# Math/Physics Guidance for Sci-Fi Story Plot Element

## Main Question or Discussion Point

Summary:: 1. At what distance (in km) would a telescope (roughly half the size of Hubble) be able to actually make out features on an approaching object with a 75m cross-section?

Hello Everyone,

New to the forum, but I've been reading a lot of other posts here and am consistently impressed with the answers here. I am writing a sci-fi short with a very high focus on realism, but I have hit a bit of a math wall in order to justify the physics logic of a particular scene.

Scenario: Without digging into the story details too much, the setting is roughly 150 years in the future, and the scenario in question is two spacecraft traveling towards each other at high velocity (relative to Earth). One of the craft (which I will refer to as bogey) is heading towards Earth and represents a possible threat, the other (which I will refer to as protagonist) is trying to intercept it (and must positively identify it before taking action).

The primary plot element here surrounds the mystery of what this approaching bogey is and having to stop it on a short time-frame of roughly half a day before it is in range to possibly do major harm to Earth or surrounding installations. The premise of the story hangs on the protagonist being forced to make life or death decisions under pressure within a very short window of time. In order for that to be true, the time between identification of the bogey and action must be a short as possible. I have done a fair amount of research on angular resolution but cannot get a good answer for myself.

Question(s):

1. At what distance (in km) would a telescope (roughly half the size of Hubble) be able to actually make out features on an approaching object with a 75m cross-section? For the sake of clarity, lets say
a) the range to reasonably make out the basic outline/shape of the object ~75m object cross-section.
b) the range at which more distinct smaller features such as armaments and modules 5-10 meters in length could be picked out?

2. Assuming the two spacecraft are approaching each other head-on, at what speed would they need to be traveling (relative to each other) to reduce the decision time to a matter of minutes between receiving a good image and overtaking each other?

Factors to consider:
-Bogey is initially powered down, not maneuvering, and already at maximum velocity for this scenario.
-Bogey is not transmitting an ID, and cannot be identified beyond its basic mass and size.
-The protagonist must visually (any spectrum) identify the object before taking further action (for plot reasons)
-The protagonist cannot launch a probe ahead of itself to spot this object (for plot reasons)
-The protagonist craft only has the use of an on-board telescope (approx. 1/2 size of hubble lens) and other instruments logical for a 150+ year time leap from today, and taking into account the hard limits of science.
-Time is of the essence, and the protagonist cannot slow down and take a more careful approach (for plot reasons)

Note on Engines: For this scenario, actual rocket rocket technology does not need to be considered. Although not explicitly stated, the craft are assumed to be using some version of a thorium (or other) powered nuclear thermal rocket engines, with some type of high-density, abundant propellant like hydrogen.

Related Writing and World Building News on Phys.org
mfb
Mentor
As estimate for when you can see structures: ##R = \frac{dD}{\lambda}## where R is the distance, d is the radius of your telescope, D is the diameter of the object or structure you are looking at, and ##\lambda## is the wavelength, something like 500 nm for visible light. With d=1 m and D=75 m this leads to R=150,000 km until the spacecraft appears as more as a single blob of light: You can make an educated guess if the cross section is circular, like a long rectangle or whatever. For the individual structures, D=5 m, the radius goes down to 10,000 km. Seeing structures on these modules needs even shorter distances. This is the ISS with 5 meters per pixel (produced from this image):

If you can make an informed decision at 5000 km and want 5 minutes to decide then the relative velocity must be 1000 km/minute or 17 km/s - a typical velocity in interplanetary spaceflight.

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Finding the mass of a spacecraft is a very difficult task - basically impossible if you can't approach it slowly.

Just a 1 meter telescope 150 years into the future is very pessimistic. If the spacecraft is intended to take images of anything you would expect an interferometer with a much longer baseline.

Thank you mfb, that definitely gives me a lot to think on.

In terms of lens diameter mounted on a spacecraft, I guess I am not sure what would be practical and realistic for a future setting. I would assume a 1 m lens with all the equipment/mirrors and some kind of directional maneuvering armature to turn it, it would still be a pretty big protrusion on a vessel's front or underbelly (assuming a 1x4m scope, and 6x6 footprint for some kind of directional apparatus).

