The Big Bang Wasn’t All At One Point (Cosmos Commentary)

I finally got around to watching the first episode of Cosmos. I quite enjoyed it, although probably not as much as I would have were I still 9 (which is the age I was back when I used to watch Carl Sagan doing Cosmos… and that, indeed, is probably a nontrivial part of why I’m an astronomer today rather than a paleontologist). I think it’s awesome that once again we’ve got a very charismatic astronomer on TV sharing the wonders of the Universe with us. Alas, I doubt it will have anywhere near the cultural impact that the original Cosmos did, simply because there is so much more out there to pay attention to now. (Not only is there more out there to pay attention to, but over time American society has become more and more ADHD.) Back in the late 1970s, there was little more than three networks of TV to choose from; it was the rare household that had cable. Now, most people have many more options for TV, never mind the ability to download stuff off the Internet on demand. Even if it’s just as high-quality, just as cool, and just as engaging as the old Cosmos was, I fear that the new Cosmos will not be noticed by as large a fraction of the population, and will be more quickly forgotten as people move on to the next shiny thing.

As for the show itself: it all seemed pretty basic to me, but then again, I’m a PhD physicist and professional astronomer who does a fair amount of astronomy outreach, so I was not the primary target audience. I liked the homage to the old show– not just the explicit one at the end (which brought a tear to my eye), but the “we are starstuff” comment, and the Ship of the Imagination (which, as Tyson points out, allows you to travel much faster the speed of light, something I’m doing all the time when I teach astronomy classes).

I did have a couple of quibbles, though. My first was when he was flying through the Solar System’s asteroid belt. The asteroid belt was thick with rocks, creating a massive hazard. The real asteroid belt is not like that. There is less mass, total, of asteroids, than there is in any single planet, and they’re spread out over a huge area in the disk of the solar system. This is why we can fly spacecraft through the asteroid belt without worrying about weaving and dodging. There, asteroids just aren’t that thick.

cosmosroids
What, is this Cosmos, or The Empire Strikes Back?

To be fair, when he was out in the Oort cloud, although yet again they were shown too thick (I know, for purposes of actually being able to see something), he did mention that the Oort cloud objects are typically as far apart as Earth is from Saturn. Still, the visual image will stick with people more than the words.

My primary quibble with the show, though, is the title of this post. One sentence of what he said promulgated one of the primary misconceptions about the nature of the Big Bang. “Our entire universe emerged form a point smaller than single atom.” GAH! No! Indeed, Tyson was (perhaps deliberately) cagey about the difference between our Universe and our Observable Universe. He did use the term “Observable Universe”, with a good description. (It’s as far away as we can see, a horizon defined by the speed of light and the 13.8-billion-year-old age of the Universe.) However, thereafter, he seemed to be conflating the Universe with the Observable Universe. While there are some good reasons why one might do this, the way in which he did it fed into a very common misconception about the Big Bang.

obsuniv
Even though our observable universe is finite, the whole universe is much bigger– indeed, perhaps (probably?) infinite.

Here’s the real story, given the Big Bang model as we best understand and use it in astronomy: the Big Bang didn’t happen all at one point. Rather, the Big Bang happened everywhere. The problem with describing it as happening at one point is that it gives you the misconception that we could identify a point in space away from which everything is rushing. This is not the description of our Universe that shows up in modern cosmological models. Every point in the Universe is equivalently the center. Any point in space you can identify: that is where the Big Bang happened. Everything is rushing away from everything else. It’s really not like an explosion, where there’s a center everything rushes away from. (I wrote about this years ago in my blog post “Big Bang”: A terrible name for a great theory.)

