B How far away can a World So Newly Born Be at relativistic velocities?

[DaveC426913]

TL;DR Summary

Queen's In the year '39 is the ballad of an interstellar journey with a huge time dilation factor. Given the specs, how far way can planet AWSNB be?

The premise (bear with me):

In the year of '39 assembled here, the volunteers
In the days when lands were few
Here the ship sailed out into the blue and sunny morn
Sweetest sight ever seen

And the night followed day, and the storytellers say
That the score brave souls inside
For many a lonely day
Sailed across the milky seas
Never looked back, never feared, never cried

[Chorus]

In the year of '39
Came a ship in from the blue
The volunteers came home that day,
And they bring good news
Of a world so newly born
Though their hearts so heavily weigh

For the Earth is old and grey
Little darling, we'll away but my love this cannot be
For so many years are gone
Though I'm older but a year
Your mother's eyes from your eyes cry to me.

In the Year of '39 - Queen


(It's a beautiful, haunting song)


Given:

1. The ship left Earth in the year (x)39.
2. The ship returned in the year (x+n)39 i.e. some multiple of a century by Earth's reckoning.
3. The Score Brave Souls Inside experienced a mere year of time passing.

Conditions:

1. Let us look at the minimum case where n=1, i.e. their round trip was one hunded years by Earth reckoning.
2. Let us assume that the ship was able to accelerate to cruising speed and back to rest in negligible time. (at 1g, I think it only takes a few months by Earth reckoning to get to relativistic speeds*.)
3. Let us assume they decelerated at planet A World So Newly Born, checked it out in negligible time, and then headed back home.

* This is flawed. It will take longer and longer (from Earth's perspective) to accelerate each subsequent segment between, say, .9c and .99c - and again to get to .999c, etc. We may have to discard the presumed accelation rate of 1g, and boost it significantly to achieve neglibly short accel/decel segments of the total 100 year journey.


Deductions:

1. The time dilation factor is 100:1. This requires a cruising speed of .999999994c.

Waitaminute ... I was going down the road of calculating the maxium possible distance of planet AWSNB from Earth (given the conditions), when realized it is much simpler than that, isn't it?

Solution:

From Earth's perspective, they traveled at (effectively) c, and made the round trip in 100 years. It immediately follows that planet AWSNB can be a maxinum distance of 50 light years. The whole relativistic time dilation aspect is a red herring.

Is my logic correct?

Followup:

If we modify Condition 1, so that n=2 (i.e. they returned in two centuries, not one), does that linearly affect the result? i.e. planet AWSNB can be 2x50 = 100 light years away?

I surmise the anwser is yes.


This seemed a lot more complex when I was working it in my head through while driving, but as I started writing, I realized it's not, is it?

 

[anuttarasammyak]

I agree. One year, one month, one minute of pilot own travel time does not matter much to almost 50 light years distance for 100 years spaceship go-return journey for the Earth.
d=c\beta t=ct\sqrt{1-\frac{\tau^2}{t^2}}

 

[PeterDonis]

at 1g, I think it only takes a few months by Earth reckoning to get to relativistic speeds

It takes about a year. A good rule of thumb is to divide $c$ by the acceleration. So 1 g is about $10 \ \text{m} \ \text{s}^{-2}$ in SI units, meaning it takes about $3 \times 10^7 \ \text{s}$ to get to relativistic speeds at that acceleration, which is just a little less than a year.

 

[PeterDonis]

Is my logic correct?

As far as giving an upper bound to how far away planet AWSNB can be, yes, the round-trip light travel distance in the given time is that upper bound.

 

[DaveC426913]

It takes about a year. A good rule of thumb is to divide $c$ by the acceleration. So 1 g is about $10 \ \text{m} \ \text{s}^{-2}$ in SI units, meaning it takes about $3 \times 10^7 \ \text{s}$ to get to relativistic speeds at that acceleration, which is just a little less than a year.

Ok, so that's "to" relativistic speeds, which is maybe .9 c.

What about from .9c to .99c? (from Earth's perspective)? And then from. 99c to .999c? Doesn't each segment take longer and longer (because Earth observes events on the ship getting slower and slower)?

 

[PeterDonis]

What about from .9c to .99c? (from Earth's perspective)? And then from. 99c to .999c? Doesn't each segment take longer and longer (because Earth observes events on the ship getting slower and slower)?

The relativistic rocket equation [1] is the go-to tool for answering such questions. From that page we obtain

$$
v = \frac{a t}{\sqrt{1 + \left( a t / c \right)^2}} = \frac{a t}{\gamma}
$$

where $t$ is the time by Earth clocks. Since $\gamma$ approaches an exponential function of $t$ for large times, the increase in $v$ with Earth clock time gets exponentially slower, which is more or less what you are describing.

