Calculating Average Time for Travel Using Acceleration and Velocity

[Mantikore]

So I am having trouble with a problem.. and this has been a thorn in my side for quite some time now. I am working on a science fiction idea where I use the rate of expansion of the universe as the rate of acceleration of a spaceship. The rate of expansion is about 70 km/sec/megaparsec.. and I think the megaparsec is throwing me off because when I reduce it to just km/sec, it's an incredibly small decimal with something like 18 zeros. Plus, for some reason.. acceleration seems to be kicking my butt.

So I was wondering if I could use a shortcut of sorts. If I knew the amount of time it'd take if I traveled at the initial velocity and knew the amount of time it would take with the final velocity, can I just take the average of those 2 times? For example, if I wanted to travel 7 billion light years and my initial velocity was 1/2 the speed of light.. if I traveled at 1/2 light speed the entire trip, it would take me 14 billion years. My final velocity is the speed of light, which if I traveled at light speed the entire trip.. it would take me 7 billion years. The average of those 2 times would be 10.5 billion years.. so my question is this, is that right? If I were to travel 7 billion light years and had an initial velocity of 1/2 the speed of light and my final velocity upon arriving was the speed of light, would it take me 10.5 billion years?

Thank you in advance.

 

[Orodruin]

when I reduce it to just km/sec

You cannot do that. The Hubble parameter has dimensions 1/time, not length/time. Note that Mpc is a length unit.

The second part of your post is difficult to decipher. It is not clear what you want to compute.

 

[Mantikore]

You cannot do that. The Hubble parameter has dimensions 1/time, not length/time. Note that Mpc is a length unit.

The second part of your post is difficult to decipher. It is not clear what you want to compute.

The Hubble constant is about 70 km/sec/megaparsec. So something one megaparsec away will be expanding away from us at the rate of 70 km/sec. Something that is 2 megaparsecs away will be moving away from us at a rate of 140 km/sec. 10 megaparsecs, it'll move away from us at a rate of 700 km/sec.. and so on.

Based off of that, if you reduce the megaparsecs (divide 70 by the number of km in a megaparsec) you would be left with the rate that is reduced to km/sec and not km/sec/megaparsec.

As for the second part.. I'll try a different example. Say I don't know my rate of acceleration and I was initially traveling at a speed of 10 mph and traveled 50 miles while accelerating to a final velocity of 50 mph, can I figure out how long it took me based off of this information? If I traveled those 50 miles at 10 mph, it would take me 5 hours. If I traveled those 50 miles at 50 mph, it would take me 1 hour. The average of those times is 3 hours, so is that how long it would take me to travel 5 miles while under a constant acceleration from 10 mph to 50 mph?

 

[Orodruin]

Based off of that, if you reduce the megaparsecs (divide 70 by the number of km in a megaparsec) you would be left with the rate that is reduced to km/sec and not km/sec/megaparsec.

No, you will be left with a number with units 1/s. This should be a very very small number. It tells you how many times its original length a distance will grow to be one second later.

As for the second part.. I'll try a different example. Say I don't know my rate of acceleration and I was initially traveling at a speed of 10 mph and traveled 50 miles while accelerating to a final velocity of 50 mph, can I figure out how long it took me based off of this information? If I traveled those 50 miles at 10 mph, it would take me 5 hours. If I traveled those 50 miles at 50 mph, it would take me 1 hour. The average of those times is 3 hours, so is that how long it would take me to travel 5 miles while under a constant acceleration from 10 mph to 50 mph?

Don’t confuse metric expansion with actual motion.

 

[Mantikore]

No, you will be left with a number with units 1/s. This should be a very very small number. It tells you how many times its original length a distance will grow to be one second later.Don’t confuse metric expansion with actual motion.

Omg.. ok.. I will show you step by step.

The rate of expansion is 70 km/sec/megaparsec. There are 3.26 million light years in a megaparsec.. so divide 70 by 3.26 million and you are left with 0.00002147239263803680981595092 km/sec/light year. There are 9,460,731,000,000 km in a light year, so divide 0.00002147239263803680981595092 by the number of km in a light year and you are left with 0.00000000000000000226963356616278486 km/sec. So for every km, space is expanding at a rate of 0.00000000000000000226963356616278486 km/sec.

 

[Orodruin]

so divide 0.00002147239263803680981595092 by the number of km in a light year and you are left with 0.00000000000000000226963356616278486 km/sec.

This is where you are wrong. 1 km/ly is about 10^-13, a dimensionless number. So 1 km/s/ly is about 10^-13 s^-1.

You cannot convert a number with units 1/time to one with units length/time. To get a speed out you need to multiply by a distance.

Omg

I suggest you drop the attitude. It is not my fault that you are doing things wrong and refusing to listen when you are told what you are doing wrong.

 

[Mantikore]

This is where you are wrong. 1 km/ly is about 10^-13, a dimensionless number. So 1 km/s/ly is about 10^-13 s^-1.

