# Solar system scale variance

#### DiracPool

Just a thought experiment. Imagine you made scale models of the sun and the planets with the relational standard of measure being a sun the size of a basketball. Now assume that the sun and the planets were all built to scale with the appropriate diameters, densities and masses of the actual 8 planets plus the sun.

Now, take this model solar system out into the flatest intergalactic space you can find and set the model-scale planets moving about the basketball sun at a scale velocity relational to the actual planets of the solar system.

My question is, would you actually get a working scale model of the solar system in this scenario. More, generally, I guess my question is are the celestial mechanics of the solar system scale invariant? Or are the dynamics dependent on scale, so you wouldn't get an exact replica of the actual solar system with the scale model.

Also, assume two different scenarios, one whereby the model system is controlled for by assuming it has the same scaling of solar wind effects and magnetic field properties of the actual solar system, and one where all you control for is the diameter, density, mass and velocity of the model spheres.

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#### phinds

Gold Member
Just a thought experiment. Imagine you made scale models of the sun and the planets with the relational standard of measure being a sun the size of a basketball. Now assume that the sun and the planets were all built to scale with the appropriate diameters, densities and masses of the actual 8 planets plus the sun.

Now, take this model solar system out into the flatest intergalactic space you can find and set the model-scale planets moving about the basketball sun at a scale velocity relational to the actual planets of the solar system.

My question is, would you actually get a working scale model of the solar system in this scenario. More, generally, I guess my question is are the celestial mechanics of the solar system scale invariant? Or are the dynamics dependent on scale, so you wouldn't get an exact replica of the actual solar system with the scale model.

Also, assume two different scenarios, one whereby the model system is controlled for by assuming it has the same scaling of solar wind effects and magnetic field properties of the actual solar system, and one where all you control for is the diameter, density, mass and velocity of the model spheres.
Do you think a basket-ball sized object, with the density of the sun (that is, with the weight of one basket-ball sized amount of sun-stuff) would have the gravitational strength to seriously attract anything?

#### DiracPool

Do you think a basket-ball sized object, with the density of the sun (that is, with the weight of one basket-ball sized amount of sun-stuff) would have the gravitational strength to seriously attract anything?
Obviously everything would have to scale proportionately. Yeah, the sun would be the size of a basketball but the Earth would be the size of a coarse pepperground or something along those lines. In any case, forget about about basketballs, etc. if you like, and just think think about the relative scale of the center of masses. Take the mass of the sun, the mass of the Earth, and the other planets, etc. and the distances between them, and scale them all down proportionately to where you could fit the whole solar system into a football stadium and set it going in intergalactic space.

It's a thought experiment about scale variance. Would the model system behave similarly to that of the actual solar system or would it not. If not, how would it differ and why?

I've actually just plugged some sample figures into Newtons equations of gravity and it looks as though scaling these figures at least along several order of magnitude give a scale invariant result for the force relations between the centers of masses. But I thought perhaps someone else may have looked into this thought experiment more deeply in terms of angular velocities, momentums,etc.

#### haruspex

Homework Helper
Gold Member
2018 Award
When juggling scales, the constraints are the constants that have dimension. In this case, there's only one: G. You have three scales you can play with independently: mass (or density), length and time. So you should be able to pick any two scales arbitrarily and deduce the third.

#### K^2

You guys are making it way too complicated.

Say I cut size of Sun in half, keeping average density constant. The mass would go down by factor of 2^3 = 8. But I also pull planets in by factor of 2, so that would compensate the decrease with a factor of 2^2 = 4 from inverse square law. So gravitational pull at orbit only goes down by factor of 2.

Now lets look at centripetal acceleration for circular orbit. (If circular orbits work, then all of the elliptic ones will too, thanks to Keppler's Laws.) Velocity of all objects is cut in half. So that would reduce centripetal acceleration by factor of 2^2 = 4. But you also reduce radius by factor of 2, which increases centripetal acceleration by factor of two, and net change is again a factor of 2 reduction.

So if I reduce scale by factor of 2, both gravity and centripetal acceleration reduce by factor of 2, so shapes of orbits remain the same. Same thing happens with any arbitrary scale.

In other words yes, if you scale sun to size of Basketball, all planets and other bodies to match, including distances and velocities, keeping average densities exactly the same, then what you'll get is a working model of solar system.

This assumes classical gravity, of course. Due to GR, you will notice some small differences. Orbit of mercury will not precess, for example, in your toy model.

#### DiracPool

This assumes classical gravity, of course. Due to GR, you will notice some small differences. Orbit of mercury will not precess, for example, in your toy model.
Nice analysis, K^2. Thanks. What is the factor that causes mercury to precess that's excluded from the the toy model?

#### K^2

Non-linear corrections due to GR. Because they are non-linear, they don't scale.