Strictly speaking, it is true that our observable universe was once upon a time compressed into a size smaller than the size of an atom. However, saying that by itself implies a misconception: that that compressed, less-than-an-atom size of extremely dense, extremely exotic matter is all there was. In fact, that’s not right. Our Observable Universe was that small… but just as today there is other Universe (filled with galaxies) outside the boundaries of our Observable Universe, at that early epoch there was more extremely exotic dense-matter Universe outside the atom-sized ball that would one day expand and become today’s Observable Universe. Indeed, if the Universe today is infinite, it was always infinite… even back at that early epoch we’re talking about.

obsuniv_endinflation
The Observable Universe (or a 2d projection thereof) at a period a tiny fraction of a second later than what I’m talking about in the text.
endofinflation
The whole Universe (or a 2d projection thereof) at the same epoch.

This may seem like a minor quibble, but the notion of the Big Bang as an explosion, something everything is rushing away from, is a very tenacious misconception that leads to other misconceptions about our Universe amongst many people I run into. It’s a little difficult to wrap your head around the real model– indeed, people find talks about cosmology that try to describe the real situation (and also the cosmology section of my current ongoing astronomy class) very brain-hurty. But, to my point of view, that’s part of the fun!

There was one throwaway comment about the Big Bang that Tyson made in Cosmos that I really liked. Just before the comment about the atom-sized Universe that got me worked up to make this post, he said about this early Big Bang epoch that “It’s as far back as we can see in time… for now.” That “for now” is great, and spot on. If you read A Brief History of Time by Stephen Hawking, he’ll talk about how the Big Bang was the beginning of time, and how it’s not even really meaningful to ask what was “before” the Big Bang. While that’s true in a purely classical General Relativity description of the Big Bang, we know that such a description can’t be right… because our Universe also has Quantum Mechanics in it, and we have huge amounts of experimental evidence telling us that we need to take Quantum Mechanics seriously. The real story is that there is an extremely early epoch in the Universe (what I tend to think of as “the beginning” nowadays) about which we can make supportable statements based on our understanding of physics. However, we also know that we don’t understand physics well enough to really know what the Universe was like before that early epoch. So, it is meaningful to talk about a before, it’s just that that before is a “known unknown”.

For now.

The Astronomical Magnitude System

Background: What and Why?

Magnitudes are a system astronomers use to talk about the brightness of objects. They’re often a headache for astronomy students, because some things about them are counterintuitive. They also leave physicists who come to astronomy scratching their heads, because said physicists may not appreciate the historical reason why it actually makes sense to use magnitudes. I go back and forth myself on whether or not I should teach magnitudes in an introductory astronomy class. Sometimes I decide not to, because they really don’t add much to the understanding of the physics of astronomical systems, they’re just one more complication when it comes to dealing with numbers. However, at the moment, my opinion has vacillated towards thinking it is worth teaching, for two reasons. First, because they are ubiquitous in astronomy, you will see them referenced a lot. For instance, if you click on an object in Stellarium, you will be told the magnitude of that object. Second, and perhaps more importantly, it’s a worthwhile intellectual exercise. The magnitude scale is a logarithmic scale, and I think it’s good brain exercise to make students wrestle with that.

The two most important things to know about magnitudes— the first of these being one of the things that makes the difficult to work with— are as follows:

  • A larger value of the magnitude represents a dimmer object
  • The magnitude scale is a logarithmic scale, so a given difference of magnitudes represents a given multiple of brightness.

I’ll explore both of these in greater depth below as I give the definition of the magnitude system.

Historically, the ancients categorized stars in brightness classes. The brightest stars in the sky (all named) were called “first magnitude” stars. The ones that appeared a step down in brightness were “second magnitude” stars, and so forth. Modern astronomers reverse-engineered this historical listing of stars to give a numerical “magnitude” that corresponds to the measured brightness of the stars such that stars would more or less have the same magnitude as they had when classified by the ancients. Because the response of our eye is much more approximately logarithmic than it is linear, this led to the magnitude scale being a logarithmic scale.