[1] https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

 

[DaveC426913]

...the increase in $v$ with Earth clock time gets exponentially slower, which is more or less what you are describing.

Alas. That's not something that can be hand waved away - not without an outrageous boost to the ship's acceleration.

Which means this cannot have a definitive answer without some additional constraining parameter.

 

[PeterDonis]

this cannot have a definitive answer without some additional constraining parameter.

In order to get an exact answer for elapsed Earth time vs. distance, you do need to decide what the acceleration will be, and what the final cruising speed will be relative to Earth. The limiting case of this is just having the ship accelerate continuously until it's halfway there, then decelerate continuously until it arrives at the planet, then repeat the process on the way back. The rocket equation page has formulas for that case (for the one-way trip stopping at the destination, but the round trip is just two copies of the one-way trip).

 

[PeterDonis]

what the final cruising speed will be relative to Earth. The limiting case of this is just having the ship accelerate continuously

Note that once the final cruising speed gets pretty close to the speed of light, additional acceleration makes practically no difference to the trip time by Earth clocks. Where it does make a difference is the trip time by ship clocks; continous acceleration can significantly shorten that time as compared to accelerating to some cruising speed and then free-falling until it's time to decelerate.

 

[DaveC426913]

I fell for the red herring again. It does have a definitive solution: planet AWSNB cannot be more than 50 light years distant.

I don't have to concern myself with the what the actual acceleration is; I simply know that there is no circumstance that can increase the maximum possible distance, only decrease it.

And there are many factors (layover delay, indirect route) that could decrease the max distance that I have explicitly ignored in my conditions, so this one is no different.

 

[DaveC426913]

In order to get an exact answer for elapsed Earth time vs. distance, you do need to decide what the acceleration will be

Yes, in my head logic, I was imagining a trip plan that spent effectively the entire voyage at top speed, less a few months at journey's accel/decel legs. But I now see that that would require an arbitrarily large acceleration.

 

[DaveC426913]

Note that once the final cruising speed gets pretty close to the speed of light, additional acceleration makes practically no difference to the trip time by Earth clocks.

Yeah, I factored that in in my OP:

From Earth's perspective, they traveled at (effectively) c

The difference in a journey at .999c versus .99999c as seen from Earth is vanishingly small. A 50 year one-way is reduced to 49.99 years - less than 4 day's savings.

 

[PeroK]

It's a beautiful, haunting song.

"Watcher of the Skies" by Genesis is a darker song with the theme of mankind having destroyed the conditions that support life on Earth and sadly turning our eyes to a planet unknown.

Watcher of the skies, watcher of all
His is a world alone, no world is his own
He whom life can no longer surprise
Raising his eyes, beholds a planet unknown

Creatures shaped this planet's soil
Now their reign has come to end
Has life again destroyed life?
Do they play elsewhere?
Do they know more than their childhood games?

Judge not this race by empty remains
Do you judge God by his creatures
When they are dead?
For now, the lizard's shed its tail
This is the end of man's long union with Earth

 

[DaveC426913]

The Loneliest of Creatures / The Lighthouse Keeper
- Klaatu

 

[Halc]

But I now see that that would require an arbitrarily large acceleration.

So that means you're asking the wrong question (How far away is the destination). That's easy: Nearly 50 light years. So the question is, what is the minimum proper acceleration required to achieve the 1yr/100yr time differential?

I get around 28¼g, accelerating the entire time round trip, but I don't know a cute formula to compute that from those two inputs specified. That gets you about 49.95 LY away, 1 and 100 years elapsed respectively.

 

[jack action]

Y'all overthinking this. It's not like the song was written by an astrophysicist or something.

 

[DaveC426913]

Y'all overthinking this. It's not like the song was written by an astrophysicist or something.

That almost went right over my head.

 

[DaveC426913]

So that means you're asking the wrong question (How far away is the destination). That's easy: Nearly 50 light years. So the question is, what is the minimum proper acceleration required to achieve the 1yr/100yr time differential?

I get around 28¼g, accelerating the entire time round trip, but I don't know a cute formula to compute that from those two inputs specified. That gets you about 49.95 LY away, 1 and 100 years elapsed respectively.

Thanks. You're right, that's the takeaway of story.

 

[DaveC426913]

Oh. One other parameter we can reasonably constrain.

Her mother's eyes in her eyes cry to him.

Unless mom's and daughter's lifespans add up to more than 200 years, we can assume they couldn't have been gone more than one century.