Intergalactic message exercises (for LightCone tutorial)

[marcus]

There's a tutorial wiki for Jorrie's calculator. It needs exercises that beginning learners can do to help them get used to using the calculator.
IF YOU HAVE SOME IDEAS, feel free to make up exercises and propose them. Anybody can suggest material for a textbook. He might not like what you offer, or it might duplicate existing stuff. But it might be helpful. Here's some I thought of today. They might or might not be accepted and go into the tutorial. Comments welcome.

Intergalactic Message Problems

These exercises are all based on one table. It runs from a=0.1 to a=10 in 20 steps, so when you open LightCone, set S_upper = 10, S_lower=0.1 and Steps=20. Then press calculate. The link is in my signature.

The situation is we discover that there's a laser message coming in from a distant galaxy that says "Please reply immediately!" We analyze the light and determine that the wavelength has been doubled. The peaks and valleys of the wave have been spread out by a factor of two.
We flash our reply immediately. When does it get to them? Or does it ever get to them?

Learners doing the exercises will have generated this table this table, as I just described. The Answers should go at the end of the chapter, but I will put the answer to this first one right after the table. *Spoiler alert*
$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.100&10.00&0.545&0.8196&30.684&3.068&4.717&2.13&3.74\\ \hline 0.126&7.94&0.771&1.1568&28.684&3.611&5.687&1.99&3.12\\ \hline 0.158&6.31&1.089&1.6308&26.444&4.191&6.804&1.84&2.57\\ \hline 0.200&5.01&1.536&2.2939&23.938&4.776&8.066&1.66&2.08\\ \hline 0.251&3.98&2.165&3.2127&21.143&5.311&9.452&1.47&1.65\\ \hline 0.316&3.16&3.041&4.4626&18.045&5.706&10.920&1.25&1.28\\ \hline 0.398&2.51&4.250&6.1052&14.651&5.833&12.396&1.02&0.96\\ \hline 0.501&2.00&5.883&8.1349&11.008&5.517&13.780&0.76&0.68\\ \hline 0.631&1.58&8.015&10.4035&7.226&4.559&14.962&0.50&0.44\\ \hline 0.794&1.26&10.669&12.6018&3.483&2.767&15.863&0.24&0.22\\ \hline 1.000&1.00&13.787&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 1.259&0.79&17.257&15.6486&3.109&3.914&16.842&0.22&0.25\\ \hline 1.585&0.63&20.956&16.4103&5.731&9.083&17.047&0.40&0.55\\ \hline 1.995&0.50&24.789&16.8364&7.890&15.743&17.153&0.55&0.94\\ \hline 2.512&0.40&28.694&17.0630&9.638&24.210&17.204&0.67&1.42\\ \hline 3.162&0.32&32.638&17.1800&11.040&34.912&17.224&0.77&2.03\\ \hline 3.981&0.25&36.601&17.2395&12.160&48.409&17.240&0.84&2.81\\ \hline 5.012&0.20&40.575&17.2696&13.051&65.411&17.270&0.91&3.79\\ \hline 6.310&0.16&44.553&17.2847&13.760&86.821&17.285&0.96&5.02\\ \hline 7.943&0.13&48.534&17.2923&14.324&113.777&17.292&0.99&6.58\\ \hline 10.000&0.10&52.516&17.2961&14.772&147.715&17.296&1.03&8.54\\ \hline \end{array}}$$

The wavelengths being enlarged by factor of S=2.0 tells us that the galaxy is NOW at distance of 11 Gly. We look down the D column to find where a target's distance is approximately the same. If we flash a message today to a galaxy that is now at distance 11 Gly it will get there in year 32.6 billion. That is, about 19 billion years from now. But yes, it will get there.

The galaxy is within communication range because its distance now (11 Gly) is less than the current value of the horizon distance Dhor = 16.47 Gly. (Look in the S=1 row.)

Incidental intelligence: our return message will be wavestretched by a factor of about 3.16.

 

[marcus]

Exercise 2: We get a similar message on light that has been stretched by a factor of 3.2. If we reply, will the message ever get to them?

Exercise 3: A similar message comes in where the wavelength has been enlarged by a factor of 2.5. If we reply, will the message ever reach the target galaxy?

Exercise 4: Again, and this time the wavelength enlargement factor is 1.26. We reply immediately. Approximately when will our return message reach them? Approximately how much stretching will our message undergo?

Exercise 5: We get a message where the carrier wavelength has doubled, and immediately send our reply. How fast was the other galaxy receding when it sent the message? How fast is it receding from us now? How fast will the distance to it be increasing when our return message gets there? By what factor will our return message by stretched while in transit?