#### TurtleMeister

I agree. Nice analysis by K^2. However, I do have one question.
This assumes classical gravity, of course. Due to GR, you will notice some small differences. Orbit of mercury will not precess, for example, in your toy model.
Why would it not precess? GR only accounts for approximately 43 arc seconds per century. The rest is caused by the gravitational influences of the other planets.

#### K^2

Sorry, yeah, only GR influence would change. I didn't think about any classical causes of precession. These won't change.

#### bahamagreen

"...a working scale model of the solar system..." to me would imply the following things:

The orbital radii are in proportion to the original
The orbital periods are in proportion to the original
these needed so that...
The relative geometric positions match the original through time
(all these to the degree that it can display past, present, and future configurations of the original).

Off hand, it looks like the manipulation of the model's parameters that makes a difference is the mass of the model Sun, this setting the scale for the orbital period of the model planets when placed at their scaled orbital radii... but would that scaling predict the original's dynamics across all the planets?

For example, the mass of the model Sun could be made small enough that the model Earth X meters away would take Y days to orbit, but scaling the radii of orbit and period, would model Venus at .72*X meters radii from the model Sun take Y*225/365 days to orbit, and would model Mars at 1.52*X meters take Y*687/365 days to orbit?

Wouldn't the model need to support that relationship in order to "work" (predict)?

#### K^2

bahamagreen, the parameters were stated. Size and velocity will be proportional. That means, all periods will be preserved. Toy model's Earth will take one year to complete its scaled orbit.

#### bahamagreen

My reading was that it was the initial conditions that were stated (scaled orbit radii and velocities)... and that the question the OP asked is if the dynamics would follow in scale (orbital periods).

Somehow I missed your post K2 - thanks, nicely done answer in Newtonian yes.

#### K^2

Well, no, the period won't scale. But the fact alone that scaling initial conditions gives you the same orbit shapes - scaled, of course, is interesting enough.

#### bahamagreen

Wait a minute..
In #11 "...all periods will be preserved."
In #13 "Well, no, the period won't scale."

I don't see the "same orbital shapes" as very interesting... does that mean any more than a model planet can be placed to circular orbit at any arbitrary orbital radii?

In my opinion, the model works if it predicts the geometry, configuration, and dynamics of the original - if the periods don't scale, I'm not seeing a working model of the Solar system.

#### K^2

Wait a minute..
In #11 "...all periods will be preserved."
In #13 "Well, no, the period won't scale."
Same statement. If period is exactly the same, it did not scale.

I don't see the "same orbital shapes" as very interesting... does that mean any more than a model planet can be placed to circular orbit at any arbitrary orbital radii?

In my opinion, the model works if it predicts the geometry, configuration, and dynamics of the original - if the periods don't scale, I'm not seeing a working model of the Solar system.
Sure, but that requires changing the initial condition. And if the masses didn't scale correctly, then ratios o periods between moons and planets would be off. You could run into whole lot of problems with that.

#### TurtleMeister

I think what K^2 meant by "the period won't scale" is that the periods will not change.

The scaling works just as K^2 described in post #5. I tried this with my n-body simulator. This simulator has the capability of running two separate simulations simultaneously. I set the initial parameters of the solar system on layer one to the ephemeris generated by the Horizons web-interface. I set the parameters for the second solar system on layer two the same, except I reduced all x,y,z parameters by a factor of 10 and all masses by a factor of 1000. The two solar systems ran in harmony. The orbital shapes and periods were the same.

#### Rap

You can do this generally by using dimensionless parameters. If mass is scaled by $\alpha_m$, time by $\alpha_t$, length by $\alpha_L$ and the gravitational constant by $\alpha_G$, then scaling means that $$\frac{\alpha_t^2\alpha_m}{\alpha_G \alpha_L^3}=1$$ In the particular case considered, we have $\alpha_G=1$, $\alpha_m/\alpha_L^3=1$ and $\alpha_t=1$, with velocities scaling as $\alpha_L$ but any scalings which satisfy the above equation will work. For example, if you require length and time to scale equally, while keeping G constant, you would need $\alpha_G=1$, $\alpha_L=\alpha_t$ which means $\alpha_m/\alpha_L=1$. In other words, if you reduce mass and length by the same factor, the periods will be reduced by the same factor. Velocities, factoring as $\alpha_L/\alpha_t$, will remain the same, but densities, factoring as $\alpha_m/\alpha_L^3$, will change.

If you wanted the sun the size of a basketball and the earth rotating every 10 seconds, you would have to have $\alpha_L=2 \times 10^{-10}$, $\alpha_t=3 \times 10^{-7}$, which would give $\alpha_m=1 \times 10^{-16}$ so that densities would increase by a factor of $10^{13}$ and the basketball sun would have a mass of about $2 \times 10^{14}$ kg.

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