Questions on stars and planets: ratio of planet radius to star, distance between the two, temperature of a star and a planet, planet rotation speed

[Hak]

A planet of mass $$M$$ and temperature $$T_{p}$$ is in thermal equilibrium with its star, of temperature $$T_{s}$$. The distance between the two celestial bodies is $$d$$. The radius of the planet is $$\frac{1}{4}$$ of the radius of the star. The planet rotates on itself with angular velocity $$\omega$$.
The planet's atmosphere is such that clouds form at a height $$h$$ above the ground. The clouds create rain that falls in a rarefied atmosphere with negligible friction. We can also assume for simplicity that $$h$$ is very small and negligible compared to the radius of the planet. Let us assume that the clouds and rain are at a $$\theta$$ colatitude.

Now, this problem serves as an example to extrapolate some really interesting physics-related information. The question posed by the problem (which I do not report) has been solved by me, but I would like some difficult implications explained. I would like to understand a little if and why the data that is given is realistic and how it is found. Meaning: how do you measure the ratio of the planet's radius to the star, the distance between the two, how do you measure the temperature of a star and a planet, the planet's rotational speed, etc...

For the temperature of the star, I think it is possible to obtain it from Wien's law if it is possible to receive the radiation emitted by the star and measure its wavelength; on the other hand, I don't think there are many other ways of obtaining information about distant celestial bodies. I had thought that the same procedure could be applied to obtain the temperature of the planet, but I don't think it is possible. Planets emit very little, that you can capture and recognise the light coming from the planet seems difficult to me. I can't think of anything.

For the planet's rotational velocity, I am reminded of the question in this year's second level competition, where it was required to apply the Doppler effect to obtain the Sun's rotational velocity. This assumes, as with temperature, that we are able to measure the radiation emitted by the celestial body with sufficient precision. However, this method does not convince me very much; in the problem we were talking about hydrogen stripes, so it may be inapplicable to a rocky planet. Alternatively, if the planet has an atmosphere, we could deduce something about its rotation by observing the motions of any clouds (assuming we have a powerful enough telescope).

For the distance between the two bodies I have thought of two possibilities: the first, simply, is that we are in a position to measure it by observing for at least one period the planet's motion of revolution (the maximum distance observed would correspond to the one sought); the second, much more improbable, is that it can be deduced from Kepler's law by knowing the planet's period of revolution (which I think is measurable) and the mass of the star. I don't really have a decent idea of how the latter could be measured in general (perhaps there is some relationship between the surface temperature and the mass of a star?), but I thought that, in the case where several planets orbit the same star, its Kepler constant (and thus its mass) can be derived by measuring the periods of revolution of the planets and the ratios of their distances from the star (I think at least the ratio can be determined by telescope).

For the ratio of planet and star radii, I throw up my hands. The only idea I have is to compare them when the planet passes exactly in front of the star (i.e. they are aligned with our view), but this only makes sense if the distance between the two is much smaller than the distance between us and that star system (which I think is true enough for every system except the Solar system) and if it is possible to obtain such high resolutions (and I already had my doubts about the distance between the two, which should be much greater than their radii anyway).

Thanks for any reply.

 

[Drakkith]

Meaning: how do you measure the ratio of the planet's radius to the star, the distance between the two, how do you measure the temperature of a star and a planet, the planet's rotational speed, etc...

These properties can be measured by different methods and instruments. Direct imaging, spectrographic analysis, doppler shift, transits, etc. Stellar temperature is trivial to measure, but planetary temperature is much more difficult. A similar situation occurs for most properties. Stellar properties are almost always (if not always) more easily determined than exoplanet properties. Mainly because we can easily see the stars and because many stellar properties are directly related to stellar mass in a relatively simple way, which is not true for exoplanets.

For the distance between the two bodies I have thought of two possibilities: the first, simply, is that we are in a position to measure it by observing for at least one period the planet's motion of revolution (the maximum distance observed would correspond to the one sought); the second, much more improbable, is that it can be deduced from Kepler's law by knowing the planet's period of revolution (which I think is measurable) and the mass of the star.

There are a couple of ways of determining this. If we know the mass of the star, then we just need to find some the orbital properties of the planet. Direct imaging can provide us with snapshots of the planet's position, from which we can then fit the planet to an orbit. Doppler measurements can give us the orbital period, which makes it trivial to find the orbital distance if we know the mass of the star. Transits can also give us the orbital period if we observe multiple sequential transits. Other methods are similar in that we usually get the orbital period and from there determine the orbital radius. The full orbital properties (inclination, eccentricity, etc) can't always be found from a single method if I remember correctly, so there is often some non-insignificant error bars in the measurements.

