladder

Measuring distances in astronomy is an important aspect of understanding the interactions between bodies in the universe and the dynamics of the universe as a whole. Since there is such a wide range of distances to consider, we need several methods to determine distances between objects. We will find that calculating the distance to "nearby" objects like the nearest stars requires a very different method than is necessary for calculating distances to stars that are farther away.

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The method we use to calculate distances to the nearest stars is called "stellar parallax." It is basically just triangulation, based on knowing the area of a triangle if your know the length of the base and the base angles. Here, distance shown for the base is the diameter of Earth's orbit about the sun.

 

We define the distance called a "parsec" using the parallax method, as shown in the above equation. Here, the angle is measured in arc-seconds. If you divide up the sky into 360 degrees, 1/60 of one degree is called an arc-minute, and 1/60 of an arc-minute is called an arc-second. One parsec is defined as the distance to an object that subtends an angle of one arc-second.

 

As shown in the diagram, one astronomical unit (AU) is defined as the distance between Earth and the sun.

 

1 parsec = 206,265 AU = 3.3 light years = 19 trillion miles = 31 trillion kilometers

 

To get a better idea of how this works, please check out the "Stellar Parallax" simulation at

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Copyright 2014 Pearson Education, Inc.

This image shows the distances to a few nearby stars. The closest star is Alpha Centauri, 4.367 light years away. We can measure distances to stars within this region using the stellar parallax method.

Scale models

Image source

 

One way to understand size scales is by considering a model. Notice that it is not possible to view the relative size of the sun and Earth and also their distance apart in this small diagram. There is just too big of a difference in the relative length scales. One thing we can do to better understand the real sizes and distances of objects is to build a scale model.

 

Let's build a generic scale model as an illustration.

In this diagram, we have a real system and a scale model system. Let's think about what needs to be true for our scale model to be accurate.

 

In the real system, the blue ball is three times as big as the diameter of the red ball. So, in our scale model, the diameter of the blue model ball needs to be three times as big as the diameter of the red model ball.

 

In the real system, the Distance between the blue and red ball is four times the diameter of the blue ball. So, in our scale model, the Distance between the model balls needs to be four times the diameter of the model blue ball.

 

We could build a smaller scale model as well, as long as we kept the relative sizes the same between the diameters and distances.

 

How do we figure out how big to make the balls and distances in our model, to keep it accurate? We need to start by choosing one model object's size and compare it to the real object. This will determine the other sizes for our model.

We need to measure the diameter of the real red ball and the distance between the real blue ball and real red ball to figure out how big to make these sizes in our model.

Now we can calculate the sizes we need for our model.

The ratio (fraction) of the model red ball to the real red ball is the same as the ratio of the model blue ball to the real blue ball.

We solve for the diameter of the model red ball, and then insert values to find the diameter we need for the red ball in our model.

We can find the Distance between the blue and red balls in our model the same way. This was easy, since we could just use a very similar equation. Once we set the scale using the blue model ball compared to the blue real ball, we have the scale for anything in the model compared to the real system. If we had nine balls, all different distances apart, we could use the same scale to find the correct sizes and distances in our model.

 

Now, let's build a scale model of our Solar system.

Let's assume that in our model, the sun is the size of a bowling ball. Comparing the size of a bowling ball will allow us to calculate the other relative distances in our scale model.

Deciding to use a bowling ball, with a diameter of about 22 cm (2.2 x 10-1 m) sets the scale for our scale model.

 

 

We also need to know the other real distances involved. Now we can use the scale set by the bowling ball to find the diameter of our model earth and how far away from our model sun to place it.

Using the scale set by the bowling ball compared to the real size of the Sun, we calculated the size of Earth in our model. In our model, Earth is the size of a small bead, a couple of millimeters in diameter.

We can use the same formula to find out how far apart the bead would be from the bowling ball in our model. Here, we calculate that the distance in our model would be 24 meters.

 

Now, if the true distance to the nearest star,  Alpha Centauri, is 4.0 x 1016 m, how far away would it be in our model?