Orbital mechanics question about Star Trek

[cybernetichero]

I'm watching the Next Generation episode Relics and Riker has just ordered the helm to "go into orbit above that point" which I took to mean geosynchronous orbit. No biggie except that the point is on the outer surface of a Dyson Sphere with a RADIUS of more than 1AU. So, assuming the sphere has to rotate to maintain 1 gee of centrifugal force on it's inner surface the Enterprise would have to move pretty sprightly to maintain position right?

 

[Ibix]

You want the centripetal acceleration, $v^2/r$ to equal $g$. So $v=\sqrt{gr}$. Plugging in Earth's orbital radius of $149×10^9\mathrm{m}$ and gravitational acceleration of $10\mathrm{ms^{-2}}$ we get about 12,000km/s, (edit: as Janus points out below, this is a typo and the correct value is 1,200km/s) or 0.4% of light speed. So way beyond anything we can do, but practically standing still for a vessel of Enterprise's capabilities.

 

[russ_watters]

So, assuming the sphere has to rotate to maintain 1 gee of centrifugal force on it's inner surface...

Why not orbital velocity around the sun?

 

[Algr]

I get the impression that in Star Trek they use "orbit" in a more general sense than we do when describing real-world satellites. So they mean "powered orbit", where the ship is hovering over one fixed point on a planet, while being much closer than geosynchronous orbit.

This would explain some of the classic series stories where turning off the engines results in the orbit "decaying" in just a few minutes and threatening to crash into the planet. The ship is actually falling rather like a helicopter would. Real geosynchronous orbits are too far from the planet for Kirk-era transporters to work. And with a low-earth orbit the Enterprize would spend too much time on the opposite side of the planet from the away crew.

 

[Janus]

I'm watching the Next Generation episode Relics and Riker has just ordered the helm to "go into orbit above that point" which I took to mean geosynchronous orbit. No biggie except that the point is on the outer surface of a Dyson Sphere with a RADIUS of more than 1AU. So, assuming the sphere has to rotate to maintain 1 gee of centrifugal force on it's inner surface the Enterprise would have to move pretty sprightly to maintain position right?

I doubt the Dyson sphere would rotate in order to provide gravity on its surface. The artificial gravity would act outward from the axis of rotation. As you moved away from the equator, it would weaken and act at a greater and greater angle to the interior surface.
Since, Star Trek universe seems to allow for some type of gravity generators ( how else would the crew be able to walk around the Enterprise the way they do), its pretty safe to assume that the builders of the Dyson sphere use the same type of tech to hold things to the interior of the sphere.

That being said, I wouldn't put too much emphasis on how Star Trek treats "orbits", as it doesn't seem to be too consistent.
However, it is also likely that as @Algr suggests, that they sometimes maintain a "forced orbit", or a "station keeping orbit", And when when such a command as "go into orbit over that point" is made, the Helmsman knows that a station keeping orbit is what is being asked for without it being expressly stated as such.

 

[cybernetichero]

You want the centripetal acceleration, $v^2/r$ to equal $g$. So $v=\sqrt{gr}$. Plugging in Earth's orbital radius, $149×10^9\mathrm{m}$ and gravitational acceleration of $10\mathrm{ms^{-2}}$ we get about 12,000km/s, or 0.4% of light speed. So way beyond anything we can do, but practically standing still for a vessel of Enterprise's capabilities.

Thankyou for that timely confirmation.

 

[Janus]

You want the centripetal acceleration, $v^2/r$ to equal $g$. So $v=\sqrt{gr}$. Plugging in Earth's orbital radius of $149×10^9\mathrm{m}$ and gravitational acceleration of $10\mathrm{ms^{-2}}$ we get about 12,000km/s, or 0.4% of light speed. So way beyond anything we can do, but practically standing still for a vessel of Enterprise's capabilities.

I think you are off by a factor of 10. I get 1223 km/sec or 764 mps.
This squares with the value given for Niven's ringworld.

 

[Klystron]

If the OP is in interested in other SF sources for achieving and maintaining an orbit around an artificial rotating body, consider the first section of the first novel "Titan" in John Varley's Gaea Trilogy.

Spoiler: An interplanetary crew attempts to 'orbit' a rotating object.

The original scientific mission requires orbiting a moon of Saturn. The crew discovers an immense 'wheel-like' cylindrical hollow body orbiting Saturn that rotates to provide artificial gravity within the rim. Can the crew modify the original mission in order to establish a stable orbit around the rim of this rotating object?

While similar to Niven's "Ringworld" in basic design, Gaea is not a section of a Dyson sphere but (possibly) an abandoned artificial spacecraft . Is Gaea a biological artifact or even a living creature?