Flux and Apparent Magnitudes

When an astronomer talks about “the” magnitude of an object, she is usually referring to the apparent magnitude. This corresponds to a measurement of how bright an object is to a given observer (almost always an observer at Earth), not to the intrinsic energy output of the object. To quantify brightness, we use the concept of flux or energy flux, which is defined as the rate at which energy is collected by a telescope with a 1 m2 aperture. Flux comes in units of W/m2 (where a W, of course, is a J/s). Notice the m2 in the denominator. This is what makes flux not depend on the telescope that’s looking at the object. If two telescopes look at the same object, the one with the larger aperture will collect more energy from the same object. To figure out the rate of energy collection, you have to multiply the flux by the collecting area of the telescope. (Real telescopes also have an efficiency that combines the reflectively of their mirrors and the intrinsic quantum efficiency of their detector, but we’ll not worry about that here.)

Suppose you’re looking at two stars, one with flux f1, the other with flux f2. The difference between the magnitudes of these two objects is then defined by:

m_1\ -\ m_2\ =\ -2.5\,\log\frac{f_1}{f_2}

Notice that this is a relative definition; it defines the difference in magnitudes in terms of the quotient of the fluxes of the two objects. The logarithm here is a base-10 logarithm. Warning: while the button on many calculators that performs this function is often labelled “log”, the log() function in many computer languages and applications actually does a natural logarithm (usually called “ln”) rather than a base-10 logarithm. To find out which you have, try taking the log of 10. If you get 1, then you’re doing a base-10 logarithm. If you’re doing a natural logarithm, you will get 2.3.

What is a log?

A logarithm is the inverse operation of exponentiation. This means that if you have

x\ =\ 10^a

then it’s also true that

a\ =\ \log(x)

You can think of a logarithm as being the operator that returns whatever you have to raise 10 to in order to get the argument of the logarithm. The natural logarithm (ln) is the same thing, only it’s what you raise e to rather than 10. Because logarithms are exponents, a few interesting properties apply to logs, which are useful when dealing with magnitudes:

\log(ab)\ =\ \log(a)\ +\ \log(b)
\log(\frac{a}{b})\ =\ \log(a)\ -\ \log(b)
\log(\frac{1}{a})\ =\ -\,\log(a)
\log(1)\ =\ 0
\log(a^b)\ =\ b\,\log(a)

Redux: Flux and Apparent Magnitude

Returning to our definition of the relative magnitude of two objects:

m_1\ -\ m_2\ =\ -2.5\,\log\frac{f_1}{f_2}

this also tells us how to get the ratio of fluxes from the magnitudes:

\frac{f_1}{f_2}\ =\ 10^{(m_1-m_2)/-2.5}\ =\ 10^{(m_2-m_1)/2.5}

The -2.5 in front of the logarithm is very important. The negative sign gives the feature that brighter objects have lower magnitudes. For instance, if object 1 has 10 times the flux of object 2, then:

m_1\ -\ m_2\ =\ -2.5\,\log(\frac{f_1}{f_2})\ =\ -2.5\,\log(10)\ =\ -2.5

Because the difference is negative, m1 is less than m2, given that object 1 has a higher flux. The 2.5 means tells you how many times one object must be brighter than another to have a certain difference in flux. It’s chosen so that a factor of 100 in flux corresponds to a difference of five magnitudes— it works out, because log(100)=2.

So far, however, I’ve only told you how to compare magnitudes. How, then, can we talk about “the” magnitude of an object? For that to happen, we must have some agreed-upon reference that is the standard of comparison. Here’s where things get sad. There two different standards in widespread usage. A more modern system, known as “AB” magnitudes, defines an object of a given flux as having magnitude 0. The more traditional system, still in widespread use by a lot of astronomers, and the system used by programs such as Stellarium, uses “Vega-based” magnitudes. In this system, the star Vega is defined to have magnitude 0. If you compare the flux of an object to the flux of Vega, you get “the” apparent magnitude for that object. The difference between Vega magnitudes and AB magnitudes matters when you’re talking about magnitudes through different filters. Note, however, that if you are comparing to objects, the difference in magnitude will be the same regardless of which system you’re on. The system matters when you cite the magnitude of an object in a given bandpass or filter.

