Scioly.org

The equation sum of masses = a^3 / p^2 is a slightly more general formula using units of solar masses, AU, and years. In fact, it's not the most general formula, which is a^3 / p^2 = G*sum of masses / (4pi)^2, all in MKS units. Unfortunately, for the first part due to constrained information you need to approximate the whole mass of the system as the mass of the star alone, which is a pretty good approximation.

What this means is you actually make an incorrect assumption in saying semi-major axis of orbit is given, it's actually derived. For part a you are ignoring that mass, and by part c that distance is still derived and not given. So basically when you add in the planet mass, m_p, you are working with different values. To work around that, you need to derive other methods with physics! Knowing to use the ratio is really just experience, if you just google something like "exoplanet formulas" along with looking through the rules (there's stuff like spectroscopic parallax and general stellar evolution comprehension) you should get an idea of math to be asked. Beyond that it's just practice.

Now to try to make the physics slightly more clear. The problem with "plug and chug" use of equations, which I know is not your intention but kind of just happened, is that you don't think as much as you can about your options and techniques, which is why I always recommend derivation (at least it brings understanding somewhat closer). If you're an astronomer in the field, you have to be VERY careful about stuff like this. I swear, there's derivations available online for various formulas.

Now you're probably wondering still, how can we use this magic ratio if we work under that assumption? Great question! The key is...well, we work under that assumption for radial velocity (along with some others). The derivation comes from the definition of the center of mass, which allows us to say m1*x1 = m2*x2 in general (x1 and x2 are respective distances to the barycenter, or center of mass). Assumed circular velocity is 2pi*r/T, so you end up getting v1 and v2 (the 2pis cancel), and so the ratio then becomes what we've been saying. An alternative way is just through conservation of momentum, which directly produces the equation. There is even something called the binary system mass ratio, but that requires ratio of masses (or velocities, distances), which again follows the same issue with keeping consistent assumptions. Both this and Kepler's third law at their base come from a derivation that can simply use Newton's third law (that is forces that form reaction pairs are equal, like the mutual gravitational force felt by both bodies in orbit for this question). If it helps you, just think of the "total mass" as being put equal to the mass of the star (so you would have a slightly less massive star to input the orbiting planet).

You may now worry all answers are wrong, but everything has error. This error from ignoring planetary mass in the equation of Kepler's third law often produces an error of about 0.001%, which is hardly as much as the errors from non-circular orbit or inclination.

Does that make more sense?

We're not getting it, but please don't give up on me! Here is what we did.

The problem states star A has the same mass, radius, and luminosity as the sun. In question A, the distance from star A to planet B was figured as .041 AU. It also states that planet B has a circular orbit. So we know that the semi-major axis is the same as the distance from the star.

Question C asks what is the mass of planet B in Jupiter masses.

The formula we used: M1 + M2 = a^3/P^2 masses should be in Solar Masses and period should be in years, according to the formula. We looked at the radial velocity curve and figured the orbital period to be 3 days, or .00822 years.

That left us with M2 = .041AU^3 / .0822 years^2 - 1SM

This gave us M2 = .01020 - 1 = negative .9898 solar masses, which is clearly wrong. We intended to convert to Jupiter masses there, but what was the point with a negative mass that was clearly incorrect.

So why would this not work, and why would think, oh, we need to figure out the velocity and use that other formula. It seemed like we had all the information we needed to use the formula we did.

(Thank you so much for any help, this is really difficult and frustrating...

sciolymom wrote: …figured the orbital period to be 3 days, or .00822 years.

That left us with M2 = .041AU^3 / .0822 years^2 - 1SM

You dropped a factor of 10 from the orbital period

This is definitely a good question that I hope is a learning lesson. In advance, if there's something that's tough to understand here, then please quote/explain what is confusing. Sorry for the long posts, but I hope they're helpful.

As I said, you're working under different assumptions. I put it as "splitting up the mass of the system to get the planet" in one case to "assuming there's extra mass in the system suddenly for the planet". You'll definitely get some error no matter what because in part A you're using Kepler's third law ASSUMING mass_planet << mass_star, then in part C you want to assume that the distance from part A applies to a case where mass_planet actually adds to the mass of the system. Does that still not make sense? I can't state enough that part A the question maker makes one assumption, and you use that assumption and data on a completely different system to solve part C as I shall demonstrate.

