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Questions? Reach out to me at jw235@rice.edu |
1. IntroductionWhat is Tidal Locking? Tidal locking in orbital systems is a phenomenon which occurs when the rotational period of an orbital body is equal to its orbital period. This causes the same side of the orbital body to face the body which it is orbiting at all times, as illustrated in Figure 1 (the smiley face represents one side of the Moon). In fact, this is why we have a "dark side" of the Moon; since the Moon is tidally locked to the Earth, its dark side will never face us. Though we use the Moon-Earth system as an example, any type of celestial orbital system can experience tidal effects, including stellar binaries quite commonly. |
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The process of tidal locking is driven by gravitational tidal forces. Tidal forces result because celestial bodies are not perfect point masses, and because of this, the part of them which is closer to the body which they orbit will experience a greater gravitational attraction according to Newton's Law of Gravitation. This stretches out the orbiting body to create a lobe. Then, when this lobe tries to rotate away from the radial direction, the enhanced gravitational force on this lobe induces a torque that slows down or speeds up the rotation of the orbiting body over time until the rotation of the orbiting body matches the period of orbit and the "lobe" of the orbiting body always faces the radial direction toward its center of orbit. This process is illustrated in Figure 2. However, tidal forces take millions to billions of years to reach tidal locking depending on the strength of gravitational interaction. 
Figure 1: A visual of a tidally-locked versus tidally-unlocked Moon orbiting the Earth, with a smiley face marking one side of the Moon. Image courtesy of LabXchange. (2021). Stellar Spectra and the Hertzsprung-Russell Diagram. Harvard University. 
Figure 2: A diagram of the effect of tidal forces in an orbital system. Image courtesy of Wikipedia Contributors. (2024). Tidal locking. Wikimedia Foundation. 2. HypothesisBy Newton's Law of Gravitation which we mentioned earlier, F = G x M1 x M2/(R^2), we know that gravitational force decreases as distance between two objects increases. We would expect tidal gravitational forces to be related to the derivative of Newton's Law, which means that tidal forces would also decrease in magnitude as the distance between two objects increases. This would make tidal locking occur more quickly at smaller distances between orbital objects, which would make tidal locking more likely to witness. HYPOTHESIS: The strength of tidal locking will decrease as the orbital distance (semi-major axis) of a stellar binary increases, and we may be able to determine a functional trend in the deviation from tidal locking behavior. Thus, the work here attempts to prove those two claims. NOTE: Because the orbits of stellar and planetary objects are elliptical and not circular, we must use the semi-major axis of their elliptical orbits as a parameter of distance between the orbital components instead of an orbital radius. a1 and a2 in Figure 3 below are examples of the semi-major axes of elliptical orbits. 
Figure 3: A diagram of the elements of two elliptical orbits. Image courtesy of Southworth, J., Smalley, B., Clausen, J. V., Helt, B. E., & Etzel, P. B. (2005). Absolute dimensions of detached eclipsing binaries – II. The metallic-lined system WW Aurigae. Monthly Notices of the Royal Astronomical Society. 3. TargetsControls Gravitational tidal forces do not only depend on the distance between orbiting bodies, but also on the mass of the objects as well as their individual radii. In order to isolate the effect of semi-major axis, the effects of mass and radius in stellar binaries must be controlled for by limiting two factors: spectral type and evolutionary stage. Since stars on the main sequence have masses that vary with spectral type by the Mass-Luminosity Relationship, we made sure to choose stellar binaries for our sample with both components of approximately spectral type A (with one exception, which will be mentioned shortly). Since radius also varies similarly as mass with spectral type, this limitation controlled for radius as well. We also limited our sample to main sequence stars or stars just coming off of the main sequence so that the Mass-Luminosity Relationship would hold. We also needed to control for the limitations of our measurements. In order to determine whether our binaries were tidally-locked, we needed to measure the rotational velocity of each star by measuring its Doppler broadening. Doppler broadening, specifically rotational Doppler broadening, increases the width in wavelength of spectral lines from the relative Doppler shifts across the star's surface due to variations in line-of-sight velocity. However, since we can only measure along the line of sight, we are restricted to measuring the rotational velocity of our stars multiplied by the sine of the inclination of the star's rotational plane, vsin(i). In order to account for this, we chose to view eclipsing spectroscopic binaries exclusively. We know that the inclination of eclipsing spectroscopic binaries is appoximately ninety degrees because we must view the orbital plane edge-on to witness an eclipsing binary, so if we assume that the rotational plane of each star in the binary is in alignment with the orbital plane, we can approximately say that we are measuring the true rotational velocity. This assumption is generally fair in the closely-orbiting binaries which we include in our sample because of the tidal forces at play, but it is not entirely accurate, and it especially might affect our measurements as semi-major axis increases in our sample and tidal forces get weaker. We discuss this consideration later in our analysis. 
