Astronomy/Physics/Math Question

[Timothy Campbell]

I read (well, skim) the journal "Nature" from time to time and while I'll admit that I don't understand most of it, there's something that keeps coming up that I think I should ask about.

In a recent article, it was stated that the median value of the estimates for the Hubble Constant is:

        -1     -1
70 km s    Mpc

I've seen this "-1" superscript in several articles, and I can't figure out what it means. My best guess is that it indicates that it is a speed that is changing over time and this "-1" indicates that they're "nailing it down" even though it isn't, umm, nailable. I'm probably wrong.

Can anybody help me out?

[hawthorne]

x^-1= 1/x
So where you see kmh^-1 it is the same as km/h

picmr

[Chronos]

picmr's explanation is correct-- it's seventy kilometers per second per megaparsec-- but it is true that it's changing with time. Really anal physicists will refer to it as the "Hubble parameter" rather than "constant". However, it's changing slowly enough that it's perfectly acceptable to "nail it down". Over a typical person's lifespan, it'll only change by a few parts in a billion, and the value is only known to a few parts in a hundred or ten or so, so its change is totally insignificant, on human scales.

[bibliophage]

The SI (Système International [d'Unités]) really frowns on division in units, so negative powers are used instead. For example, the unit of density is kg•m^-3, not kg/m³.

In the example you gave, the Hubble constant is 70 kilometers per second per megaparsec. ("Mega" means million and a parsec is about 3¼ lighyears.) It means that a galaxy one megaparsec away would be expected to be receding from us at about 70 kilometers per second. But a galaxy two megaparsecs away would be expected to be receding at 140 kilometers per second.

[Timothy Campbell]

Oh! Thanks for explaining the notation. I was worried it was something to do with imaginary numbers, which I'm pretty sure couldn't be properly explained in a single post!

Of course, if anybody wants to take a crack at it, I'd be a very interested reader! Yes, I know it's the square root of minus one and I know that you "get rid of it" before you get to the end of what you were doing. And I think I've heard that it helps you calculate phase angles in oscillations, or something like that, but ...

Who USES it!? With a side order of "Why the heck does it work!?"

If you can explain that, maybe you can explain some REALLY tough questions, like why soccer fans don't fall asleep in the stands.

[threemae]

Imaginary Numbers in a Nutshell (Or at least one of dem' big-ass Brazil nuts):

The defenition of an imaginary number is a number expressed as a produect of a real number and i. i^2 is defined as i^2=-1. Unlike any real number, i can be raised to an even integer and be negative unlike either 3 or -3 for instance. 3^2=9, (-3)^2=9. Just remember this definition and you can basically do everythig that you regularly do with powers, except assume that an even power is positive. The only real use that i holds is for simplifiying thigs like (-9)^(1/2) to 3i. Occasionally, i will combine with other is to create an i to an even power which can be turned into a real number.

How exciting. Just like soccer.

No, really.

[Punoqllads]

Originally posted by Timothy Campbell
Who USES (i)!? With a side order of "Why the heck does it work!?"

i is very useful in describing wave equations. Think of the imaginary numbers as numbers 90^o out of phase with real numbers. Instead of just a number line, in one dimension, you have an imaginary number line perpendicular to it, sort of like a y axis on a graph.

As to why in the heck it works, I'm not sure how I can answer that. It's defined that way.

[scr4]

Originally posted by Chronos
Really anal physicists will refer to it as the "Hubble parameter" rather than "constant"

I thought it's because the value changes every time someone publishes a new article on the topic...

[Timothy Campbell]

So imaginary numbers are a number line perpendicular to our regular, boring kind of numbers. X and Y axis. Hmm.

Do I get to name the numbers that appear on the Z axis?

[Lance Turbo]

Originally posted by bibliophage
The SI (Système International [d'Unités]) really frowns on division in units, so negative powers are used instead. For example, the unit of density is kg•m^-3, not kg/m³.

Ok, I'm cruising down the highway at 60mph. That converts to the metric 96 kph (approx). Which equals 96000 meters per hour. Divide by 3600 and you have around 27 m/s. This however, would make the SI guys frown so we change it to 27 m s^-1. But wait s^-1 is the same thing as hz. So I am actually cruising down the highway at 27 mhz.

I think I'll go convert my car's speedometer from "miles per hour" to "meter hertz".

[dtilque]

Originally posted by Lance Turbo
I think I'll go convert my car's speedometer from "miles per hour" to "meter hertz".

While you're at it, compute your fuel usage per distance travelled in hectares.

[scr4]

You might also note that in Hubble Constant (around 70 km s^-1 Mpc^-1), Mpc and km are both units of distance, so they cancel out. You're left with s^-1=Hz, so the Hubble Constant is actually about 2.2x10^-18 Hz.

Of course, the better way to think about it is "(kilometers per hour) per Mpc". The farther away it is, the faster it's moving.

[micilin]

Originally posted by Timothy Campbell
Do I get to name the numbers that appear on the Z axis?