As finagle said, your answer isn't negative: http://www.wolframalpha.com/input/?i=%2 ... ter+masses just to reconfirm what s/he got. But this is a farcry from the real answer, 0.495 Jupiter Masses based on appropriate assumptions. Momentum, and physics in general, can be weird at first, and you have to spend some work to understand. The most awesome thing here is that momentum comes very simply from Newton's Laws. It's most powerful ability is that it demonstrates net differences before and after changes occur in a system, and you don't have to care at all about anything in between. One other piece of advice is in part B you find the velocity of the planet, and the questions here are meant to be somewhat guided (especially for this question). Maybe some unit weirdness, but that's just from commonly used units.

In case you still don't feel satisfied, let me show you something. If you understand the law of conservation of momentum (which I can prove in another post if you don't), then we know m_1*v_1 = m_2*v_2. We know implicitly that the answer from the key would work. Let's test your answer, and we can try to get v_planet from it by v_planet = m_star*v_star / m_planet (all using your response):
http://www.wolframalpha.com/input/?i=%2 ... 000+km%2Fs

I am doing this quickly, and I hope this isn't too ramble-y. As you can see this velocity is tremendously lower than what we have in Part B. First, we have the logic that 21 jupiter masses is within range for a low-mass brown dwarf. Second, why would it be so low for a planet? Wouldn't the lower-mass planet be moving a lot faster (of course that all depends on initial parameters, but even then think momentum)?

What went wrong? How is it that this system with a different mass can satisfy Kepler's Third Law to produce the same semi-major axis and period, yet not satisfy what we'd expect in terms of mass and momentum? The answer lies, as I want to emphasize, in working with really different systems. There's a few things going on:

1) The question states outright you HAVE to let mass_system = mass_star for Kepler's Third Law. This IS an approximation to allow us to simplify things, and it produces relatively little error (there are probably better methods, but I can't remember off the top of my head). Do you want to add the mass of the planet into the system and see what semi-major axis we get given the period? Do you think it'd be different? You'd be right, here it is: http://www.wolframalpha.com/input/?i=%2 ... ical+units
It's different. Why is that? Because we weren't following the approximation! This may sound silly, but science uses approximations all the time, and it's incredibly important to know what works and what doesn't.

2) The velocities are different. This is in fact directly from how you're treating the masses, which is a really cool concept to show. You can have really massive objects and less massive objects orbit around the same point based on velocity, though stability of the system is another, more complex story. But you should understand your mass does still apply to a completely different binary system because the velocity definitely doesn't follow our original assumptions. Based on the semi-major axis and period, part B must follow the formula for circular velocity, v_circular = 2pi*r/T. Using your mass, it clearly does not.

I hope this has definitively shown you are effectively making a different system and thus not using given data appropriately. Don't change your assumptions, and use formulas which flow correctly with the question. Outside that it's just reading, practice, and asking questions! There are many websites online, and the nats supervisors are actually incredibly kind in giving help to prepare. I am quite confident that your students can search around stuff like "radial velocity method formulas", "radial velocity method derivation", or other associated concepts. Not to say don't questions, please ask stuff that's confusing. It's a new topic, so it is tough, but I guarantee it's like any year having to search up formulas (in fact it's better since it's more level for everyone who has done basically pure stellar evo for years on end).

Specifically for Kepler's Third Law, if you make the assumption mass_system = mass_star, then that's that and you can't change it. You have given the system some condition, and you have declared that the mass of the star is far greater than that of the planet, so you can't add the planet back in like you want (or else you change the system). You can still use conservation of momentum because error is fairly low with this ratio. A surface argument to why you get so much more error is that you're basically just adding stuff into the system, while using the ratio is using already derived quantities (rather than going back and changing the original assumption). You WILL get some error, but clearly it could be worse. Here's hoping this just makes it "cool" instead of "magic"

Oh my gosh...my tiny brain is starting to understand! It really requires an understanding of the concepts rather than just a knowledge of the formulas, which is something that we are gaining slowly but surely. Thank you for your patience and assistance.

