What are the biological effects of a toe-to-head force gradient upon a human over the long term?

This question pertains to the notion of constructing rotating spacecraft and habs for human use in space in order to mitigate the effects of microgravity. Such craft have been a staple of many sci-fi works and pop films, but no craft to date has employed this on at least a serious scale.

The basic concept is to imagine two capsules at the ends of a long cable or, perhaps, if one really wants to make it expensive, a giant ring, like a wheel, rotating about a central axis. The setup rotates so that the apparent force vectors along the cable / out from the wheel center, are enough to provide "good" gravitational conditions - about one gee, ideally - inside the habitation.

The size of the needed habitation can be found from elementary mechanics: if the desired acceleration is $$a$$, then the criterion for the rotation is

$$a = \frac{v^2}{r}$$

where $$r$$ is the ring radius and $$v$$ the velocity at the rim. If rotation is measured in angular velocity $$\omega$$ instead, then, $$v = r\omega$$ and

$$a = r\omega^2$$

Taking $$a$$ to be $$9.8\ \mathrm{m/s^2}$$ and $$r = 5\ \mathrm{m}$$ shows that for a ring with 5 m radius (10 m diameter), you need an angular velocity of 1.4 rad/s, or 0.22 rev/s (one revolution every 4.5 s). This is a respectable velocity, commanding a decent amount of rotational kick (kinetic energy), but not impossible.

However, an interesting problem with a tight ring seems to be the following: note that from the formula just given, there is a very obvious dependence on $$r$$ for any given rotational speed $$\omega$$. This means that, within the rotating reference frame established by the rotating spacecraft, there is a gradient of force depending on the distance from the center, and that means also a gradient of force over any extended object, such as a human, within that craft. In particular, the figures above assume a point object, but a human is extended, perhaps considerably, e.g. a guesstimate for the world average figure (remarkably hard to find, just a guesstimate of such from a long time ago) is 1.70 m height.

The force gradient between two points is easily seen to be in fact even independent of the size of the ring: it is

$$\Delta a(\Delta r, \omega) := (\Delta r)\omega^2$$

and thus solely a function of the "height" of the human $$\Delta r$$, and their rate of rotation, $$\omega$$. Of course, a larger ring can rotate more slowly to produce a given "gravity": twice the radius, and you quarter $$\omega$$. Taking the above figures with $$\Delta r := 1.70\ \mathrm{m}$$, $$\omega := 1.4\ \mathrm{rad/s}$$ gives a $$\Delta a$$ of about 3.3 $$\mathrm{m/s^2}$$, which seems like a VERY significant delta-gravity - that's from roughly one gee at your feet to two-thirds of a gee at your head!

That sounds alarming, so lets instead imagine we make the ring/rope 100 m large/long (50 m radius), hence this becomes 1/100 ($$10^2$$) and is 0.33 $$\mathrm{m/s^2}$$.

The question is: what is the biological effect of such a gradient in force, as it seems quite likely there is going to be one? How does it compare to the effects of microgravity, especially over the long term? If you have to live with an essentially permanent $$0.33\ \mathrm{m/s^2}$$ accelerative gradient from your toes to your head for, say, tens of megaseconds (hundreds of days), what would be the effects? (You can assume 1/3 of the time spent sleeping though, if you want) I'm especially thinking of a duration equal to one Mars launch window distance, i.e. the minimum time between an arrival in Mars orbit and departure for the first minimal-energy return to Earth possible. How low would you have to get it to prevent something serious?

Have any tests been done on this? One effect I'm wondering about is altered blood flow throughout the body. Could you get some kind of chronic hypoxia in the brain as a result of the blood being "pushed" toward their feet and away from their head? Would this be compensated by, say, increasing heart rate (and would that lead to severe levels of cardiac stress)? At what levels does it become critical? At what levels does it lead to chronic neuronal death?