You cannot convert a number with units 1/time to one with units length/time. To get a speed out you need to multiply by a distance.I suggest you drop the attitude. It is not my fault that you are doing things wrong and refusing to listen when you are told what you are doing wrong.

There are 9,461,000,000,000 km in a light year, how is that dimensionless? If you can calculate the rate of expansion by megaparsec, there is no reason you can't do it other ways. The rate of expansion between 2 objects 1 megaparsec away from one another is 70 km/sec. For 2 objects that are 2 megaparsecs, 140 km/sec. If there were 2 objects that are 0.5 megaparsecs apart from one another, the rate of expansion between them would be 35 km/sec. For 2 objects that are 0.25 megaparsecs apart, the rate of expansion between them would be 17.5 km/sec. So why can't you not look at it in other units? Based off of the rate of expansion for one megaparsec, you should be able to convert it to find out the rate of expansion per parsec, per light year, per km, or even per mile if you wanted to.. and that's what I did, I found the rate of expansion between 2 objects that are only 1 km apart.

 

[PeterDonis]

There are 9,461,000,000,000 km in a light year, how is that dimensionless?

@Orodruin didn't say light-years were dimensionless. He said the ratio km/light-year is dimensionless. That's true. One km/light-year is the dimensionless ratio 1/9,461,000,000,000.

Similarly, 1 km/megaparsec is the dimensionless ratio 1/(9,461,000,000,000 * 3,260,000), since there are 3.26 million light-years in a megaparsec.

Therefore, 1 km/sec/megaparsec is the same as 1/(9,461,000,000,000 * 3,260,000) inverse seconds. So it has units of inverse seconds, just as @Orodruin said.

 

[PeterDonis]

Moderator's note: Thread moved to the cosmology forum.

 

[PeterDonis]

I suggest you drop the attitude.

The moderators suggest the same thing.

 

[Mantikore]

Don’t confuse metric expansion with actual motion.

I missed this the first time around, but I am not. Like I said in my OP, I wanted to use the rate of expansion as my rate of acceleration for my science fiction idea. I'm not saying that they are the same, just that I want to use the rate of expansion to play around with. That's why I wanted to break it down the Hubble constant and move it away from megaparsecs to something more manageable. I figured if I found the rate of expansion between 2 objects that are only 1 km apart, it'd be more manageable.. but it kind of isn't.

 

[PeterDonis]

That's why I wanted to break it down the Hubble constant and move it away from megaparsecs to something more manageable.

The value I gave in post #8, 1/(9,461,000,000,000 * 3,260,000) inverse seconds, can be multiplied by any distance to get a speed. If you multiply it by 1 km, you get 1/(9,461,000,000,000 * 3,260,000) km/s. So that's the speed at which the current expansion of the universe would cause two objects 1 km apart to move away from each other.

If you want an acceleration, though, the above is not sufficient, because it doesn't tell you at what rate that number 1/(9,461,000,000,000 * 3,260,000) inverse seconds is changing. For that, you need something called the "deceleration parameter" (the name of which is a holdover from the period before it was discovered that the expansion is accelerating). Because of the way it's defined (see my comment on the name just now), what cosmologists call "acceleration" of the expansion of the universe corresponds to this parameter, denoted $q$, being negative. The best current value that I can find for it is $q = - 0.55$, from here:

https://ned.ipac.caltech.edu/level5/Sept02/Reid/Reid6.html

(Note that this value comes from "theory"--which as far as I can tell just means it's model-dependent, not something we can observe directly.)

The equation for $q$ in terms of $H$ and its time derivative $\dot{H}$ is:

$$
q = - \left( 1 + \frac{\dot{H}}{H^2} \right)
$$

We can invert this to get an equation for $\dot{H}$:

$$
\dot{H} = - \left( q + 1 \right) H^2
$$

When I plug in the numbers, I get a value of $- 4.73 \times 10^{-40}$. Note that this value is negative--$H$ is decreasing, even though cosmologists use the word "acceleration" to describe what's going on.

Fortunately, there is at least one quantity that is in fact increasing: it's $\ddot{a} / a$, which is the second time derivative of the scale factor, divided by the scale factor itself (so that its units are inverse seconds squared, just like $\dot{H}$). This is given by:

$$
\frac{\ddot{a}}{a} = \dot{H} + H^2
$$

Plugging in numbers again, I get $\ddot{a} / a = 5.78 \times 10^{-40}$. This is the thing you would multiply by some distance to find out at what acceleration the accelerating expansion of the universe is "pulling" two objects separated by that distance apart. So two objects separated by 1 km would be accelerating apart at $5.78 \times 10^{-40} \text{km/s}^2$.

 

[rootone]

objects that are only 1 km apart

Expansion is calculated or observed in light years, nothing interesting happens on a kilometer scale.

 

[mfb]

(Note that this value comes from "theory"--which as far as I can tell just means it's model-dependent, not something we can observe directly.)

Not yet. ELT could do it, although the required observation time (simulations work with 4000 h) is probably unreasonable.

