Lecture 3: Binary Stars PDF

Lecture 3: Binary Stars Prof. Evan Keane PYU22X20 — Observing the Universe [email protected] Thursday 23rd Jan 2025 12:00  L2 Recap In L2 we looked at: 1. Spectral classi cation of stars 2. The origin of spectral lines 3. Physical processes r...

Lecture 3: Binary Stars Prof. Evan Keane PYU22X20 — Observing the Universe [email protected] Thursday 23rd Jan 2025 12:00  L2 Recap In L2 we looked at: 1. Spectral classi cation of stars 2. The origin of spectral lines 3. Physical processes relevant to spectral lines 4. The Hertzsprung-Russell diagram fi  L3 Learning Outcomes Goal: To understand the manifestations of binarity in the stellar population, and what we can learn from quantifying these. Lecture 3 Outline 1. Binary parameters, Kepler’s Laws 2. Types of observed binaries 3. Doppler Effect 4. Stellar masses  Aside: Ellipses P (x, y) The equation of an ellipse is: x2 y2 2b + = 1 F2 (−ae,0) F2 (ae,0) a2 b2 which if you’ve not encountered before should look familiar in the 2a case where a = b = r The area of an ellipse is πab For a circle the two focii coincide  Aside: Ellipses The distance from the origin to P (x, y) a focus is: c = ae 2b For any point on the ellipse P F2 (−ae,0) F2 (ae,0) | F1P | + | F2P | = const = 2a —> this is how you draw an ellipse 2a Can see that the eccentricity is given by: b2 a e= 1− 2 b a ae https://upload.wikimedia.org/wikipedia/commons/e/eb/Orbit1.svg Binary Parameters An orbit can be de ned by 6 ‘orbital elements’ Closed orbits are elliptical so the shape is de ned by 2 params. a — size of semi-major axis e — eccentricity [e = 0 circle, 0 < e < 1 ellipse, e = 1 parabola, e > 1 hyperbola] b — is the semi-minor axis. We need either (a, b) or (a, e) to de ne shape of ellipse fi fi fi  Binary Parameters An orbit can be de ned by 6 ‘orbital elements’ You can do a lot of calculations with just a and e but if you want to explain observations you have to factor in the orientation of the orbit and the orbital plane as these are also entangled in what we observe. i — orbital plane’s inclination wrt us the observer Ω — longitude of ascending node, measured in observer’s plane fi  Jargon Buster You will hear/read terms like periapsis, periastron, perihelion, perogee. These are all the same thing geometrically The slightly different names refer to the system under discussion helios == sun —> perihelion is the closest point the Earth gets to the Sun on its orbit perogee —> closest point something (e.g. some natural/arti cial satellite) gets to the Earth on its orbit periastron is a general term for a system involving star(s) Ditto for apapsis, apastron, aphelion, apogee for furthest point on the orbit fi  Binary Parameters An orbit can be de ned by 6 ‘orbital elements’ i and Ω de ne orientation of the orbital plane. There are 2 nal orbital elements needed for relative orientation of orbit within that plane ω— (argument) of periastron ν — true anomaly. This is the angle that de nes the epoch of the orbital body, i.e. where is it during the observation. A function of time cyclic in the orbital period Pb fi fi fi fi  Theories of Gravity For any binary system observed in the sky these orbital elements are each independently measurable. If we had a theory of gravity that de ned how interactions/ motion happen (e.g. through a central potential), then we would know how these elements should be related to each other, and to physical constants like G, and to any ‘charges’ related to this like masses. Different orbital relations exist for different theories of gravity, but can all be expressed in terms of the orbital elements. fi  Kepler Orbits As you know, if we use Newtonian gravity we get Kepler’s Laws K1: Orbits are ellipses dA K2: = const, where A is area swept out by line joining 2 masses dt 2 4π K3: Pb2 = a 3, where M = m1 + m2 GM  Theories of Gravity In general the orbital elements may change with time and we have additional parameters for these. Then we, very generally, just model things like: · −t )+… ω(t) = ω(t0) + ω(t · 0 Pb(t) = Pb(t0) + Pb(t − t0) + … Again we can measure these ‘post-Keplerian parameters’ independently of any model of the physics (i.e. gravity) of the system, and then use these to test any theory of gravity we like. More on this later in the course when we discuss pulsars. Ferdman et al., 2020, Nature, 583, 211.  Binary Parameters PROBLEM 3.1 (a) ** What are the most and least likely angles of inclination to observe and why? Take the ‘plane of reference’ in earlier slides to be the plane of the sky. (b) ** A binary system has a total system mass of 3 M⊙ and orbital period of 5 hours. What is the size of the orbit in metres? and in light-seconds? Is the orbit relativistic, i.e. what is v/c and is it signi cant? (c) ** A binary system has a total system mass of 1 M⊙ and orbital period of 365 days. What is the size of the orbit in metres? and in light-seconds? Is the orbit relativistic, i.e. what is v/c and is it signi cant? fi fi  Aside You probably have heard this many times but just in case I suggest you are familiar with using tools such as Google calculator and Wolfram Alpha. They can be very powerful and very convenient. Be careful with AI LLMs for calculations as they might give you nonsense  Types of Binaries Binary systems are very numerous and arise naturally everywhere. Thankfully! Binary systems are often classi ed based on how they are observed. Clari cation: a ‘double star’ is not (usually!) a binary system. This is terminology that means that 2+ stars are very close in angular separation, typically unresolved using very high spatial resolution observations. fi fi  Visual Binary This is, as you might expect, one where both components in the system can be detected and spatially resolved (at some λ, doesn’t have to be optical) # Pb is such that it can be noticed. Angular separations can be determined. If we have distance (e.g. from parallax) then we know the actual distances, e.g. we know a and hence we can determine the total system mass from K3. This is basically a scaled-up stellar version of Jupiter and the Galilean moons.  