 

[marcus]

If anybody wants to help, I'd really appreciate some feedback. Do you find these too hard? Are they too easy? Is the wording OK, or do you find it awkward? Is more explanation needed? Meanwhile I'll post some more Intergalactic Message problems, in case Jorrie finds some of them suitable for the "beginner's LightCone textbook" that's to be put together.

The next problem, or set of exercises, asks the learner to generate the same table but with 32 steps instead of just 20. Same upper and lower limits S_upper=10, S_lower=0.1, just change Steps to 32 and recalculate.

Exercise 6: We get a message where the carrier wavelength has been stretched by a factor of 1.54.
They want an immediate reply and they promise that they will respond to OUR message immediately when it arrives. Can we expect to hear from them a second time?

Here's the table that the person doing the exercises would have generated:

$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.100&10.00&0.545&0.8196&30.684&3.068&4.717&2.13&3.74\\ \hline 0.115&8.66&0.677&1.0167&29.461&3.402&5.306&2.05&3.35\\ \hline 0.133&7.50&0.840&1.2607&28.148&3.754&5.952&1.95&2.98\\ \hline 0.154&6.49&1.043&1.5624&26.738&4.117&6.656&1.86&2.64\\ \hline 0.178&5.62&1.293&1.9349&25.226&4.486&7.418&1.75&2.32\\ \hline 0.205&4.87&1.604&2.3932&23.605&4.847&8.233&1.64&2.03\\ \hline 0.237&4.22&1.987&2.9549&21.870&5.186&9.096&1.52&1.76\\ \hline 0.274&3.65&2.460&3.6386&20.017&5.482&9.996&1.39&1.51\\ \hline 0.316&3.16&3.041&4.4626&18.045&5.706&10.920&1.25&1.28\\ \hline 0.365&2.74&3.752&5.4413&15.957&5.827&11.848&1.11&1.07\\ \hline 0.422&2.37&4.615&6.5792&13.761&5.803&12.755&0.96&0.88\\ \hline 0.487&2.05&5.653&7.8634&11.474&5.588&13.616&0.80&0.71\\ \hline 0.562&1.78&6.883&9.2557&9.124&5.131&14.402&0.63&0.55\\ \hline 0.649&1.54&8.319&10.6896&6.752&4.384&15.091&0.47&0.41\\ \hline 0.750&1.33&9.958&12.0787&4.404&3.302&15.666&0.31&0.27\\ \hline 0.866&1.15&11.789&13.3364&2.133&1.847&16.124&0.15&0.14\\ \hline 1.000&1.00&13.787&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 1.155&0.87&15.923&15.2443&2.002&2.312&16.726&0.14&0.15\\ \hline 1.334&0.75&18.165&15.8793&3.810&5.080&16.905&0.26&0.32\\ \hline 1.540&0.65&20.485&16.3368&5.429&8.361&17.028&0.38&0.51\\ \hline 1.778&0.56&22.860&16.6558&6.866&12.209&17.109&0.48&0.73\\ \hline 2.054&0.49&25.274&16.8734&8.130&16.695&17.162&0.56&0.99\\ \hline 2.371&0.42&27.713&17.0194&9.236&21.902&17.195&0.64&1.29\\ \hline 2.738&0.37&30.170&17.1162&10.201&27.934&17.214&0.71&1.63\\ \hline 3.162&0.32&32.638&17.1800&11.040&34.912&17.224&0.77&2.03\\ \hline 3.652&0.27&35.114&17.2217&11.769&42.979&17.227&0.82&2.50\\ \hline 4.217&0.24&37.594&17.2490&12.402&52.299&17.249&0.86&3.03\\ \hline 4.870&0.21&40.078&17.2668&12.951&63.065&17.267&0.90&3.65\\ \hline 5.623&0.18&42.564&17.2785&13.426&75.500&17.278&0.93&4.37\\ \hline 6.494&0.15&45.051&17.2859&13.838&89.861&17.286&0.96&5.20\\ \hline 7.499&0.13&47.539&17.2908&14.195&106.446&17.291&0.99&6.16\\ \hline 8.660&0.12&50.027&17.2940&14.504&125.598&17.294&1.01&7.26\\ \hline 10.000&0.10&52.516&17.2961&14.772&147.715&17.296&1.03&8.54\\ \hline \end{array}}$$

Exercise 7: Similar to exercise 5, the incoming wave is stretched by a factor of 2.37. We send an immediate reply. How fast was the distance to the galaxy increasing when they sent the message? How fast is it increasing NOW? How fast will the galaxy be receding when our message finally reaches its destination?
And what year will that be (when the message gets there)?