(for lack of a better way to demonstrate the issue)

Are there other theoretical ship-mounted designs you are aware of which would create a larger interferometer? Maybe two or more linked lenses spaced apart?

As a bit of background: the protagonist's spacecraft in in the scenario is a military craft maybe 25m across at its widest / 80m long. The closest existing naval analogy would be an Arleigh Burke class destroyer. It would be expected to have some of the best equipment, but its mission is military, so it is very cramped already. Perhaps, lore-wise, this type of ship would normally utilize a reconnaissance probe or vessel ahead of it to identify targets and would have a less extensive "sensor suite".

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mfb
Mentor
If there is enough light then a set of 4 small (10 cm?) telescopes in a square with 10 meters side length would give a much better image than the single 1 meter telescope if you can combine their light at a single plane.

If this spacecraft was made to identify other spacecraft then it can come with much more. It's not impossible to make something equivalent to the lunar laser ranging project - the background will be very low, so even low photon counts can help. It would give you a "distance distribution" of the front view. The big telescope helps here as this will be limited by the photon count. Once the spacecraft gets closer this becomes a 3 D image. Downside: It is active, the spacecraft can know that you study it.

Klystron
Klystron
Gold Member
Concur. Given the OP's story synopsis, the bogey would likely use active (transmit or TX) radar for navigation. The radar acronym describes using radio waves to Detect and compute changing Range of an object.

Protagonist can scan expected EMF (electromagnetic fields) spectrum for signals using passive search: receive (RX) only. Once the bogey is detected the protagonist has the option of transmitting their own radar signals and/or tracking the bogey by its EMF (electromagnetic fields) radiation. Active TX provides much more information to the protagonist at the expense of announcing their presence and capabilities to the bogey.

In either case the telescope(s) can be pointed automatically to the computed bogey track.

If the bogey travels without emitting, the protagonist can acquire the target using active radar sector scans and/or detecting emissions from the bogey drive. For realism the OP can mention the speed-of-light limit to EMF transmission and reception and how relative motions of the protagonist and bogey target effect range information. Depending on the expected audience the story can describe the Doppler effect on the received signals due to relative motions. See Doppler RADAR for more information.

The essential difference between a radar receiver and (visible light) telescope is the EM wavelength. The excellent suggestion to use active lasers to identify the bogey implies an active transmitter of coherent light. Twentieth and current 21st Century technology commonly uses radar (radio/microwave) wavelengths for Search depending on the expected geometry -- size and shape of the search objects -- to acquire and track targets and direct telescopes to the object for identification.

-The protagonist craft only has the use of an on-board telescope (approx. 1/2 size of hubble lens) and other instruments logical for a 150+ year time leap from today, and taking into account the hard limits of science.
That's a lot of time for scientific breakthroughs and technological development, so you need to figure out how to either lock your social structure to something familiar to us now, or lob in some high-tech grenades that will place the action 15 decades upstream. It is also worth considering whether such physical realism really matters to the story. Very few readers are going to calc the resolving power of any telescope you mention, so just describe the constraint and get on with the story.

I'm about halfway through my second novel and have spent a ton of time (too much really), working out correct orbital positions for various solar system objects. It's accurate, but a pain to do and I know readers won't care but it's way easier to calc spacecraft transfer times between planets and asteroids using their actual positions at time X than create and track a whole artificial solar system. Whereas, the inventive side, is way more fun and what the reader is there for. So, somewhat on point regards detecting things, I needed a high-res view of Betelgeuse for the plot, and just created a new telescope in the narrative:

"But you’ve probably never heard of ORT. It came online early in 2035, three hundred telescopes orbiting between Mars and Earth. Each of them has a three-meter mirror for visible and near-visible spectrum work, and a two-hundred-meter radio wave lattice. Collectively, they have a three hundred and sixty million klick baseline, and that gives you really impressive resolution."

If anyone does the hard yards of working out whether my mythical ORT (Orbital Refactoring Telescope, by the way) can actually image Betelgeuse, good for them, because I've no idea. But it seems feasible that such a device could make out details on a red giant 600-odd lightyears away and that's what 99% of the audience is after. Readers value accessible characters that they can care about over a technically correct but 'so what' plot, and it sounds like you have the basis for a taut, high-stakes decision making situation, which means it is worth keeping in mind that too many facts can get in the way of good storytelling.