 

[Algr]

I've always wondered how those Dyson spheres are supposed to work. If you rotate it, you could have gravity at the equator, but what is holding up the poles? Wouldn't everything there colapse into the star?

Spoiler: An interplanetary crew attempts to 'orbit' a rotating object.

That strikes me as being impossible. "Geosynchronous" orbit would always be far inside the ring, and might not exist anywhere if the hub is not very heavy. In other words, the rotation of the ring is producing 1G away from the center, so a ship touching the ring would also "fall" away from the ring by moving in a straight line away from the ring.

 

[Ibix]

I think you are off by a factor of 10. I get 1223 km/sec or 764 mps.
This squares with the value given for Niven's ringworld.

Correct - must have been a transcription error because 0.4% light speed is correct.

 

[DrStupid]

I've always wondered how those Dyson spheres are supposed to work.

Dyson spheres as in the Next Generation episode Relics (with a habitable inner surface) work with magical technology only. @Janus already mentioned gravity generators which are commonplace in the Star Trek universe. Real Dyson Spheres wouldn't be habitable for biological life. How they could work depends on the design.

 

[cybernetichero]

They found the sphere in the first place because it was creating a gravitational anomaly so maybe they were implying the mass of the sphere created it's gravity by good old Einsteinian methods however, after using a tractor beam to pull Enterprise inside (disabling her engines in the process) she began plunging towards the sun inside the sphere on momentum which baffled me since either way she should have "fallen" towards the inner surface of the sphere and not a fairly distant star (inverse square rule of gravity).
Still, what's the point of expecting proper science in Star Trek, Spock's mother has iron based blood and his father copper but somehow he was not only conceived but carried to term (I wonder if he is a mule, maybe T'Pring knew something we don't).

 

[Janus]

They found the sphere in the first place because it was creating a gravitational anomaly so maybe they were implying the mass of the sphere created it's gravity by good old Einsteinian methods however, after using a tractor beam to pull Enterprise inside (disabling her engines in the process) she began plunging towards the sun inside the sphere on momentum which baffled me since either way she should have "fallen" towards the inner surface of the sphere and not a fairly distant star (inverse square rule of gravity).
Still, what's the point of expecting proper science in Star Trek, Spock's mother has iron based blood and his father copper but somehow he was not only conceived but carried to term (I wonder if he is a mule, maybe T'Pring knew something we don't).

The net gravitational effect caused by a hollow sphere on something inside the sphere is zero for anywhere inside the sphere. Newton proved this with his shell theorem. So the only gravity the ship could feel was that of the star.

 

[cybernetichero]

The net gravitational effect caused by a hollow sphere on something inside the sphere is zero for anywhere inside the sphere. Newton proved this with his shell theorem. So the only gravity the ship could feel was that of the star.

So we are back to wondering how any inhabitant would stick to the outside.

 

[Ibix]

So we are back to wondering how any inhabitant would stick to the outside.

Spin, or artificial gravity. Or else the physics model underlying this episode sucks, and sucks hard enough to hold people to the inside of the sphere.

 

[cybernetichero]

So we are back to wondering how any inhabitant would stick to the outside.

Obviously I mean the inside of the outside .

 

[cybernetichero]

Or else the physics model underlying this episode sucks

And that would NEVER happen.

 

[Algr]

And that would NEVER happen.

Oops! Dr Crusher activated everyone's DNA and the whole crew turned into fish!

 

[Janus]

Spin, or artificial gravity. Or else the physics model underlying this episode sucks, and sucks hard enough to hold people to the inside of the sphere.

I got thinking about this.
The main purpose of a Dyson sphere is to capture and make use of the entirety of the star's output. That being said, does the whole of the interior surface need to be habited?
What if we consider a Ringworld/Dyson sphere hybrid? You only live in latitudes not too far from the equator (something like the between the Tropics for Earth.)
You spin the sphere for gravity.
You would need to step the interior surface so that the ground stays "level under your feet. You would also need something like The ringworld's rim walls on the to keep the air from Each step from pooling at the equator. something like this cutaway view:

Gravity at steps further from the equator would be less, and how much variation you can adapt to would determine how wide the habitable band of the Sphere would be. As would the fact that the sunlight would hit at a lower angle further from the Equator, with less effective surface heating.
The rest of the interior of the sphere would be your solar energy collection system that provides power for your civilization. as shown here with green being habitable and black being energy collection.

The other issue would be getting rid of waste heat, which I assume would have to be done by radiating it away from the outer skin. If this system can be made "tunable", you could also use it to create "weather". Turn it up at this point and down at another, and you can produce and control wind patterns.
Just more of an SF idea than a real life practical one.

 

[Ibix]

I got thinking about this.