Astrometric Binary This is a binary system where only one component is detectable. But we see the system oscillate spatially. The signal is the same as seen for parallax if circular orbit, i.e. sinusoidal motion. But with system Pb rather than a 1-year cycle. We thus infer the existence of the second undetected object. Distance again allows us to determine a and K3 then gives M. Based on luminosity (and limiting luminosities) we could even separate the masses (see later slides).  Eclipsing Binary These are systems which are edge-on, or close to edge-on, i.e. i ≈ 90 deg When one component star crosses in front of the other it blocks some light. A dip is seen in the light curve (i.e. brightness vs time plot ) with a period of Pb Based on size, shape, duration of the dip can infer relative radii # BB temp # absolute values if distance known High-precision photometry enables the detection of even very small and cool binary companions. Lee et al., 2020, AJ, 160, 49.  Spectroscopic Binary These are systems where the components emit spectral lines As they both orbit the centre of mass of the system the lines will each be Doppler shifted in the opposite sense In many cases only one source is visible but with lines that shift in both senses with a period of Pb. Again we can infer angular distances, and thus absolute values of a (and L⋆ and R⋆ and …) if distance known  Types of Binaries PROBLEM 3.2 (a) What fraction of main sequence stars are in binary, or higher multiple, systems? [This is a ‘go look it up’ question!] (b) A circular orbit will produce sinusoidal spectral variations as a function of time. What would an eccentric orbit’s variations look like? (c) If 2 stars at a distance of 1 kpc were in a 1000-year binary system would we be able to tell?  Doppler Effect The Doppler Effect is one that you are no doubt very familiar with and I have already been referring to it with no explanation. For completeness I’ll remind you of the short-cut relation: Δν v = ν c So if the relative velocity of 2 moving systems is v then a frequency emitted at ν will be detected at ν ± Δν NB. I have the letters nu and vee here,  Doppler Effect There is always relative motion and so the Doppler effect is always in play. However if v/c ≪ 1 the effect can be undetectable So too if the spectral resolution of your instrument is too coarse, e.g. if you have spectral channels that are 1 MHz wide you won’t notice Doppler shifts that are 1 kHz in magnitude.  Doppler Effect Credit: NASA PROBLEM 3.3 (a) If you were to transmit a signal at 123.456 MHz into the cosmos, what is the maximum Doppler shift that an observer of the Earth would see for this spectral line? (b) The Mars Rovers transmit radio signals to Earth at a frequency of about 8 GHz. What spectral resolution do the receiving systems need to have to notice the Doppler shift for the relative motion of Mars and Earth, so as to be able to identify signals from the Rovers?  Stellar Masses Using the methods outlined above we can determine stellar masses. The method is by no means exclusively applicable to just stars however. We will see this again later when we discuss exoplanets.  Stellar Properties Overall now, combining the info from L1—L3 inclusive, we realise that we can nd out a lot about stars. Measure ux in some band, measure distance by parallax —> bolometric luminosity Measure spectral shape and/or spectral lines —> get BB temperature. Lines also give composition info. fl fi  Stellar Properties Can plot H-R diagram. The Stefan-Boltzmann relation then means we can work out stellar radii If in a binary —> can get mass Mass + radius —> can work out the star’s density. All together —> interior chemistry and structure We can also see the so-far empirical relation between mass # luminosity  Mass-Luminosity Relation As simply asserted in L2 the relation is L(M) ∝ M ∼3.5 Note that this result can in fact be derived, but we need the Virial Theorem and a couple of other things for that, so we will revisit that later. For now we can just take it as an observational fact. Many examples in the textbooks but let’s look at the original plot from 101 years ago. Mass-Luminosity Relation Eddington, 1924, MNRAS, 84, 308.  Mass-Luminosity Relation From this result we can actually do a lot. We can infer things like the lifetime of stars iff we know the energy source for the star. In fact it was this that led to the realisation that stars are nuclear fusion reactors.  Stellar Ages Remember that the following gives a timescale: anything time = (anything) d dt For energy then the age of a star is approximately: age ≈ E/L. To determine how long a process can fuel a star we try different values for the energy supply budget and see how long we get. We can compare to other data and see if it makes sense.  Stellar Ages For gravitational contraction the associated gravitational potential energy (i.e. the energy stored due to the work done by gravity to combine the star) is: 3 GM⋆2 U=− 5 R⋆ If this is the source of energy then age = tKH = | U | /L, which is known as the Kelvin-Helmholtz timescale. Checking for solar values gives tKH ∼ 20 My. When rst calculated nuclear reactions were unknown but the geological # biological data suggested Earth was much older than this so realised another source must be fuelling the sun. fi  Stellar Masses PROBLEM 3.4 (a) By looking at a H-R diagram determine the ratio of the average density of a white dwarf to that of the sun. (b) Verify the calculation for the Kelvin-Helmholtz timescale for the sun.  End of Lecture 3 L4 next Tuesday Jan 28th at 12:00