Ryan_m_b
Staff Emeritus
As a bit of background: the protagonist's spacecraft in in the scenario is a military craft maybe 25m across at its widest / 80m long. The closest existing naval analogy would be an Arleigh Burke class destroyer. It would be expected to have some of the best equipment, but its mission is military, so it is very cramped already. Perhaps, lore-wise, this type of ship would normally utilize a reconnaissance probe or vessel ahead of it to identify targets and would have a less extensive "sensor suite".
At 25m in diameter it could easily fit a gimballed 10m diameter scope on the front, which would begin resolving features as small as 5m wide at 100,000km. But if the vessel is 80m long it could have an interferometer running along its length. With some axial sensor booms it would effectively be a telescope 80m in diameter. It would have the same resolving power of 5m features up to 800,000km.

Out of interest what is the protagonist meant to do if it is identified as an enemy? If they’re meant to engage then presumably they have a long range weapon such as a missile. If they have missiles they have probes, the missile could be launched to fly by the vessel whilst transmitting back what its seeing. If it appears to be an enemy an order to change course to hit can be sent.

This all depends on how good rocket propulsion is in your setting. If it’s just nuclear engines then long range engagements will be less of a thing.

mfb
Mentor
If anyone does the hard yards of working out whether my mythical ORT (Orbital Refactoring Telescope, by the way) can actually image Betelgeuse, good for them, because I've no idea.
We have a non-point-images of Betelgeuse with existing telescopes. Here is one in the millimeter range, here is one in the UV range. With 3 AU baseline and an optical telescope you can resolve features as small as 10 meters if you have enough integration time. At this point the limit won't come from the baseline, but from the light collection and calibration. A 10 m x 10 m spot on its surface leads to 1-3 photons in ORT per day, roughly. The surface changes much faster than that. Take a 100 km x 100 km area and you get thousands of photons per second, and the surface doesn't change that rapidly any more. You might see structures of that size. Seeing 100 km structures on a star with a diameter of 1.25 billion kilometers...

Tghu Verd
If there is enough light then a set of 4 small (10 cm?) telescopes in a square with 10 meters side length would give a much better image than the single 1 meter telescope if you can combine their light at a single plane.

If this spacecraft was made to identify other spacecraft then it can come with much more. It's not impossible to make something equivalent to the lunar laser ranging project - the background will be very low, so even low photon counts can help. It would give you a "distance distribution" of the front view. The big telescope helps here as this will be limited by the photon count. Once the spacecraft gets closer this becomes a 3 D image. Downside: It is active, the spacecraft can know that you study it.
Yeah, that makes a lot of sense. Definitely 1m is not plausible given all of these potential options.

Concur. Given the OP's story synopsis, the bogey would likely use active (transmit or TX) radar for navigation. The radar acronym describes using radio waves to Detect and compute changing Range of an object.

The essential difference between a radar receiver and (visible light) telescope is the EM wavelength. The excellent suggestion to use active lasers to identify the bogey implies an active transmitter of coherent light. Twentieth and current 21st Century technology commonly uses radar (radio/microwave) wavelengths for Search depending on the expected geometry -- size and shape of the search objects -- to acquire and track targets and direct telescopes to the object for identification.
Agreed - I would expect radar and lidar is being used in conjunction with scopes.

I know that radar creates a cross-section which could be extrapolated to be an object of a certain size and physical composition, and that lidar can create a more 3d image, but maybe at a shorter range… (I’m showing my limits of knowledge here).

What I don’t know is at what distance these tools would actually play a role in indentifying the unique shape and characteristics of an object. If it is less than what could be determined from a 2-4m telescope observing multiple spectrums of radiation, then I am not worried about it. If it is greater, then that is a problem for me.

The ultimate objective here is for the protagonists’ vessel to get enough data – both direct imagery and 3d modeling data, to put together a profile of the bogey and hopefully match it against something in a database.

That's a lot of time for scientific breakthroughs and technological development, so you need to figure out how to either lock your social structure to something familiar to us now, or lob in some high-tech grenades that will place the action 15 decades upstream. It is also worth considering whether such physical realism really matters to the story. Very few readers are going to calc the resolving power of any telescope you mention, so just describe the constraint and get on with the story.