As a few examples, on the Vega system the Sun has an apparent magnitude of -26.74. Sirius, the brightest star in the night sky, has a magnitude of -1.46. Vega has a magnitude of 0.03. (What? you cry. Why not exactly 0? It turns out that “Vega magnitudes” are based on a historical idealized Vega rather than on the real thing.) The North Star, Polaris, has a magnitude of 2.02. The dimmest stars you can see under a good dark sky, on a moonless night and away from city lights, will be magnitude 5 or 6. (Numbers are visual magnitudes, and are from SIMBAD, except for the Sun’s magnitude which is from this page from GSFC.)

Filters and “Color Index”

Up to now, we’ve been talking about the flux of an object, implicitly including all the flux at all wavelengths. The technical term for this is the bolometric flux. In reality, we generally do not collect the flux at all wavelengths. Every detector we use is sensitive only to a finite range of wavelengths. For example, our eyes are only sensitive to wavelengths in about the range 450-650 nm, but stars emit light at wavelengths outside that range as well as inside that range. What’s more, there is a benefit to talking about the flux in even smaller ranges of wavelengths; this allows us to quantify the color of an object, by comparing (say) its flux at red wavelengths to its flux at green wavelengths.

In order to talk about colors and fluxes through different filters, we have to choose a filter set to use. A filter is defined by its transmission function; that is, what fraction of the photons it lets through as a function of wavelength. Two of the common filter systems in use are the Johnson-Cousins filters and the Sloan Digital Sky Survey filters. This image shows the Johnson-Cousins passbands:

ubvri_400
UBVRI passbands from Bessel, 1990, PASP, 102, 1181

This system is sometimes called the “UBVRI” system. U stands for Ultraviolet, B for blue, V for visual, R for red, and I for infrared. All of these letters make sense except for V; V is not violet. V is more like green or yellow-green. It corresponds to more or less the peak of the eye’s sensitivity.

You can then define the flux through a filter as the flux coming from an object including just the photons transmitted through that filter. This will be less than the flux of the object as a whole (unless the object is very perverse and all of its light is emitted exactly at the wavelength where the transmission of the filter is exactly 1.0). A color, then, corresponds to the ratio of fluxes an object shows through different filters. For a given object, fB/fV would be the ratio of its B-band flux to its V-band flux. This ratio will be higher for bluer objects, because more of the light will be emitted at shorter wavelengths. If f0V and f0B are the fluxes of the reference object (i.e. Vega), then we have:

m_B\ -\ m_{0B}\ =\ -2.5\,\log\frac{f_B}{f_{0B}}
m_V\ -\ m_{0V}\ =\ -2.5\,\log\frac{f_V}{f_{0V}}

However, remember that the magnitude of Vega is defined to be 0, so:

m_B\ =\ -2.5\,\log\frac{f_B}{f_{0B}}
m_V\ =\ -2.5\,\log\frac{f_V}{f_{0V}}

If we subtract these two numbers, we get:

m_B\ -\ m_V =\ -2.5\,\log\left(\frac{f_B}{f_{0B}}\,\frac{f_{0V}}{f_V}\right)

That’s a bit of a mouthful, but it corresponds to the “color index” B-V of an object, defined by:

B\,-\,V\ =\ m_B\ -\ m_V

Notice that because a higher magnitude is a dimmer object, as B-V gets larger, it means that the flux in the B filter gets lower compared to the flux in the V filter. This is backwards from what might be intuitive.

Astronomers will very frequently cite the color index (something like B-V, B-R, V-I, or similar) as a way of talking about the color of stars or other astronomical objects.