Figure 4: A diagram of the inclination of a star and how it relates to rotational velocity. Image courtesy of Wikipedia Contributors. (2024). Stellar rotation. Wikimedia Foundation. Targets: With our control limitations and the magnitude limitations of our telescope and the lovely Houston sky, we were able to locate six eclipsing spectroscopic binaries, all roughly A-type stars, with varying orbital periods. (We relate these periods to semi-major axis in our analysis and find that semi-major axis increases similarly with orbital period.) Alpha Coronae Borealis is the only binary which has a component that deviates drastically from our A-type standards: a G star. However, because of the drastic flux difference between the primary and the secondary, we could only measure the rotational velocity of the primary, which we expected to use as a reference for clear unlocked tidal behavior. We were also unable to measure the secondary of 21 Hydrae because of the flux difference between the primary and the secondary, and though there was not such a drastic flux difference in the V1143 Cygni binary, we caught it during its relatively long secondary eclipse so that the signal was only present from the primary component. Thus, we were only able to measure three complete binaries. The orbital period information, spectral lines we examined, spectral types, and tidal locking expectations for each system are displayed in Figure 5. 
Figure 5: Details for each of our six binary targets. 4. MethodologyData Methods: We performed the standard data reduction process: subtracting biases and darks, dividing by the normalized flat, wavelength calibration, and extracting a 1D spectrum for each of our exposures. We took three exposures for each of our binary systems and examined 2-4 lines per exposure for each star in order to lower the statistical uncertainty in our measurements. Then, to relate the Doppler broadening of our spectral lines to the rotational velocities of the stars, we fit Gaussian profiles using the rotational velocity model on the left of Figure 6, where the total broadening of each spectral line is equivalent to a function of that line's wavelength, the rotational velocity, and the speed of light, with the inherent instrumental broadening added in quadrature. The factor of 1.66 present accounts for the difference between a Gaussian fit and a rotational kernel fit; rotational kernel is more accurate, but we can relate the broadening of a Gaussian profile to the broadening of a rotational kernel with this factor. The 2.355 factor accounts for the relationship between standard deviation and full-width-half-maximum of a Gaussian profile. Example fitted Gaussian profiles appear in Figure 8 for Beta Aurigae A and V1143 Cygni A. Then, in order to account for our error, we took four sources of error into account: statistical error (how much our measurements varied between trials), variation in the instrumental broadening measured from our wavelength calibrations, error from the signal-to-noise ratio, and error from our pixel resolution, wich all add in quadrature. Extracting Semi-Major Axis & Ideal Rotational Velocity: In order to find the deviation from expectation of our measured rotational velocities, we would need to calculate the necessary rotatonal velocity in order to achieve tidal locking for each of our stars. Since rotational velocity is equal to the circumference of the star divided by its rotational period and we know tidal locking equates rotational to orbital period, we could simply calcualte the ideal rotational velocity to be the circumference of the star divided by orbital period of the binary as displayed on the left of Figure 7. We estimated the radius of each star using spectral type. Then, in order to relate semi-major axis to tidal locking deviation, we would first need to calculate the semi-major axis of each binary. We did so by estimating the total mass of each of our binaries based on the spectral types of both components, and since we knew the orbital period of each binary, we could then relate mass and period to the semi-major axis of each binary's orbit using Kepler's Third Law of orbital motion as displayed on the right of Figure 7. 
Figure 6: An equation for the broadening of a Gaussian based on rotational velocity on the left, and an equation for total uncertainty in our measurements on the right. 