Dyson apparently considered the solid sphere implausible, although it's very popular in fiction. Multiple ringworlds, or swarms of hovering solar sails he thought were more reasonable.

A problem with the solid sphere is what you do about the solar wind. Where do all those ions go, long term?

 

[Janus]

Dyson apparently considered the solid sphere implausible, although it's very popular in fiction. Multiple ringworlds, or swarms of hovering solar sails he thought were more reasonable.

A problem with the solid sphere is what you do about the solar wind. Where do all those ions go, long term?

Plus, there is nothing to keep the Sun at the center of the sphere ( shell theorem again) It is free to drift anywhere within the volume.

Ring worlds have a worse problem. If it drifts just a bit off center, gravitational effects between Ring and star will tend to pull it even further off center. Eventually leading to the star brushing the ring. This is how James Maxwell showed that Saturn's rings could not be solid structures as they would not be stable.

Niven missed this in his first Ringworld novel, but after having it pointed out to him addressed it in the second one by having Bussard ramjet type engines mounted on the outside of the rim walls to stop any drift.

 

[Klystron]

That strikes me as being impossible. "Geosynchronous" orbit would always be far inside the ring, and might not exist anywhere if the hub is not very heavy. In other words, the rotation of the ring is producing 1G away from the center, so a ship touching the ring would also "fall" away from the ring by moving in a straight line away from the ring.

Essentially correct. The characters in the book "Titan" attempt to orbit spinning Gaea. While impossible to land perhaps they can enter the hollow object at the hub? Disaster intervenes when their proximity to the strange device/creature activates an automatic retrieval system that 'accidentally' destroys their spaceship while bringing everything and everyone aboard the alien vessel or life form. Adventures ensue.

The crew metamorphose in various ways according to the whims of Gaea who has learned about humans from early RF broadcasts, mainly entertainment. In later novels Gaea uses her angular momentum to launch probes across the solar system from openings in her rim. The modified crewmembers explore the interior of the strange alien object; climbing many kilometers 'up' transverse cables from the rim interior to the central hub. The reader eventually discovers how the gigantic craft grows from an 'egg' to balance the incredible forces that threaten to tear mother Gaea asunder.

[Edit 20210214: The author has Gaea rotate to produce roughly 1/4 G at 'sea level' at the interior. Even so, her old bones weaken and strands of support cable begin to unravel. The alien craft/creature also features gigantic IR radiating fins to help maintain a livable interior.]

 

[Ibix]

Ring worlds have a worse problem. If it drifts just a bit off center, gravitational effects between Ring and star will tend to pull it even further off center.

I must admit that I hadn't realized it was actually unstable - I thought it was merely gravitationally neutral like the full sphere.

 

[DrStupid]

The other issue would be getting rid of waste heat, which I assume would have to be done by radiating it away from the outer skin.

That would mean that the outer temperature needs to be below the inner temperature if you don't want to waste additional energy to get the heat out. With an inner temperature of around 293 K and a Sun-like star in the center it would need to be a rigid sphere with a radius of at least 1.8 astronomical units. That's far from beeing realistic but maybe good anough for SF.

 

[cybernetichero]

I got thinking about this.
The main purpose of a Dyson sphere is to capture and make use of the entirety of the star's output. That being said, does the whole of the interior surface need to be habited?
What if we consider a Ringworld/Dyson sphere hybrid? You only live in latitudes not too far from the equator (something like the between the Tropics for Earth.)
You spin the sphere for gravity.
You would need to step the interior surface so that the ground stays "level under your feet. You would also need something like The ringworld's rim walls on the to keep the air from Each step from pooling at the equator. something like this cutaway view:
View attachment 277960Gravity at steps further from the equator would be less, and how much variation you can adapt to would determine how wide the habitable band of the Sphere would be. As would the fact that the sunlight would hit at a lower angle further from the Equator, with less effective surface heating.
The rest of the interior of the sphere would be your solar energy collection system that provides power for your civilization. as shown here with green being habitable and black being energy collection.
View attachment 277959

The other issue would be getting rid of waste heat, which I assume would have to be done by radiating it away from the outer skin. If this system can be made "tunable", you could also use it to create "weather". Turn it up at this point and down at another, and you can produce and control wind patterns.
Just more of an SF idea than a real life practical one.

I figured it would be terraced but didn't think further than that beyond thinking that there would be places where zero gee manufacturing could take place, interesting.
You could make it more than one gee at the equator and live in the higher latitudes for industrial purposes. I'm not sure which processes would be improved by more gravity but I'm sure there are some. Dirty water would settle faster at higher gravity. You could also use it for resistance training (John Carter of Mars).

 

[cybernetichero]

Of course the more you jack up the gravity the more stress on the structure.