Radius of the smallest exoplanet detectable with the transit method

[QuantumX]

Please help me with this astronomy problem. I am supposed to calculate the smallest planet that is detectable with the transit method, given a signal to noise ratio and a star's radius:

Suppose the star is seen at its distance D with a signal to noise ratio of S/N = 10^4. This means that in the light curve for the star (i.e. flux vs. time), the ‘noise’ is 1/10000 the mean of the star’s flux). What is the smallest planet that is detectable? (Hint: assume that the noise has a standard deviation σ and that detection of the transit requires at least a 3σ dip in the flux.) Consider a star exactly like the Sun and evaluate your expression in kilometers by using the known radius of the Sun (5*10^5 km).

Any help is appreciated!

 

[Matterwave]

Do you have any guesses on how to approach this problem? For example, what does the hint tell you?

 

[QuantumX]

Well, my guess is this - I know how big the star is and the SNR tells me the resolution of the instrument, which is to say how small an object it can distinguish from the background noise - in this case a dip in the flux of 3 standard deviations of the noise. So I need to use this information to come up with the smallest radius of a planet that can be detected, but I just don't know how to do that...

 

[Matterwave]

Ok, you're definitely going in the right direction. Do you know in what way (mathematically) the planet will reduce the flux coming from the star?

The planet basically will just block star-light right?

If the planet was the same size as the star, one would expect 100% of the starlight to be blocked during transit right? If the planet was half as big as the star (i.e. radius of the planet is 1/2 radius of the star), how much starlight do you think will be blocked?

 

[QuantumX]

Well an area of a circle is pi*r^2, therefore the area blocked would decrease with the square of the radius. So if the planet has 1/2 the radius of the star, then it would block 1/4th of the starlight?

I just don't know how to convert the S/N ratio with the 3 standard deviation sensitivity into flux blockage

 

[Matterwave]

Ah, so that's where your question is. Well, the signal to noise is simple:

$$SNR=P_{signal}/P_{noise}$$

Which you have given as 10,000. This means if I block 1/10,000 of the signal, then I could attribute this to the noise. But simplistically, if I block 3/10,000 of the signal, then I can no longer attribute this to the noise.

Do you know if your instructor is asking specifically about SNR issues dealing with the noise, or if they just want a rough answer? To give a full answer some statistical analysis is needed, but that might not be in the spirit of this class (unless it's a class on signal processing or some such).

 

[QuantumX]

Ah, I'm starting to understand. No, it's an astronomy class, definitely not a signal processing/engineering or anything like that. However, I do need to use all of the information given.

Okay, so how do I get the smallest planet detectable from a 3/10,000 flux blockage minimum. I need to know the flux of the star, don't I? How do I get the flux from the radius? Don't I also need the luminosity? The problem does say assume the star is the same as the Sun, but I really don't think I'm supposed to look up solar luminosity and use it for the problem, I'm pretty sure all I need to use is 3/10,000th and the radius 5*10^5 km.

 

[Matterwave]

You don't need to know the flux of the star, you just need to know that you have to block 3/10,000 of it. In post #5, you stated that if the radius of the planet is 1/2 the radius of the star, then you block 1/4 the flux. This is true. Now just try the statement, if the radius of the planet is 1/x the radius of the star, then you block 3/10,000 the flux of the star. What is x?

 

[QuantumX]

Ah, I see. So the relationship is radius^2 ∝ area (or is it radius ∝ area^2? I get confused). So it's x^2 = 3/10,000. x = 0.0173 the area of the Sun, which is 0.0173*5*10^5 km = 8650 km (or slightly bigger than Earth).

Is this correct?

 

[Matterwave]

Looks OK to me, except x is not the area, but the radius (I assume it's just a typo on your part).

 

[QuantumX]

Yes, I meant radius :) Thank you so much for walking me through it! If I may ask you for a brief advice on the next part of the question:

Three different flux levels can be measured (assuming the star’s luminosity does not vary): (1) when neither the star nor the planet block one another; (2) when the planet transits the star; and (3) when the star occults the planet. Sketch the three geometries, showing the line of the sight to the observer. Flux from the planet is reflected or re-radiated starlight. Assume it does not vary with orbital phase. Let F* be the flux incident on the Earth from the star and Fp that from the planet. For the three cases, write expressions for the fluxes using these quantities and, as needed, other quantities. Sketch a time sequence where you show schematically these flux levels vs. time over one orbit of the planet. In some cases the planet’s flux does vary with orbital phase because the planet is tidally locked to the star, presenting the same side to the star. When in the time sequence would you then expect the planet’s observable flux to be maximum?

I think I have it all figured out except for the bolded part. I'm confused as to how I'm supposed to write expressions for the fluxes in the three cases. I'm guessing that when they don't block one another the flux is F* + Fp, when the planet transits the star the flux is F* - Fp and when the planet is behind the star the flux is just F*. Are those really "expressions" though, or do I need to write equations? Am I to extrapolate from the flux incident on the Earth to figure out the entire flux of the star-planet system? Nothing else is given, so I am a little confused, what's your take on it?

