The Hubble constant is an important
constant in astronomy. It is often measured in units of
kilometers per second per megalight-year. Which of the following units could
Hubble constant also be expressed in? A) Meters per second, B) meter
squared per second, C) per meter, D) per second, E) meters.
Now in this question, we’ve been
told that the Hubble constant — which we’ll call 𝐻 subscript zero — is given in
units of kilometers per second per megalight-year. By the way, this convention is
often used to represent units. In other words, the square
parentheses around the 𝐻 subscript zero represent the units of 𝐻 subscript
So anyway, we’ve been given the
units of the Hubble constant and we need to find which of the following units could
the Hubble constant also be expressed in. In other words, this set of units
has to be equivalent to one of these. Now to work out which one it is,
let’s first think of these units as kilometers per second being a fraction and then
being divided by megalight-years.
In other words, what we have is a
fraction kilometers per second being divided by megalight-years. But then, this is the same as
multiplying by one over that thing. In other words, what we’ve got is
kilometers per second multiplied by one divided by megalight-years. At which point, we can multiply the
numerators and multiply the denominators. What this leaves us with is just
kilometers in the numerator because we had kilometers multiplied by one and in the
denominator, we’ve got seconds multiplied by megalight-years.
Now at this point, we can recall
that a kilometer is a unit of distance and a second is a unit of time. So what does the unit
megalight-years represent? Well, a light-year is the distance
travelled by light in one year. And hence, a light-year is a unit
So a megalight-year is simply 10 to
the power of six light-years because remember the prefix mega just means 10 to the
power of six. And hence, what we have is another
unit of distance in the denominator. So at this point, we’ve got the
units of the Hubble constant, which are equal to the units of distance divided by
the unit of time and unit of distance.
So we could feasibly convert
kilometers into megalight-years or megalight-years into kilometers; they both
measure distance. At which one, we’d have some
numerical value to multiply or divide by depending on which we converted to
which. But that’s not relevant.
The point is that we could convert
megalight-years, for example, to kilometers, at which point they would both cancel
out of course leaving the numerical factor. But the unit that would remain is
one divided by seconds because at this point when the kilometers cancel, we’ve got
nothing in the numerator. So we stick a one there. And in the denominator, we still
have a unit of seconds.
Now, once again, we’re ignoring the
numerical factor that we had earlier because it’s not relevant. It’s only going to change the
number in front of the unit, which we could conventionally write of course as the
numerical factor multiplied by whatever 𝐻 nought was in kilometers per second per
megalight-years. But then the remaining unit is one
divided by seconds.
And of course, another way to write
one divided by seconds is seconds to the power of negative one because anything
raised to a negative power is the same as one divided by that object to the positive
power. A way to say this mathematically is
that one divided by 𝑎 to the power of 𝑛 is the same thing as 𝑎 to the power of
And so at this point, we’ve arrived
at the final answer. We can see that out of the options
we’ve been given, the correct one is option D: the Hubble constant is often given in
units of kilometers per second per megalight-year. However, another way to express the
Hubble constant is in units of per second.