Sorry for the math error, that was just a careless mistake. I believe we did it correctly at one point, which as you pointed out, still did not give us the correct answer. We must have worked that problem a hundred times....

sciolymom wrote:Oh my gosh...my tiny brain is starting to understand! It really requires an understanding of the concepts rather than just a knowledge of the formulas, which is something that we are gaining slowly but surely. Thank you for your patience and assistance.

Sorry for the math error, that was just a careless mistake. I believe we did it correctly at one point, which as you pointed out, still did not give us the correct answer. We must have worked that problem a hundred times....

Haha, it's fine, that's why we ask questions . I am just extremely glad you now see what I mean about "really getting it". Physics is well-known as not being an easy major for a reason, and astronomy definitely uses physics nowadays (and of course I am right now doing both...). Also, you and your students don't have such tiny brains, I swear it's just practice (for some odd reason you throw math in the mix and people get frightened/confused, including me). As for math errors we all make them. I've made many while making test questions. Oops. I promise us test makers TRY to check each other when possible! Sorry for the ginormous posts, and, yeah, if you do it a hundred times or there about, then it's definitely a good idea to ask for help. To everyone, if there's any issues/confusion, then please ask.

Hi guys! Wow, lots of discussion on calculations here.

Yeah, the mean absolute magnitude of RR Lyrae stars is 0.75.

Formulas... Three years ago, I was blindly formula chugging. It was only in the last few years that I learned enough physics / actually devoted time to figure out some derivations of the stuff on my formula sheet. I mean, formula chugging works, as long as you know your units and unit conversions and don't mess up while entering into you calculator. Some practice is necessary.

About the planetary equilibrium temperature calculations, what do you do if they ask that kind of problem, but don't give you albedo at all? No "assume it's a blackbody" - just nothing. I sort of used an estimate for Earth's albedo...

Is it possible that they give you an "exoplanet-orbiting-a-star" problem where you can't assume that the star has almost all of the mass?

Also, do you guys know where to find info on Herbig Ae/Be stars and FU Orionis (not the variable type but the star itself)? The Tiger test asked for how much magnitudes HABe tend to vary by, and ... I can't find a number for that.

For planetary equilibrium temperature, if you look at the formula there's a value, a, for albedo. If they make this 0, you just plug in 0, and the term is really (1+a)^something, so really it just goes to multiplying by one. Nothing too much to worry about. Think about it in context, it's like when emissivity is one (we basically ignore it in the formula, blackbodies make for convenient/appropriate simplifications when possible). Of course, if they somehow figured out the albedo, then you should know to just plug it in. If they don't mention it...I guess try to ask the proctor, because it does make a large difference depending on what the albedo is.

Well, planets generally are less than what, 13 Jupiter Masses? So you'd have to have A REALLY massive Hot Jupiter to have the assumption break down (or just plainly a brown dwarf). Assuming you mean this question to be a radial velocity question, then in that case I'd assume the radial velocity graph would show the velocities of BOTH objects, but I guess I can't guarantee that. I hope you know if they give velocities for both objects in the system, then the problem becomes a good deal easier. Commonly, though, you can get the data for the star more easily than for the plane too, so that's why the approximation is so common. That said, you should also understand the transit and radial velocity methods can have MANY complications that can trip you up .

HAEBE star info can be tough to find, but not impossible. FUOR stuff is definitely online. I'll leave it to others if they want to share, but it's all about search terms . Probably for States, Cicc wouldn't use a question quite like that, and nats definitely doesn't focus on that sort of question. That's because it's not about throwing around random numbers as much as really getting down to understanding what makes these objects tick! At the very least, if you don't want to search too much, I give the friendly reminder to check out AAVSO.

Hey everyone! I have a few questions about the practice test this year. First of all, is there any particular way to determine a star's spectral type from its temperature, or do I just have to approximate it with an H-R diagram? If it's the latter, could someone link me to an H-R diagram good enough for accurate approximations?
Anyways, I have several questions about the practice test. First of all, I have no idea how to do #2. What is a J-K color index, and how can I use it to answer the questions? I'm also a bit confused about the whole map concept in questions 16-19. I also am not sure how to do 25e. I don't know how to determine the period of the star given the graphs - any help with this? I couldn't get 26c or 28 due to the problem I had above - any help with these? Also, I don't really understand spectra - could someone tell me how to read them?