Absolute Magnitudes and Distance Modulus

Apparent magnitude (or just “magnitude”) corresponds to flux, or the observed brightness of an object. We also have a magnitude defined that corresponds to the luminosity (intrinsic energy output) of an object, and we call that “absolute magnitude”. The absolute magnitude is defined as the magnitude that would be observed for an object by an observer 10pc away from the object. (pc means parsecs; one parsec is equal to 3.262 light-years, or 3.086×1016 meters.) This may seem arbitrary… and it is. It’s just the reference distance. We had to pick something. So, hey, why not 10pc?

Imagine we’re looking at two stars, the Sun, and another star just like the Sun that is exactly 10pc away. The flux of the Sun is f, and the flux of the other star we shall call fG. Both have the same luminosity (which we’ll just call L). We know that flux depends on distance by:

f\ =\ \frac{L}{4\pi\,d^2}

So, we have:

\frac{f_\odot}{f_G}\ =\ \left(\frac{L}{4\pi\,{d_\odot}^2}\right)\,\left(\frac{4\pi\,{d_G}^2}{L}\right)

Simplifying:

\frac{f_\odot}{f_G}\ =\  \left(\frac{d_G}{d_\odot}\right)^2

From the definition of magnitudes, we also have:

m\ -\ M\ =\ -2.5\,\log\frac{f_\odot}{f_G}

where m is the magnitude of the Sun and M is the magnitude of the Sun as observed 10pc away. Because the magnitude of the Sun as observed 10pc away is the definition of absolute magnitude, M is also the absolute magnitude of the Sun. By convention, we use capital letters for the absolute magnitudes of stars. (However, be careful! We’ll often use capital letters for color indexes even when we’re talking about fluxes. It turns out that (discounting redshift effects) the absolute and apparent color index of an object will be exactly the same, so we can afford to be sloppy.)

Putting these two equations together, we have:

m\ -\ M\ =\ -2.5\,\log\left[\left(\frac{d_G}{d_\odot}\right)^2\right]
m\ -\ M\ =\ 5\,\log\frac{d_\odot}{10\,\mathrm{pc}}

This last equation defines the distance modulus for the Sun. To get there from the previous equation, I’ve used a couple of the properties of logarithms given earlier. In general, for an object a distance d away from us, its distance modulus is the difference between its apparent magnitude m and absolute magnitude M, and is:

m\ -\ M\ =\ 5\,\log\frac{d}{10\,\mathrm{pc}}

For the Sun, the distance from the Earth is 1AU. Putting this in to the distance modulus equation, together with a unit conversion from AU to pc, we can figure out the absolute magnitude of the Sun:

m\ -\ M\ =\ 5\,\log\frac{d}{10\,\mathrm{pc}}
M\ =\ m\ -\ 5\,\log\frac{d}{10\,\mathrm{pc}}
M\ =\ -26.74\ -\ 5\,\log\left[\left(\frac{1\,\mathrm{AU}}{10\,\mathrm{pc}}\right)\,\left(\frac{1\,\mathrm{pc}}{206265\,\mathrm{AU}}\right)\right]
M\ =\ 4.83

As a final note, you can, of course, turn around the distance modulus equation to write the distance in terms of the absolute and observed magnitudes of an object:

d\ =\ (10\,\mathrm{pc})\,10^{(m-M)/5}

Summary

The magnitude system is a system astronomers use to talk about the brightnesses of stars. It is a logarithmic system, so a step of magnitude corresponds to a factor of energy output or energy detected. It is also backwards, so that larger magnitudes correspond to dimmer objects. Magnitudes can be “bolometric magnitudes” (taking into account all flux at all wavelengths), but more commonly we talk about magnitudes through a given filter. The color index is the difference of two magnitudes; by convention, we subtract the magnitude through the redder filter from the magnitude through the bluer filter. Absolute magnitudes are defined as the magnitude that would be observed by an observer 10pc away from the object; absolute magnitudes correspond to the luminosity of an object, while apparent (or observed) magnitudes correspond to the flux of an object. Finally, the distance modulus is defined as the difference between the observed and absolute magnitudes of an object, and corresponds directly to the distance between you and that object.