Math Modeling – Odd STEM

Could Extraterrestrials Visit Earth?

“We need some idea of their starting point.”

Could intelligent beings from a planet outside our solar system visit Earth in the near future? I’m not asking whether they have visited us or will visit us, but whether they technically could. Could? I technically could do cartwheels across the floor, but I won’t for various reasons (including my age and an aversion to hospital bills). But I’m still able to do so if I wanted to; therefore I could. So, could extraterrestrials visit our planet soon, as in could it technically happen (assuming they exist)?

More specifically, have extraterrestrials had enough time to evolve, acquire intelligence, and develop sufficient space travel technology to make their way here in (approximately) our time? We humans are the only intelligent life we know of, and our planet Earth is the only planet we know to have ever had life on it. So we’re going to use the history of humans and Earth as a yardstick for this question. Admittedly, our current space travel technology seems quite challenged by the scale of the cosmos.

It took 4.6 billion (Earth) years for our Solar System to form and for us humans to make it into space, so we might expect a similarly long time for the extraterrestrials too.[1] That 4.6 billion years includes the entire history of life on Earth so far, including the development of our space technology. In fact the past couple of millennia leading up to our development of space technology is insignificant compared to a billion years and could be ignored anyhow. Even another million years wouldn’t matter much, mathematically. On the bright side, this simplifies our calculations. Naively, we could just look for a planet reasonably like Earth that is orbiting a star at least as old as our sun, that isn’t too far away.

Unfortunately, this also means that Earth may very well be visited for the first time by extraterrestrials after everyone who is currently alive, has died of old age. Or maybe our visitors showed up long enough ago that no one was here to adequately greet them. It certainly would be nice if someone alive today (including me) could meet space aliens visiting Earth, assuming they are friendly. If they aren’t friendly, it’s just as well to leave that problem for a future generation, who has sufficient technology to defend Earth against them. The calculations here won’t be accurate enough to indicate how good our visitor’s timing might be for the current residents of Earth.

Now the question is how long would it take extraterrestrials to actually travel here from their own solar system. There’s the standard distance formula of r*t = d (rate times time equals distance), which can be rearranged as t = d/r to calculate travel time. Starting with r for rate of speed, for simplicity we’ll assume the extraterrestrial spacecraft is as fast as ours. How fast is that? The spacecraft with the fastest reported speed is the Parker Solar Probe, which reached a record speed of 192 km/s on December 24, 2024.[2] Admittedly, the craft used gravity to help achieve that speed, but at least it’s possible. And why not take advantage of gravity to cover a large distance more quickly? So this speed will work for our purposes.

Next is d in the formula, the distance our visitors would have to travel to us. We need some idea of their starting point. Reportedly the closest Earth-like exoplanet (a planet outside our solar system) is Proxima Centauri b, which orbits the closest star, Proxima Centauri.[3] We hear that Alpha Centauri is the closest star system to Earth, and that’s true, but that star system has 3 stars. Proxima Centauri is the closest of those stars, and Proxima Centauri b is a planet orbiting that star. Anyhow, Proxima Centauri b is 4.2 light years away from our solar system. A light year, as you may recall, is the distance that light can travel in one year.

Now we just need to divide the distance by the rate, to determine the travel time. But the distance is in light years, and the rate of speed uses kilometers. It’s fairly easy to find on the web that a light year is roughly 9.46 * 10^12 kilometers (9.46 times ten raised to the twelfth power).[4] Proxima Centauri b (our chosen starting point of our interstellar visitors) is 4.2 light years away, which comes out to 3.97*10^13 kilometers. Dividing that by our spaceship speed of 192 km/s comes out to about 2.1*10^11 seconds. But how many years is that? A year is roughly 3.16*10^7 seconds, so the space craft would take about 6,600 years to make the journey. That is not a short amount of time for our visitors (or us) to travel.

Back to the question of whether having visitors is technically possible at all, we again see that what’s really significant is the age of the star and planet they come from. Is the star even old enough to have evolved life like Earth has? The age of the host star Proxima Centauri is estimated at 4.85*10^9 (4.85 billion years), which is slightly older than our sun. The age of the planet Proxima Centauri b is unknown, so we don’t know whether there has been enough time for life to develop, acquire intelligence, develop space flight, and find it’s way to us. We don’t even know if the planet supports life, though it’s believed to be the right distance from its star to have liquid water, if there is any.

Looking at a good alternative just in case, there’s TRAPPIST-1e, which is 40.7 light years away.[5] That comes out to 3.85*10^14 kilometers. Dividing that by our spaceship speed of 192 km/s comes out to about 2.01*10^12 seconds. That in turn is equivalent to about 63,300 years. That’s roughly 10 times longer than the other, which was long to begin with. But what about the age? TRAPPIST-1e orbits TRAPPIST-1, a star that is estimated to be 7.6 billion years old, which is older than our sun, which is only 4.6 billion years old. That gives our hypothetical extraterrestrial visitors an extra 3 billion years to evolve, develop adequate technology, and come visit us. Again there’s the question of whether life could start and survive there, but that question will exist until we actually find extraterrestrial life.

How long of a trip is too long? Maybe whatever makes a difference compared to 4.6 billion years, arbitrarily because it would affect our calculations. So maybe 100 million years might be too long for a trip. That means we could expect visitors from 1000 times as far away. It turns out, the center of our Milky Way galaxy is about 27,000 light years from us, which is less then a thousand times as far as TRAPPIST-1e.[6] That alone indicates there probably has been enough time for us to have extraterrestrial visitors, even if their space technology is as slow as ours currently is. But our galaxy seems to be over 12-13 billion years old, which seems to give them even more time to come knocking on our proverbial door. Of course, our solar system has only been around for 4.6 billion years. Still, there are probably plenty of stars that are roughly our sun’s age, that could have planets with life that could visit us.