ImNotSmart wrote:Hey everyone! I have a few questions about the practice test this year. First of all, is there any particular way to determine a star's spectral type from its temperature, or do I just have to approximate it with an H-R diagram? If it's the latter, could someone link me to an H-R diagram good enough for accurate approximations?
Anyways, I have several questions about the practice test. First of all, I have no idea how to do #2. What is a J-K color index, and how can I use it to answer the questions? I'm also a bit confused about the whole map concept in questions 16-19. I also am not sure how to do 25e. I don't know how to determine the period of the star given the graphs - any help with this? I couldn't get 26c or 28 due to the problem I had above - any help with these? Also, I don't really understand spectra - could someone tell me how to read them?

This seems to tell me you may not have 100% put this all together (who knows, maybe 99%. though ). Spectral type, temperature, and color index are all correlated, which you can see if you look at enough HR diagrams. You could also look up tables with spectral class and luminosity class, and those generally have temperatures. Those will give it a bit more precisely. But yes, at competitions worst case I just took out an HR diagram and approximated. I would say google "Hipparcos HR Diagram" as I don't see one from GAIA or anything else sadly! Hope that gives some ideas for 26c and 28. Again, tables are a great option if you can find some reliable ones. I think something from the Carroll and Ostlie textbook appendices may be online, though.

Next, have you at all googled "brown dwarf color-magnitude diagram"? J-K is just another color index like B-V or U-B. It uses 2MASS colors, and I've used them for research actually (not on brown dwarfs, for other stuff). It's just in the IR region. If you've looked up color-magnitude diagrams in general, then you'd see it simply is not a major worry. The important bit isn't in the colors or the magnitudes (though, I guess if you know it's all IR, then that could help know spectral type in general), it's in the SHAPE. It's just like stellar tracks, we can show brown dwarfs evolve along these lines based on various evolution parameters.

Have you looked up "exoplanet atmospheric temperature map", "substellar point", or some combination? I would say as you're looking it up, keep in mind there's multiple models, and I think some NASA and ESA sites explain how this one works, http://www.nasa.gov/mission_pages/spitz ... 95_prt.htm is an example of another one also for finding the hot spot. I think that one is more related to the observational diagram. Perhaps if you think of how transit light curves work, then you'll be thinking better about this stuff. Transmission spectra there's plenty of links online referencing exoplanets and their general explanation. Outside learning all this stuff conceptually, as you are trying to, you should be looking up the DSOs themselves. If you googled all the DSOs and got good notes, those questions shouldn't be as much of a problem.

25e: Nobody has asked 25 I think, for some reason. There's a few ways to do this one I think. There's some whole formula to find primary eclipse time. You could look that up and use that I guess. Or we can do another way! They give orbital phase on the bottom, right? If you find deltat from the beginning to end of the eclipse, then we can use that PHASE difference to figure out what the time is utilizing the given period. Phase is basically just (portion of period)/(total period), so multiplying the deltat by orbital phase is like doing ((end time)/period - (start time)/period)*period = (end - start), or the amount of time for the transit. Hope that helps, if it doesn't I'll do exact work (you asked a lot of questions, so I'm not trying to write too long an essay ).

Spectra take a trained eye, and others are better at figuring it out than me. There's those bars, you could be given characteristics, or you could be given digital spectral (as in an intensity vs. wavelength or frequency plot, etc etc). There's many options. My recommendation is to spend time looking up picture examples. You can try testing yourself if you really want, and you should be able to understand at least what, for example, spikes or dips would mean on a graph in general (which isn't too hard to look up). Some examples of stuff to read would be absorption lines (dips), emission lines (peaks), doppler shifted spectra, broadened/narrowed lines (based on luminosity class or really many reasons), and a blackbody spectrum (continuum, understanding Wien's law, etc). If this is all alien, first thing is to look it up! I admit I still am no genius at spectra-reading, but preparing diagrams and practicing concepts is fair game to help get ready.

I say a lot of this as "have you looked some of this up first" because your questions were fairly general, and I have the tendency to give more general answers to general questions. If you can't figure it out from looking it up, could you show a link or something to show a paragraph you just plainly don't get?