Readers value accessible characters that they can care about over a technically correct but 'so what' plot, and it sounds like you have the basis for a taut, high-stakes decision making situation, which means it is worth keeping in mind that too many facts can get in the way of good storytelling.
While I wouldn’t expect the general audience to know the math, I don’t want to write around any plot elements that would contradict basic physics or logic – hence the reason for this post. I want to create a realistic setting that stands up to at least some basic scrutiny. I need to understand the distances and timing so I can create a logical chain of events in my own mind that I can write about – at that point, the science and math can fade into the background and I can just focus on the characters acting around that framework.

For further background, I am trying to create a hyper-realistic “snapshot” of a future space engagement in the form of a short story that is meant to be as close to reality as possible without being an info-dump. I know this has been done before, but I wanted to strip out any “invented” technology and go with what we know is theoretically plausible. In essence, I want to speculate in detail on how the future of space travel and combat would look, and then throw some characters in the middle of this very scary nuclear submarine-like setting.

The tension of the story revolves around not knowing what this bogey is, but they are hurtling towards one another and will have a short window of time to decide what to do. However, a lot of the responses here confirmed my suspicions that this mystery around the bogey would be quite difficult to maintain. There is just no way that a future military would not be able to get a good three-dimensional or even direct image of this bogey long before the protagonist gets to it. I’ve spent a lot of time trying to come up with reasons to explain away the “why couldn’t they just do X?” and the conspiracy plot becomes too convoluted. I will need to go back to the drawing board to make this more plausible.

At 25m in diameter it could easily fit a gimballed 10m diameter scope on the front, which would begin resolving features as small as 5m wide at 100,000km. But if the vessel is 80m long it could have an interferometer running along its length. With some axial sensor booms it would effectively be a telescope 80m in diameter. It would have the same resolving power of 5m features up to 800,000km.

Out of interest what is the protagonist meant to do if it is identified as an enemy? If they’re meant to engage then presumably they have a long range weapon such as a missile. If they have missiles they have probes, the missile could be launched to fly by the vessel whilst transmitting back what its seeing. If it appears to be an enemy an order to change course to hit can be sent.

This all depends on how good rocket propulsion is in your setting. If it’s just nuclear engines then long range engagements will be less of a thing.
The technology of this future universe uses nuclear engines, because I think they are the most practical and likely. I speculate that our engines will hit a rough upper limit in terms of thrust and efficiency and we’ll probably just continue to enhance nuclear reaction engines but with dimishing returns. I don’t like the idea of “magic” drive systems.

But you are right that I think a bigger interferometer has to be used. I think 10m is a little too much, but certainly 2-4, and even then we’ve now pushed the identification distance out a ways. So either I make the vessels moving even faster so they cross that distance quicker, or I think of way to maintain the mystery and tension even though they have a good picture of what is coming at them.

The ultimate objective here is for the protagonists’ vessel to get enough data – both direct imagery and 3d modeling data, to put together a profile of the bogey and hopefully match it against something in a database.
Another aspect of this to consider is that 150+ years from now, everything in the system will be comprehensively tracked and traced. So, whenever/wherever your bogey launches will be known, and that'll allow the military to know all about it before it gets close enough to do any damage. Not sure how to get around this, perhaps there can have been some event that knocked tech back many, many decades, or perhaps a war has split the solar system and compromised such capabilities...or you just don't worry about all that and concentrate on the characters.

There's a great sci-fi novel that sadly I can't recall the name of which describes light-speed limited space battles and it is about as tense as anything you'll read. Orbital mechanics, relativity, light delay, limited fuel, hard-to-hide ships, 3D aspect of space warfare - it's all taken into account without getting bogged down in technical details...and posting this, I want to read it again, so I'll have to dig around and find it, now.

Ryan_m_b
mfb
Mentor
Lidar is subject to the same diffraction limit as simple optical observation. Its depth information can be better, but with a complex structure it can be difficult to interpret. The required power might be unreasonable, too. Radar has similar constraints but an even worse angular resolution.

The spacecraft could be quite dark and close to the Sun as seen by the other spacecraft, that can make it harder to see structures.