For the ratio of planet and star radii, I throw up my hands. The only idea I have is to compare them when the planet passes exactly in front of the star (i.e. they are aligned with our view), but this only makes sense if the distance between the two is much smaller than the distance between us and that star system (which I think is true enough for every system except the Solar system)

Stellar radius is pretty easy to determine if the star is main sequence, at it is mostly a function of stellar mass. I'm not sure about stars off the main sequence track. Planetary radius is much, much more difficult to determine since there isn't a simple relationship to go by. Transits can directly give us the size of the planet relative to the size of the star, but only if we also know the distance between the two at the time of the transit. Direct imaging can get you a rough estimate, but other methods are even less accurate or not useful at all in determining the planetary radius.

 

[PeterDonis]

is in thermal equilibrium with its star

This is impossible; it would mean that the planet and star were at the same temperature.

I think what you mean here is that the planet's temperature, on average, is stable, taking into account incoming radiation from its star and outgoing radiation emitted by the planet. But that is not the same as being in thermal equilibrium with the star.

The question posed by the problem (which I do not report) has been solved by me, but I would like some difficult implications explained.

That's going to be tough if we can't see the problem statement.

I would like to understand a little if and why the data that is given is realistic and how it is found.

Was the data claimed to be realistic in the problem statement?

 

[Hak]

That's going to be tough if we can't see the problem statement.

Was the data claimed to be realistic in the problem statement?

The statement of the problem is not related to my questions, it is merely a pretext to introduce the quantities in respect of which I have made subsequent enquiries. By 'realistic' I mean: "are the questions I asked possible? Can these quantities be calculated?" Thanks.

 

[Drakkith]

Everything is realistic and calculable except perhaps cloud height, but I don't see a problem with the question simply stating these values.

 

[Ken G]

The radius of the planet is $$\frac{1}{4}$$ of the radius of the star.

That's a pretty big planet!

The planet rotates on itself with angular velocity $$\omega$$.

This will not have much impact on general properties, and is very hard to measure.

Meaning: how do you measure the ratio of the planet's radius to the star,

That's most easily done by "transit" measurements, where you watch how much of the starlight is occulted by the planet as it passes in front (if it passes in front. You have to be satisfied with what you get).

the distance between the two,

This is usually obtained from Kepler's law, where you watch the orbital period and use your knowledge of the mass of the star to determine the distance to the planet. You can then check that against the expected orbital speed of the planet and the duration of the transit to build a consistent picture.

For the temperature of the star, I think it is possible to obtain it from Wien's law if it is possible to receive the radiation emitted by the star and measure its wavelength; on the other hand, I don't think there are many other ways of obtaining information about distant celestial bodies.

It is pretty easy to get the temperature by looking at spectral lines that tell you the degree of ionization of various ions. The temperature and spin rate of a planet are much harder to know. If you see the

I had thought that the same procedure could be applied to obtain the temperature of the planet, but I don't think it is possible. Planets emit very little, that you can capture and recognise the light coming from the planet seems difficult to me. I can't think of anything.

Yeah, I don't think this is usually possible to directly observe, you can however calculate the temperature you expect, given the distance to the star and other things you would need to know (essentially how well the planet reflects vs. absorbs light, called its albedo).

For the planet's rotational velocity, I am reminded of the question in this year's second level competition, where it was required to apply the Doppler effect to obtain the Sun's rotational velocity. This assumes, as with temperature, that we are able to measure the radiation emitted by the celestial body with sufficient precision. However, this method does not convince me very much; in the problem we were talking about hydrogen stripes, so it may be inapplicable to a rocky planet. Alternatively, if the planet has an atmosphere, we could deduce something about its rotation by observing the motions of any clouds (assuming we have a powerful enough telescope).

Rocky planets are small and it is hard to see light reflected off them or emitted by them. You might see some absorption from their atmosphere, but basically they just transit the star and occult starlight and not much else. Sometimes you can see light reflected off them, but it might not give much detailed information about the planet. Mostly they get the size of the planet and have to kind of guess the rest. If the planet is large enough (like your planet with 1/4 the radius of the star), you can look at its gravitational influence on the star, creating an orbital wobble you can see in its spectral lines.

For the distance between the two bodies I have thought of two possibilities: the first, simply, is that we are in a position to measure it by observing for at least one period the planet's motion of revolution (the maximum distance observed would correspond to the one sought); the second, much more improbable, is that it can be deduced from Kepler's law by knowing the planet's period of revolution (which I think is measurable) and the mass of the star.