Sorry Tim, a guy named William Hamilton got to do that, he called it j, I think. In order to make the sums work out, though, he had to add another axis, so apart from i, there's also j and k. The deal is that i² = j² = k² = ijk = -1

This is part of a theory called 'quaternions', and apparently it's very handy for working out how robots move. It's also very vaguely remembered from a single college lecture, so I might have got the axes mixed up. I'm sure somebody will be along in a while to correct me, and to bore you about it more.

If, while they're at it, they could explain how galaxies receding at faster than the speed of light doesn't violate relativity, I'd much appreciate it.

[Achernar]

Quoth dtilque: "While you're at it, compute your fuel usage per distance travelled in hectares."

Well that wouldn't work. Hectares are units of area, not distance. You might be able to do it in hectare^½, though.

Quoth micilin: "If, while they're at it, they could explain how galaxies receding at faster than the speed of light doesn't violate relativity, I'd much appreciate it."

Well, I can't explain it, but I don't think it's an issue. The Hubble Constant, as I understand it, can also be used to determine the approximate size of the Universe. Just figure out how far away something moving at c would have to be, and that's the Universe's upper size limit. Certainly, if light hit us from a galaxy receding at faster than c, it would reach us with a negative wavelength.

[Timothy Campbell]

Quote: "Just figure out how far away something moving at c would have to be, and that's the Universe's upper size limit."

Umm, since c is a constant, are you saying the universe will grow to a certain size then hit a wall!? I think you mean "the observable universe".

Quote: "Certainly, if light hit us from a galaxy receding at faster than c, it would reach us with a negative wavelength."

Oh, so that's what imaginary numbers are used for!

[panamajack]

Another discussion of the uses of imaginary numbers :

Have you ever used the square root of -1 in real life?

Also, while 1 Hertz = 1 s^-1, Hertz refers to frequency (i.e. cycles per second). I don't think it makes much sense to apply it to a quantity it doesn't fit. It would seem even more strange to claim you were moving at 25 meter-becquerels. And Radians are essentially dimensionless; doesn't mean you can apply affix rad to any quantity.

Of course, if it's a rental car, then everything's different.

panama jack

[DrMatrix]

The system of complex numbers is just another system of numbers. We use different systems of numbers for different applications without thinking about it. If you are counting eggs in a basket, you use whole numbers. If you are measuring butter in pounds, you need non-integer numbers. For some applications complex numbers are more useful.

As to how galaxies can recede faster than light:
Special Relativity states that the speed of light is constant and nothing with mass can travel faster than light. SR applies only in inertial reference frames (no acceleration and no gravity).

General Relativity states that nothing travels faster than light (in its neighborhood). The speed of light is not constant in GR nor is there an upper limit to speed of an object.

Light from a galaxy receding faster than the speed of light will never reach us. There is an event horizon where the recession velocity equal the speed of light. The observable universe is what is within the event horizon.

[micilin]

Originally posted by DrMatrix
Special Relativity states that the speed of light is constant and nothing with mass can travel faster than light. SR applies only in inertial reference frames (no acceleration and no gravity).

General Relativity states that nothing travels faster than light (in its neighborhood). The speed of light is not constant in GR nor is there an upper limit to speed of an object.

In it's neighbourhood? Do you mean within a couple of blocks, or is it a ZIP code thing? Seriously though, what's a neighbourhood? The observable universe?

[Lance Turbo]

Originally posted by panamajack
Also, while 1 Hertz = 1 s^-1, Hertz refers to frequency (i.e. cycles per second). I don't think it makes much sense to apply it to a quantity it doesn't fit.

Thanks for tip panamajack. You saved me some embarrassment because I was totally serious about using meter hertz as a way to measure velocity.

[DrMatrix]

Sorry. I should have left out "(in the neighborhood)". I just muddied the waters with that.

In GR, light is the fastest signal. The speed of light (in GR) is not a constant. At the event horizon of a black hole, light directed outward stands still. The speed of light outward is zero. This is not a contradiction of SR, because any inertial frame would be falling into the hole at the 186,000 mps at the horizon and even faster inside.

[dtilque]

Originally posted by Achernar
Quoth dtilque: "While you're at it, compute your fuel usage per distance travelled in hectares."

Well that wouldn't work. Hectares are units of area, not distance. You might be able to do it in hectare^½, though.

Fuel usage per distance travelled is in liters/kilometer. In base SI units that would be m³/m = m² and can easily be converted to hectares. Of course, we usually measure kilometerage (is that a word?) in kilometers per liter so perhaps I should have said hectares^-1.

[Achernar]

Ah, that's interesting - I never thought about it like that. Cool. I admit that I was misreading your post:

• compute your fuel usage per (distance travelled in hectares).

instead of:

• compute your (fuel usage per distance travelled) in hectares.

Stupid language.

As for the GR thing, like I said, I can't explain it, but I do want to apologize for using "Universe" for "observable Universe" - I never really understood the distinction between the two. Also, when I referred to the Universe's "upper size limit" I should have used the term "current upper size limit". As I understand it, the size of the (observable) Universe is only a constant in the same respect that Hubble's Constant is a constant.

[Achernar]

Er, the Hubble Constant. Not Hubble's Constant. Sorry for any confusion potential.