 

[Edem]

Hi, I came upon this site while doing personal research. I'm a layman and don't get much of the math, but I do enjoy contemplating the theories. This thread is the most recent relating to topic, and allowing a reply post. Please pardon any violation of guidelines.
I have a question about acceleration.

Do astronomers and cosmologists take relative time into account when deducing the accelerating expansion of the universe?

Something about this current scientific dictum never seemed right to me.
If I understand correctly, this is based on the fact that red shift is observably greater when looking at more distant galaxies. Which leads astronomers to conclude that; farther galaxies "are" moving away faster than closer ones. Therefore, the expansion "is" accelerating.
It seems that the conclusion should be that; farther galaxies "were" moving away faster in the very distant past than closer ones "were" moving away in the less distant past.
Everyone knows that light has a speed and is not simultaneously observed, so when we look at anything we are seeing it as it was in the past. The farther we look in distance, the farther we see into the past. So when we look at galaxies billions of light years away we are seeing them as they were billions of years ago. Are we not also seeing their red shift as it was billions of years ago? This seems not necessarily indicative of their current red shift (speed), and does not inform us on the current state of the universe.
To my logic it would be expected that everything would be moving faster when looking back in time towards the Big Bang. I would think that looking at closer galaxies would be a better indication of the current rate of expansion. And in fact, expansion doesn't seem to exist locally at all.
So we know it was expanding, but how can we know if the universe is still expanding? If far distant galaxies stopped moving away from us billions of years ago, it wouldn't be observable to us for billions of years in the future. What the current state of the extended universe is appears to be a question that can't be answered by looking at the past.
In conclusion, since closer galaxies are moving away slower than farther ones I would infer that the expansion of the universe has been up to a certain point in time decelerating. Which I believe had been previously postulated, based on relativity and gravity theories.
As an aside, there would also be no need for dark energy, which was created to explain the accelerated expansion.
Any help in understanding why my logic is wrongminded is much appreciated. Thanks

 

[PeterDonis]

If I understand correctly, this is based on the fact that red shift is observably greater when looking at more distant galaxies.

No, that's just ordinary "expansion"--it will be true whether the expansion is accelerating, decelerating, or neither.

What tells us the expansion is accelerating (more precisely, that it started accelerating a few billion years ago and has been ever since) is the specific relationship between redshift and other observables--the key ones are the brightness of the galaxies and their angular size. Different "expansion profiles" of the universe over time give different relationships between these three observed quantities, so the actual relationship can distinguish between different possible models of the universe. The model that best matches the actual data is one in which, for the past few billion years, the expansion has been accelerating.

 

[PeterDonis]

red shift is observably greater when looking at more distant galaxies

It's also worth noting that the "distance" to a particular galaxy is not something we directly observe. The "distances" quoted by cosmologists are in fact outputs of the particular model they are using--which, as I noted in my previous post, has to be obtained by looking at the relationship between redshift, brightness, and angular size over many galaxies. Unfortunately, most cosmologists present their results in a way that obscures all this--which is fine for them, since they all understand what the actual data is and how the process of using the data to distinguish between models works, but makes it very difficult for a lay person to correctly understand what is being claimed, and to distinguish between what we actually observe, what key things allow us to arrive at a specific model based on what we actually observe, and convenient numbers (like the "distances" to various galaxies) that are outputs of the models once they have been selected.

 

[Edem]

No, that's just ordinary "expansion"--it will be true whether the expansion is accelerating, decelerating, or neither.

What tells us the expansion is accelerating (more precisely, that it started accelerating a few billion years ago and has been ever since) is the specific relationship between redshift and other observables--the key ones are the brightness of the galaxies and their angular size. Different "expansion profiles" of the universe over time give different relationships between these three observed quantities, so the actual relationship can distinguish between different possible models of the universe. The model that best matches the actual data is one in which, for the past few billion years, the expansion has been accelerating.

Thank you Peter,
I knew I was missing something. So it's a combination of observations, and not just the difference in red shift between near and far galaxies.
But I still don't understand how any observation that looks that far into the past can inform us on the current state of the universe.

 

[Orodruin]

But I still don't understand how any observation that looks that far into the past can inform us on the current state of the universe.

As Peter said, it is not a direct observation. You cannot observe what is happening "now" in any other part of the Universe, you have to infer it from the model. However, there are good reasons to believe that the model is reasonably accurate.

As an analogy, if you know the initial velocity and position of an object moving in a gravitational field, you can predict where it is going to be after some time. If you trust your theory sufficiently well, you do not need to observe where it is.

 

[Edem]

As Peter said, it is not a direct observation. You cannot observe what is happening "now" in any other part of the Universe, you have to infer it from the model. However, there are good reasons to believe that the model is reasonably accurate.

As an analogy, if you know the initial velocity and position of an object moving in a gravitational field, you can predict where it is going to be after some time. If you trust your theory sufficiently well, you do not need to observe where it is.

So, the model is an extrapolation of observational data. Which can then be checked against more observed data? That makes some sense out of it. Thanks

 

[Edem]

Thank you both for your prompt replies. I do have a better understanding of this topic after reading your posts. Even if it does give me more to think about.