Video: Finding the Recession Velocity of a Distant Galaxy

An absorption line normally at 552 nm appears at 585 nm in the spectrum of light coming from a galaxy. How fast is the galaxy moving away from Earth? Give your answer in kilometers per second and in scientific notation to 3 significant figures.

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Video Transcript

An absorption line normally at 552 nanometers appears at 585 nanometers in the spectrum of light coming from a galaxy. How fast is the galaxy moving away from Earth? Give your answer in kilometers per second and in scientific notation to three significant figures.

Okay, so in this question, we’re looking at an absorption line or, in other words, a line on an absorption spectrum, which we have been told normally appears at 552 nanometers. Now, when we say normally, what we mean is that in an absorption spectrum taken on Earth, this particular absorption line appears at 552 nanometers. However, in the absorption spectrum taken from the light coming from a galaxy, this same absorption line appears at 585 nanometers. In other words then, we can see that the wavelength of light coming from this galaxy has increased relative to the wavelength of light that we would see on Earth.

Now this increase in wavelength of light coming from the galaxy is known as red shift because the light is shifted to the red end of the spectrum. And this red shift occurs because the source of the light in question is moving away from the observer. In other words, when the absorption line that we said was normally at 552 nanometers was found, the source of that light was on Earth itself. And therefore was not moving relative to the observer. However, the galaxy from which we’re gathering light does seem to be moving away from the observer, which in this case is a scientist on Earth. And because of that, the light coming from the galaxy is red shifted.

Now, we’ve been given the original wavelength of a particular absorption line as well as the wavelength it moves to in the light coming from this particular galaxy. So based on these numerical values, we need to work out how fast the galaxy is moving away from Earth. To do this, we need to recall the equation that tells us that Δ𝜆, the difference in wavelength of the absorption line in the light from the galaxy and the light from a source that is stationary relative to the observer, divided by the wavelength of the spectral line, from the source of light that’s stationary relative to the observer, is equal to the velocity of the galaxy away from Earth divided by the speed of light.

And at this point, we can see that we’ve got enough information to answer this question. Because Δ𝜆, the difference in wavelengths of the absorption line, is simply given as Δ𝜆 is equal to 𝜆 subscript galaxy, the wavelength of the absorption line from the light from the galaxy, minus 𝜆 subscript rest, which is once again the wavelength of the spectral line in the light coming from a source that is stationary relative to the observer. And we have values for both 𝜆 galaxy and 𝜆 rest. So we can work out what Δ𝜆 is in the numerator of this equation. And additionally, we have 𝜆 rest and we know that the speed of light, 𝑐, is about three times 10 to the power of eight meters per second.

So we can take this equation and rearrange it to solve for 𝑣, the velocity of the galaxy relative to the earth. So let’s go about doing that. We can do this by multiplying both sides of the equation by the speed of light, 𝑐. This way, the speed of light on the right-hand side will cancel out. And what we’re left with is that the difference in wavelength between the two spectral lines multiplied by the speed of light 𝑐 divided by the spectral line’s wavelength, when coming from a source that’s stationary relative to the observer, is equal to the speed of the galaxy away from Earth, 𝑣.

So now that we know this, let’s plug in some values. We can say that the speed of the galaxy 𝑣 is equal to firstly Δ𝜆, which we know is 𝜆 sub galaxy minus 𝜆 sub rest or, in other words, 585 nanometers minus 552 nanometers. And then we multiply this parenthesis by the speed of light, which we know is three times 10 to the power of eight meters per second. And then we have to divide this whole thing by 𝜆 sub rest, which once again we know is 552 nanometers.

Now, firstly, when we carry out this subtraction, we see that it becomes 33 nanometers. And then we can see that the unit of nanometers in the numerator cancels with the unit of nanometers in the denominator. Therefore, our final unit is going to be meters per second. And to find the numerical value, all we need to do is to find 33 multiplied by three times 10 to the power of eight divided by 552. Altogether then, we find that the velocity of the galaxy away from Earth is equal to 0.17935, dot, dot, dot times 10 to the power of eight meters per second. But remember, we need to give our answer in kilometers per second and in scientific notation and to three significant figures.

So let’s start by putting our answer in scientific notation. To do this, we can recall that a number in scientific notation is written as 𝑎 times 10 to the power of 𝑏. Where 𝑎 is any number between one and 10, one is included in this range but 10 is not, and 𝑏 is an integer. So based on this information, we can see that our number over here is not in scientific notation. Because our value of 𝑎 does not lie between one and 10.

So to remedy this, what we can do is to think about what this 10 to the power of eight means. 10 to the power of eight simply means 10 times 10 times 10, and so on and so forth. We’re multiplying this eight times. And so, what we can do is to transfer one of these times 10s into our value of 𝑎. What that leaves us with then is 1.7935. That’s our value of 𝑎 now. And we multiply this by 10 to the power of seven. Because remember, we took this times 10 and moved it into our value of 𝑎. So there’s one fewer powers of 10 left in this part here. And so, we can say that the value of the velocity of the galaxy moving away from Earth is 1.7935 times 10 to the power of seven meters per second.

So we’ve dealt with the scientific notation part. Now, let’s convert this to kilometers per second. To do this, we can recall that one meter is equivalent to one thousandth of a kilometer. And therefore, one meter per second is equal to one thousandth of a kilometer per second. And hence, to convert from meters per second to kilometers per second, we need to divide this number by 1000. Or, in other words, we need to divide it by 10 to the power of three, which is 1000. And so, we can recall that 10 to the power of seven is equal to 10 times 10 times 10 times — seven times. And 10 to the power of three is 10 times 10 times 10. So three of the powers of 10 are going to be cancelled from the numerator. And we’re going to be left with 10 to the power four. At which point, we’ve converted to kilometers per second. So tidying everything up a bit, we see that our velocity is 1.7935 times 10 to the power of four kilometers per second. So we’ve now converted to kilometers per second as well.

The last thing we need to worry about is rounding our answer to three significant figures. So here’s significant figure number one, number two, and number three. Now we look at the next significant figure, which is this value here. It’s a three. And so, because three is less than five, our third significant figure is going to stay the same. It’s not going to round up. And so, at this point, we’ve found the answer to our question, to three significant figures as well. We can say that the speed at which the galaxy’s moving away from Earth is 1.79 times 10 to the power of four kilometers per second. Given in the correct units of kilometers per second and in scientific notation and to three significant figures.

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