Also, orbital phase is to be ignored, but for the very last part of the question - if the planet is tidally locked, I think it would be brightest when it is right on the edge, right before it goes behind the star, as then you would see half the planet illuminated. Make sense?

And for the next part where I am to sketch a time sequence of the flux, I'm thinking it's a straight line when the planet is behind the star, then there's a dip when it transits and then it goes up when both are radiating.

Thoughts? Sorry if I overwhelmed you, thanks for your help so far.

 

[Matterwave]

1) For the fluxes during different periods. When you can see both the star and the planet, it makes sense that the flux is F*+Fp (correct). When you can't see the planet, then the planet is entirely blocked by the star, and therefore the flux is F* only (correct). But when the planet is blocking the star, why do you say the flux is F*-Fp? We just did this problem earlier, and the answer did not have anything to do with Fp!

2) Your answer here makes sense to me. The surface of the planet pointing directly at the star will be much hotter and brighter than the other side.

3) Most of the time will not be transits (of either star blocking planet or planet blocking star). This is because the orbit of a planet is usually much larger than the radius of the star. So most of the time you should have both the planet and the star's light shining on you. That should be your "straight (horizontal) line". Can you figure out what happens when the planet blocks the star, and then when the star blocks the planet? A harder question (which cannot be answered with the variables given to you) would be which transit event gives you a deeper transit depth? Can you make a guess?

Lastly, for a bonus question, the flux that we have been considering in this question has always been the integrated flux over all wavelengths. What do you think, though, is the difference between the light emitted by the star, and the light emitted by the planet? And therefore, what do you think we will see in terms of transits if we looked at a spectroscopic image (resolving all the wavelengths) instead of a total light curve?

 

[QuantumX]

1) Okay, so my logic there was that when the planet is blocking the star, the flux is F* - the area of the planet, and I thought that that would be F* - the planet's flux, but now that you've pointed it out, that doesn't make much sense. So is it just F* - area of the planet?

2) Actually now that I think about it, what difference does it make whether it's tidally locked or not from our perspective? The planet's rotation has nothing to do with what percentage of it is illuminated? The only thing that would make a difference is, as you pointed out, that it would be much hotter, so we'll see more infrared radiation, but as far as visible light, it would be the same whether or not it's tidally locked, correct?

3) Okay, I think I see where you're going here. So it would be a straight line when both are radiating (ignoring orbital phase), then I'm guessing when the planet blocks the star you would see a big dip (big is relative here of course) as now you have the star's radiation - the area of the planet, and when the planet goes behind the star, the line would go up a little as now you only have the star's radiation, and then when the planet comes around again, the line would go up to its original height...?

Bonus - I would definitely think the light emitted from the star would be much richer in terms of wavelengths, as in you would have a lot more thermal, x-ray, maybe radio? And the planet would be mostly visible light with some thermal, depending on orbital phase here...?

 

[Matterwave]

1) Yea, except I definitely wouldn't call it "area of the planet". I'd call it "flux blocked by the planet" which is proportional to the area (but not equal to it of course, the units wouldn't work that way).

2) Most of the light would be reflected light. So really, the light curve should be slightly different depending on orbital phase, all the time (because we'll see phases of the planet, just like phases of the moon, or phases of venus), but this is presumably a small effect.

3) Straight line during both radiating. When the planet is blocking the star, you get a large dip as you said (correct), but why do you think the flux goes up when the planet is behind the star (is that just a typo)?

Bonus: The star emits more light at all wavelengths than the planet, of course (it's much much brighter after all). But the planet's emitted light would be mostly in the infrared. This means when you have the star blocking the planet, you see a dip mostly in the infrared portion of the light curve. From the nature of this dip, you can actually derive the temperature of the planet!

 

[Mordred]

Looks like Matterwave has helped enough, I didn't want to add interference, or confusion. So I held off posting a couple of useful papers. These don't apply directly to your question however related. Figured they may help in your studies. Though both will help relate to your questions.

I particularly like this one for the visual correlations.

transit detection algorithms
http://www.tls-tautenburg.de/research/artie/lectures/TDA.pdf

TRANSITS OF EARTH-LIKE PLANETS
http://arxiv.org/ftp/arxiv/papers/0903/0903.3371.pdf

 

[QuantumX]

1) I see. Is there a way to express the scenario when the planet is in front of the star in terms of F* and Fp?

2) Agreed.

3) Well, I think you have a straight line when both are radiating, and then a large dip means the planet is now not radiating as it is in front of the star AND the star's flux is blocked, and then you go back to a straight line when the 2 are radiating again, then when the planet goes behind the star, only the star is radiating, so you have a smaller dip, and then when both are radiating again, you go back to a straight line. Does that not make sense?

 

[Matterwave]

1) No, not in general. The only way to do it would be if you knew the temperature of the planet, and obtained the surface area via Fp. But that's really just a round-about way. The essential information you need is the radius of the planet.

3) That's fine. You had a typo in your previous post where you said the light curve goes UP when the planet is behind the star, when in fact it goes down.

 

[QuantumX]

Okay, well thank you so much for all your help, I really appreciate you taking the time!