Exercise 8: It's estimated that the first living cells (prokaryotes) appeared on Earth when cosmic distances were 75% of their present size. Imagine that light from a distant galaxy emitted just when the first life appeared is arriving today. By what factor will its wavelengths be enlarged? How far was the galaxy when it emitted the light? How far is it from us today?

 

[Mordred]

I've been thinking of how to have a few examples that show the function of all 7 inputs.

The communication example is a good one.

the format on the tutorial I've been considering is in example one use the first two inputs and fine tune the ranges.
example two use S_upper and lower to do the same.
example 3 work these into graphing.
The Seq and omega are more advanced functionality.

Setting up the appropriate ranges of calculations, in time line and stretch, follow each example to graph
seems to be the most efficient short means of showing how the calculator works.

Do you concur Marcus?

Fine tuning the matter dominant to lambda dominant would be a good fine tuning exercise as well
You and I posted that procedure in other threads on my phone atm but I suggest that as one of the exampled

 

[marcus]

I think were at a stage where we can't say what exercises there SHOULD be. I think rather than top down we are sort of accumulating examples of what COULD be. We need a bunch of possible samples to look at and pick from.

Something that just occurred to e was there could be a bunch of exercises that emphasize using the calculator as a ONE-SHOT. that is where you put Steps=0 and you just get one line.

So confront the reader with some straightforward questions that can be answered WITHOUT table, or rather it's a table with just one row. simply by putting in the right S_upper and pressing "calculate"

You set the Steps=0 at the outset, and keep it that way while put in a succession of different S_upper to answer each question.
S_lower is irrelevant to this mode of operation. You can keep it at the default value of 0.01.

It might be a poor idea, or might not. Ill make up some tomorrow (sleepy now) and we can see how they play.

=================

Also there is the "time machine" type of problem where you construct a table that shows how history looked to someone in our galaxy who lived before the solar system existed, say. that could be fun.
=================

Sometimes its good for a problem to have a bit of narrative. Just a minimum amount. Like the message exercises. Something is happening that the reader can imagine, but it is not gummed up with unnecessary details. It helps to motivate the calculation.

these are just some ideas. Have to get to bed and get back to this in morning.

 

[marcus]

I've been thinking of how to have a few examples that show the function of all 7 inputs.
...

I see what you are driving at. We can distinguish between worked EXAMPLES that show by example the various features and functions (but don't challenge the reader by posing problems or asking questions) and then on the other hand problem EXERCISES which give the reader a chance to figure something out on hizzer own.
From your post I can see you have been thinking about former. Probably we need to begin with some fairly detailed step by step examples, maybe with partial screen-shots, which I already see in some of the user guide and related material.
And then once the basic layout and functions are explained, there might be what I was thinking about, namely some easy problem exercises which give the reader some practice and a way to see how well he/she understands.

I'm still not sure I get what you mean by "fine-tuning", but I get the idea of beginning with worked examples.

I wonder how some worked examples would look if we set the limits at S=12 and S=.333 or S=.25. When you have done everything else and proceed to demonstrate the CHART feature you don't want to go beyond .333 or .25, I think. that may or may not be relevant.

 

[Mordred]

Something along this line is what I meant by fine tuning.
I should have added fine tuning the range of calculations lol.

The new tool tips and columns really helps in usage. I took one of Marcus previous examples and tried to narrow down just how long the Universe was close to static. This is the period when the matter/dark energy was close to being balanced.

First I kept the inputs as default, turned all my column selections on, increased the number of decimal places to 6 and set steps at 100. Click calculate.

then I looked for the period of time where A R^o was lowest. This showed around 7.4 to 7.8 G yr.

I looked over on the S column and picked two S values surrounding that period in time. In this case 1.68 which I set for S_upper, S_lower I set for 1.64. As I didn't need as many rows I set steps to 30.

Between S 1.653333 and 1.650667 A R⁰ is the smallest values. so the universe was almost balanced for a very short period of time cosmologically speaking. Roughly 200 million years.

to post on the forum I simply click PF format tab. then click calculate and then copy the results and post on the forum.