So, yes, extraterrestrial visitors are possible, when looking at the aspect of time alone. The theoretical possibility is exciting, inspiring, scary, and humbling, all at the same time. Why would they even visit us though? The center of the galaxy is said to have a very massive black hole, which is an object so dense, that even light can’t escape it, due to gravity. I would guess that it didn’t start out that big, meaning that matter around it could eventually fall into it. Being literally crushed by strong gravity could very well motivate extraterrestrial life to find us, very far from the galactic center. Radiation given off by matter that is being pulled in, could also offer motivation. So we very well be getting some extraterrestrial company some day, after all.

References

[1] https://en.wikipedia.org/wiki/Sun

[2] https://www.guinnessworldrecords.com/world-records/66135-fastest-spacecraft-speed

[3] https://en.wikipedia.org/wiki/Proxima_Centauri_b

[4] https://earthsky.org/astronomy-essentials/how-far-is-a-light-year/

[5] https://en.wikipedia.org/wiki/TRAPPIST-1e

[6] https://en.wikipedia.org/wiki/Milky_Way

Witches’ Brew

“But what if there’s more to it than that?”

Like other aspects of culture, witches have stereotypes that are fueled by tradition, fiction, and modern media. A well-known aspect of witch stereotypes is the brewing of potions or witches’ brew, often in a large black cauldron. The purpose of a potion could be to cure illness, obtain immortality, or try to “induce love”, according to Wikipedia.[1] Other historic purposes for witch’s brew can be found on the web as well. The ideas for tradition and fiction are often based on truth. In fact, the mixing of natural ingredients in potions eventually helped bring about modern medicine. Could there be any truth to other supposed uses of witches’ potions, beyond potential use as medicine?

The Universe Doesn’t Care What We Think

“Reality is simply not that convenient.”

Some (most?) people may not care what I think, and that’s fine. But I think the physical universe doesn’t care what any of us thinks. In attempting to explain the universe or the part we experience, just about any theory we throw at it, has limitations that make the theory inaccurate under certain conditions. The experimental results and theories from physics can be different enough from our everyday experience, to make it seem presumptuous to expect the universe to behave the way we normally perceive it or to understand why it does what it does. Relativity and quantum physics come to mind.

The Distance Between Heaven and Hell

“You can calculate just about anything, if you really want to.”

Around Halloween time, you may see someone dressed up like Satan or maybe watch scary movies that involve Satan or demons. But the day after Halloween is All Saints Day, which celebrates people of a more holy disposition, particularly the Christian Saints. And then there is Christmas in December, celebrating the birth of Jesus with images of singing angels. That’s quite a reversal over night or even in two months, as if Heaven and Hell aren’t so far apart. Granted, they may be metaphysical places that we can’t travel to physically, what if we could? How far apart might they be?

Human Design and Reproduction

“Is the quality a defect?”

Every species – whether brought about by evolution or divine creation – has a biological design that strives for the survival of that species. Part of that biological design is reproduction, which is necessary for the survival of our species too, assuming humans will never be immortal. What if someone never reproduces? Biologically, such people seem at first glance to not be contributing to the survival of our species. Are they deviating from our design?

The Lowly Distance Formula

“…the definition of a light year is a direct application of the formula…”

The classic distance formula of rate×time=distance (or r×t=d) may not look like much. I wasn’t particularly impressed with it in elementary school when the teacher started discussing it. I wanted to learn new math that day, and this was just multiplication (or division if rearranged). Was it a new profound concept? No, just boring, mundane stuff. If you travel at this rate of speed for this amount of time,you end up traveling this distance. Being good in math and a fan of science fiction and science, I was hoping for something more interesting. But there’s more to this simple distance formula than meets the eye.

Santa’s Flight-Path Challenge

“This might take some trickery.”

Though there has been much speculation, no one really knows how Santa Claus does it all. Traditionally, he and his crew spend most of the year making and gathering toys for children all over the world. Then he delivers them all on Christmas Eve and early Christmas morning. The delivery itself creates some alarming statistics for how fast Santa’s sleigh must travel and how much time Santa has at each house, for delivering presents and consuming whatever milk and cookies the children leave for him. But I’m wondering, how can Santa even figure out a flight path? How does he choose the first house or the last house or any of them in between?

Continue reading “Santa’s Flight-Path Challenge”

The Physically Unknowable

“The universe is keeping secrets from us….”

People say that no one has all the answers. In fact, no one can have all the answers, even if they have some place to store all that information. Some things are simply unknowable. This isn’t just my opinion – this is according to accepted theory. Strangely, some unknowable information occurs in physics, which is contrary to the idea that we can learn as much as we want about something, as long as we have adequate technology. Continue reading “The Physically Unknowable”

Space Pilgrims

“Nomenclature aside, how much energy is required to get every living human off the Earth and relocated elsewhere?”

When we Americans talk about “the Pilgrims,” we usually are talking about the group of English settlers who came to North America in 1620 in search of religious freedom. They celebrated a good fall harvest the following year, which became the first Thanksgiving.[1] In general, a pilgrim is someone who makes a journey for religious purposes.

What then is a space pilgrim? Suppose the entire population of the Earth needed to evacuate the planet and relocate elsewhere. Continue reading “Space Pilgrims”

The Hidden Circle in a Simple Harmonic Oscillator

“Yes, there is definitely a circle there.”

When I was in college, it didn’t make sense to me that the position of an object on the end of a spring should be described by a function like sine or cosine, as in:

It’s just a simple harmonic oscillator. All the thing does is move back and forth. Why is there a function from trigonometry when I don’t see any circles or angles involved? Continue reading “The Hidden Circle in a Simple Harmonic Oscillator”