## The Cosmic Distance Ladder, part 2…

January 11, 2013 Leave a comment

Let’s continue our explanation of how astronomers are able to gauge the distances to celestial objects. In my first post on this topic, we talked about how radar ranging and stellar parallax are used to determine distances in the solar system. Now we’ll continue working our way up the Cosmic Distance Ladder.

### Beyond our solar system

As we leave our local planetary neighborhood, our solar system, we can still use *stellar parallax *up to a point. In fact, I received an email from Scott in Placentia, CA, asking *“What maximum distance an object can be reliably determined by parallax measurement using the latest technology?” *As I explained to Scott in my response, it’s hard to answer that question with a single number. Ideally, stellar parallax should always work, because the stars close to us will always move at least a little bit with respect to the background stars. Unfortunately as we look at stars farther and farther away, the angle that the near-ground star moves gets smaller and smaller. As you can imagine there is going to be a point where that slight angular movement is almost imperceptible. However, technology is always improving. In 1989, the European Space Agency (ESA) launched the *Hipparcos* mission which was specifically designed for astrometry- determining the position and location of stars. The *Hipparcos* mission found the positions of nearly 100,000 stars as far away as 1,600 lightyears. Expanding on that mission’s success with new technology, another mission named *Gaia* is slated to launch this summer and claims to have the ability to find distances to stars tens of thousands of lightyears using stellar parallax. Some sources claim that stellar or trigonometric parallax is limited by current technology to around 1,000 parsecs or roughly 3,260 lightyears.

### Examining other Suns

Once we reach that upper limit for the use of stellar parallax, where we can no longer accurately measure the angular shift of the star with reference to the background stars, we have to turn other methods for determining distances. The next technique we need to rely on is called *spectroscopic parallax. *Now, confusingly this new type of parallax is totally different from and unrelated from actual (stellar) parallax, it just uses the word “parallax” because it’s a technique used to find distances, like parallax. (That’s not confusing at all…) Okay, so how does this spectroscopic parallax work? Well you might remember in *The Cosmic Distance Ladder, part 1…* I talked about an important instrument used by astronomers called a spectrometer (a.k.a. “spectrograph” or “spectroscope”). These extremely useful devices break white light up into all the colors of the rainbow- this resultant array of colored light is called a *spectrum*. You’ve probably seen this phenomenon before. Well spectroscopic parallax, as the name implies, is all based on being able to take a good spectrum of a star. This is what sets the upper limit on the maximum distance of objects that we can determine using this technique; with current technology, spectroscopic parallax is limited to roughly 10,000 parsecs (1 parsec = 3.26 lightyears = 19.2 trillion miles). For distances where both spectroscopic and stellar (trigonometric) parallax can work, we generally use stellar parallax because it’s more precise- you’ll see why in a bit.

You might also remember in my recent post about stellar classification, that I explained how each stellar class has a spectrum that is unique to that class.So that means that if you look at a nearby B-type star and a far away B-type star, they should have the spectra that are nearly identical. The difference between the two measured spectra will be their intensity, which correlates to the brightness or luminosity of a star. But now we have to be careful because there are two types of brightness: *apparent magnitude* and *absolute magnitude*. Absolute magnitude is how bright an object actually, intrinsically is, while apparent magnitude is how bright it appears to be at Earth if you ignore the atmosphere. So basically absolute magnitude is how bright an object would be if you were at it and apparent magnitude takes into account how far away you might be from the object. Think about a lightbulb: absolute magnitude would be the physical amount of light that the bulb puts out and apparent magnitude is the amount of light you measure from the bulb when you stand on the opposite side of the room. As you can imagine, the bulb will appear less bright the farther away from it you are. Same with stars.

Now, once you’ve measured the apparent magnitude (using some sort of light collecting device) to see how bright the star appears to be, you have to figure out the absolute magnitude. This can be a little tricky and involve some guesswork, but the main tool we use is the Hertzsprung-Russell diagram.

As you can see, when you plot the H-R diagram, you can easily determine a range of absolute magnitudes that apply to each spectral class. Unfortunately that means there isn’t one right answer, so you can’t get an exact distance for the star…but you can get a pretty good range. Once you determine an appropriate value of absolute magnitude (*M*), you need to use some math and a mathematical formula known as the *distance modulus*.

### Yay math!

Now that we have the apparent magnitude (*m*), which we measured, and the absolute magnitude (*M*), which we inferred from the H-R diagram because we know the star’s spectral type, we can now use both to get the distance modulus: *m – M*. You might ask, “How does that help?” Well if you think about it, the difference between the absolute brightness and apparent brightness of a star is determined by distance; naturally then there is a mathematical relation that proves that.

*Woah, that probably seems like a lot of math…but it’s not so bad. The lefthand-side of the equation is our distance modulus (**m – M*), that gives us a dimensionless number (since neither *m* or *M* has a unit of measurement like feet or grams) that’s the difference between the absolute and apparent magnitudes. Since the lefthand-side has no units, then that means the righthand-side can’t have them either, that works out as long as we say that the *10* in the denominator is in the unit of parsecs, hence the “*pc*“. Then our distance (*d*) will be in units of parsecs and everything is balanced out correctly. So rearranging this equation to solve for the distance, we get:

And calculating that gives us the distance to the star, *d*. Now granted, we might need to do that for a high and low value of *M*, based on how we read the H-R diagram, meaning we’d get a range of distances and not a single answer, but it still gives us a much better idea than we had about these far away stars. This is why we’d choose trigonometric parallax over spectroscopic parallax for distances where both can work, but for distances beyond the reach of stellar parallax, then spectroscopic parallax is better than nothing! Somewhere “between A and B lightyears away” is much better than “really far”!

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