$${\scriptsize \begin{array}{|c|c|}\hline R_{0} (Gly) & R_{∞} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.92&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize \begin{array}{|r|r|} \hline S=z+1&a=1/S&T (Gy)&R (Gly)&D (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly)&a'R_{0} \\ \hline 1.680000&0.595238&7.425739&9.821697&8.187321&4.873405&14.687083&22.664831&0.872703\\ \hline 1.678667&0.595711&7.433501&9.829625&8.174286&4.869511&14.690984&22.690599&0.872692\\ \hline 1.677333&0.596184&7.441368&9.837561&8.161085&4.865512&14.694792&22.716505&0.872681\\ \hline 1.676000&0.596659&7.449143&9.845502&8.148050&4.861605&14.698704&22.742355&0.872671\\ \hline 1.674667&0.597134&7.457022&9.853451&8.134849&4.857593&14.702524&22.768345&0.872661\\ \hline 1.673333&0.597610&7.464908&9.861406&8.121648&4.853575&14.706351&22.794376&0.872652\\ \hline 1.672000&0.598086&7.472702&9.869366&8.108612&4.849648&14.710282&22.820350&0.872644\\ \hline 1.670667&0.598563&7.480600&9.877334&8.095411&4.845617&14.714120&22.846464&0.872636\\ \hline 1.669333&0.599042&7.488505&9.885309&8.082210&4.841579&14.717964&22.872620&0.872628\\ \hline 1.668000&0.599520&7.496417&9.893290&8.069008&4.837535&14.721815&22.898818&0.872621\\ \hline 1.666667&0.600000&7.504334&9.901278&8.055807&4.833484&14.725671&22.925058&0.872615\\ \hline 1.665333&0.600480&7.512259&9.909272&8.042605&4.829427&14.729534&22.951340&0.872609\\ \hline 1.664000&0.600962&7.520189&9.917273&8.029404&4.825363&14.733403&22.977664&0.872603\\ \hline 1.662667&0.601443&7.528126&9.925280&8.016202&4.821292&14.737278&23.004031&0.872599\\ \hline 1.661333&0.601926&7.536070&9.933293&8.003000&4.817215&14.741159&23.030440&0.872594\\ \hline 1.660000&0.602410&7.544119&9.941314&7.989633&4.813032&14.744947&23.056990&0.872591\\ \hline 1.658667&0.602894&7.552075&9.949341&7.976432&4.808942&14.748841&23.083484&0.872588\\ \hline 1.657333&0.603379&7.560038&9.957374&7.963230&4.804845&14.752740&23.110021&0.872585\\ \hline 1.656000&0.603865&7.568106&9.965414&7.949863&4.800642&14.756547&23.136700&0.872583\\ \hline 1.654667&0.604351&7.576082&9.973461&7.936661&4.796531&14.760459&23.163322&0.872582\\ \hline 1.653333&0.604839&7.584164&9.981514&7.923294&4.792315&14.764278&23.190087&0.872581\\ \hline 1.652000&0.605327&7.592252&9.989573&7.909927&4.788091&14.768102&23.216895&0.872581\\ \hline 1.650667&0.605816&7.600247&9.997639&7.896725&4.783961&14.772033&23.243647&0.872581\\ \hline 1.649333&0.606306&7.608348&10.005712&7.883358&4.779724&14.775871&23.270541&0.872582\\ \hline 1.648000&0.606796&7.616456&10.013791&7.869991&4.775480&14.779714&23.297480&0.872583\\ \hline 1.646667&0.607287&7.624570&10.021876&7.856624&4.771229&14.783564&23.324462&0.872585\\ \hline 1.645333&0.607780&7.632691&10.029968&7.843257&4.766971&14.787420&23.351487&0.872588\\ \hline 1.644000&0.608273&7.640819&10.038066&7.829890&4.762707&14.791282&23.378557&0.872591\\ \hline 1.642667&0.608766&7.648953&10.046171&7.816523&4.758435&14.795151&23.405670&0.872594\\ \hline 1.641333&0.609261&7.657093&10.054283&7.803156&4.754157&14.799026&23.432828&0.872599\\ \hline 1.640000&0.609756&7.665341&10.062400&7.789624&4.749771&14.802807&23.460130&0.872604\\ \hline \end{array}}$$

 

[marcus]

I see what you mean. And I seem to recall you SAID something about fine-tuning the range, I just did not get what you meant the first time you said it.

That could be an interesting example, or something like that. To show the reader that he/she can adjust the range so as to bring out a certain feature in the universe's history. Get more exact about a specific interval of the past, and such.

It seems to me you found the epoch S = 1.652000 when the expansion speed of every comological distance reached a MINIMUM.

It is a pretty idea, a pretty moment in time. All the distances did it in unison. Each one's speed curve took a swoop, or dip, and the all reached minimum at the same time and then their expansion speed began to rise.

So that juncture in history S=1.652 is a nice thing to identify. I wouldn't call it "static" because the distances were still expanding---the sample one tabulated was still expanding at 0.87 c even at its slowest. But maybe that was "static" in some technical, or calculus, sense. Might confuse a beginner though.

It was just the time when everybody reached their slowest expansion speed. Good example!

Looks like year 7.59 billion.

You could make the range tighter around 1.652 and then the table would be shorter and not take up so much space, but it would still show the minimum speed.