That's not more improbable, that second way is the easiest way available.

I don't really have a decent idea of how the latter could be measured in general (perhaps there is some relationship between the surface temperature and the mass of a star?), but I thought that, in the case where several planets orbit the same star, its Kepler constant (and thus its mass) can be derived by measuring the periods of revolution of the planets and the ratios of their distances from the star (I think at least the ratio can be determined by telescope).

You get the period of the orbit, and you think you know the size and mass of the star (by looking at the star's spectrum carefully). That's enough to infer the distance to the planet via Newton's modification to Kepler's law. Then you also have the duration of the transit, so that tells you if it crosses the star's equator or not, and gives you a basic check you have the right picture.

For the ratio of planet and star radii, I throw up my hands.

That has to do with how much the starlight dims during transit, which tells you the fraction of the stellar area covered by the planet area. You also get information about how quickly the dimming occurs as the planet first starts to cover the star. It's pretty straightforward to understand the geometry of the planet in front of the star if you get a repeating transit (the basis of the Kepler and TESS missions).

The only idea I have is to compare them when the planet passes exactly in front of the star (i.e. they are aligned with our view), but this only makes sense if the distance between the two is much smaller than the distance between us and that star system (which I think is true enough for every system except the Solar system) and if it is possible to obtain such high resolutions (and I already had my doubts about the distance between the two, which should be much greater than their radii anyway).

What makes it possible is you "resolve" it in the time domain, i.e., you see a "light curve". Then you don't have to try to resolve anything spatially, which would be very hard to do.

 

[Drakkith]

That's a pretty big planet!

Or a moderately small star. Jupiter-to-Proxima Centauri diameter ratio is about 0.652. But if we assume the planet is not a gas giant, then the ratio does seem unrealistically large unless rocky planets get much larger than I thought.

 

[PeterDonis]

The statement of the problem is not related to my questions

How do we know that if we can't see the problem statement? I have already pointed out one item in your OP that is impossible. Was that really in the problem statement?

For that matter, you claim to have solved the problem. How do we know you did so correctly?

 

[Hak]

How do we know that if we can't see the problem statement? I have already pointed out one item in your OP that is impossible. Was that really in the problem statement?

For that matter, you claim to have solved the problem. How do we know you did so correctly?

The statement was: By how much is a raindrop deflected eastwards as it hits the ground? It is not linked with my subsequent questions. Let's pretend that the problem has nothing to do with it: all I'm interested in are the questions.

For the rotational speed of the planet, I had actually thought that if the planet is close to the star probably the tidal forces make it spin around itself at the same speed as it spins around the planet (like Moon and Earth)... But that's by no means guaranteed. I'd be curious to know how this is done.

 

[Ken G]

You are referring to "tidal locking", which depends on the age of the system and is not common for planets unless they are very close to the star, but it sounds like the planet spin is a given, not something you need to calculate. And are you sure the question is not how much is it deflected as it falls, rather than as it hits? The former would involve the coriolis effect, so is fairly straightforward if given the planet's spin rate and gravity. How much it deflects "as it hits the ground" requires you to know how a moving droplet smashes into the ground, I doubt that's the issue.

 

[Hak]

You are referring to "tidal locking", which depends on the age of the system and is not common for planets unless they are very close to the star, but it sounds like the planet spin is a given, not something you need to calculate. And are you sure the question is not how much is it deflected as it falls, rather than as it hits? The former would involve the coriolis effect, so is fairly straightforward if given the planet's spin rate and gravity. How much it deflects "as it hits the ground" requires you to know how a moving droplet smashes into the ground, I doubt that's the issue.

Of course, you are absolutely right. It is just a matter of semantics, since I translated from Italian and did not think of this possibility. It is as you say, of course.

I didn't understand what you say about my thinking about the rotational speed of the planet. Could you explain further? Thank you very much. Anyway, as I said, I solved the statement of the problem, maybe I was wrong to take the latter as a pretext, but I am mainly interested in the subsequent questions. Thank you again.

 

[Ken G]

I didn't understand what you say about my thinking about the rotational speed of the planet. Could you explain further?

I'm saying it sounds like you are to take the rotational speed as a given, not something you are supposed to figure out or observe. I think in most cases, we would not know the rotation speed of the planet, but that's not essential to the question.

 

[Drakkith]

The OP has been banned for a serious violation of PF rules. Further questions on this topic can be posted in a new thread. Thread locked.