• I think you partly answer this yourself - imagine a normal human (on earth) lying down on a hospital bed for an extended period (such as a coma patient) - the gravitational force then runs perpendicular to the spine. We hear plenty about how the muscles will atrophy, but I've never heard about it causing skeletal issues – Mike Brockington May 29 '19 at 9:04
• @MikeBrockington There have been cases of osteoporosis caused by extended bed rest of pregnant women. – Uwe May 29 '19 at 9:41
• @Mike Brockington : That sounds more like an answer as to the question of the effects of the absence of a "gravitational" force, which are already well-known and are what the proposed remedy is about. My question is about the effects of the presence of an uneven "gravitational" force across a human body. – The_Sympathizer May 29 '19 at 11:00
• Uwe as a point about osteoporosis, but my overall point was that if there are no significant issues due to extended bed rest, (complete lack of gravity) then why would there be effects from partial lack? – Mike Brockington May 29 '19 at 14:17

We can know a little of this effect. The insight comes form another answer here.

"unless the rotation forces the human somehow to rotate with the same angular velocity as the coordinate frame/spacecraft" Is a correct caveat however it is of course relevant. People naturally do force themselves to rotate at the same speed rate as the ground!

The force experienced is the sum of the force upwards form the ground and the rotation centered at the point of contact. As per the question we can separate out the constant upwards acceleration and the rotation, but this is a bit tricky to simulate exactly on earth.

However the rotation is the cause of the differential rate of acceleration. Hence we can test it in isolation. Simply lie down at the centre of a spinning disk and you will have differential rates of observed acceleration acting on you. You could compare this to someone laying on a non rotating disk.

If this is fast enough the magnitude of those forces will be large and potentially problematic. But if its slow, I doubt there would be a problem.

This question over at Worldbuilding gives a pretty good answer. The bottom line is that dizziness from the Coriolis effect is more of a concern than the force gradient, and the radius really needs to be rather large (it lists 220 meters) to avoid all discomfort. The minimal tolerable radius is said to be around 25m, which is consistent with what I have read elsewhere.

Too long for a comment, so I'm posting an answer:

This question is based on a wrong premise: $$r$$ is not a variable. $$r \omega^2$$ is a constant for everything in the rotating coordinate frame, and will not affect the human body, unless the rotation forces the human somehow to rotate with the same angular velocity as the coordinate frame/spacecraft.

One can easily see that there should be no effect on a human from the inertial frame: The spacecraft rotates, but the human tries to move forward at any given moment with its CMS momentum according to Newton.
The spacecraft's rotation will impart a force on the feet though, which will translate into a torque about the humans CMS. Thus, back in the rotating frame, the human will simply feel a force pushing the head backwards.

This is different from the tidal effects which you described.

• If $r$ is not a variable but is set for the whole "rotating coordinate frame", what sets $r$ for such a frame? – The_Sympathizer May 29 '19 at 12:56
• @The_Sympathizer: It is the radius of the rotating cylinder. But a connected human body (let's imagine for arguments sake 2 mass-blobs connected with a solid rod) will have different inertial behaviour than 2 unconnected mass-blobs on orbits of different radii. The subtle difference is in the centripetal force: Consider 2 satellites viewn in the inertial frame. Their centripetal force is the gravitational force, and has gradients. This is not true for 2 satellites inside the rotating spacecraft, as there is no centripetal force. They only have inertia. So connecting them, creates a torque. – AtmosphericPrisonEscape May 29 '19 at 13:11
• Isn't it a variable if there are multiple floors in the rotating environment? – Chris B. Behrens May 29 '19 at 14:36
• @ChrisB.Behrens: Sure, but a discrete one, not a continuous one, and only defined for the rotating parts of the spacecraft and those in direct frictional contact with it. – AtmosphericPrisonEscape May 29 '19 at 20:25
• I'm still not buying this. Let's consider the barbell scenario. What I am imagining (assumptions) is the barbell is at rest, standing straight up, when viewed from the rotating reference frame. In that case, the free-body diagram looks like this: drive.google.com/file/d/19U1dKyKlubrrcI0nQ8qaCXg-qBPxKZEl/… – The_Sympathizer May 30 '19 at 5:10