Figure 7: An equation for the tidally-locked rotational velocity on the left, and an equation for the semi-major axis on the right. 
Figure 8: Example Gaussian fits of Beta Aurigae A (left) and V1143 Cygni A (right). 5. AnalysisAfter performing the calculations described above, we found results for estimated mass, estimated radius, semi-major axis, measured rotational velocity, ideal rotational velocity, and locking status for each of our stellar binaries. The results are listed in Figures 9 and 10. Our results follow the general trend we wished to see: Beta Aurigae is clearly tidally locked as we expected, and though one of the components of WW Aurigae has error slightly below that which would include the ideal rotational velocity we wished to observe, this could be due to slight inaccuracies from our inclination approximation of ninety degrees, since inclination of the orbital plane need not be precisely ninety degrees to eclipse. The rotational plane of this star could also be slightly different from the orbital plane, although it should not vary much because of the strong tidal forces at such a close semi-major axis in the binary. Thus, we can conclude that both of our Aurigae binaries are likely tidally locked. Alpha Coronae Borealis is clearly unlocked, since the rotational velocity of the primary is much, much faster than that of its ideal tidally-locked rotational velocity, and inclination variation would only mean that the true rotational velocity is even faster. Our results in-between our extremes are a bit murkier, though it appears as though V1143 Cygni is definitely tidally unlocked; its rotational velocity is much faster than necessary for locking, and inclination would only worsen this as for Alpha Coronae Borealis. 21 Hydrae also appears to be unlocked, though since it lies below our rotational velocity resolution limit, there could be more error in our measurement than we may account for. However, though this case is also true for RR Lyncis, RR Lyncis provides a bit of worry: it appears to be incredibly close to tidal locking, and one component appears to be almost certainly tidally locked from the results of our data. 
Figure 9: Results for estimated mass, radius, and semi-major axis of each stellar binary. 
Figure 10: Results for measured rotational velocity, ideal rotational velocity, and locking status of each stellar binary. In Figure 11, one can see more clearly how our results vary from expectations. Alpha Coronae Borealis lies clearly above expectation over in the top left corner, and on the right side, WW Aurigae and Beta Aurigae lie incredibly close to expectation. However, though 21 Hydrae and V1143 Cygni stray further from the expectation line, one can see that the darkest blue points (RR Lyncis A and B) lie incredibly close to the expectation line as well, unexpected for the second-highest semi-major axis of our sample. We then plot the percent residuals of each data point against semi-major axis in order to attempt to characterize the variation. In Figure 12, the data appears to simply be a wiggle of points close to zero with one extreme outlier at the far right (Alpha Coronae Borealis), but in Figure 13 where we fit a quadratic, you will find that with an R^2=0.980, the fit matches our data quite well. However, you can also see from both Figure 12 and Figure 13 that there is a wide range of semi-major axis values which our data does not cover, so we cannot determine whether the function truly matches stellar binary behavior in that region with solely our data. You may also notice that we denoted age of our binaries with a colorbar, where darkest blue is the oldest and lightest blue is the youngest. Why did we include this? More to come... 
Figure 11: Graph of measured vs. tidally-locked rotational velocity. The brown dashed line indicates a line of perfect match between the two values. 
Figure 12: Graph of percent residual between ideal and measured rotational velocity versus semi-major axis. 
Figure 13: A second-degree polynomial fit to the data displayed in Figure 10, expluding data from the RR Lyncis binary. Reasons for exclusion discussed. 6. ConclusionsWhat DIDN'T We Account For? We already mentioned that our assumption about vsin(i) contains a degree of inaccuracy. But even considering that and the potential risks of measuring below our instrumental broadening limit; if those were't driving our unexpected RR Lyncis results, what would? Well, remember what was mentioned before about tidal locking occurring more quickly with stronger tidal forces. There lies a key point here: tidal locking would occur MORE QUICKLY. It takes TIME to achieve tidal locking. So, even if tidal forces are weaker, if a binary has more time to achieve tidal locking, then it will ALSO be more likely to exhibit tidal locking. And, as seen from our plot, RR Lyncis is by far the oldest binary present here. Is this responsible for our unexpected result? Given other sources of error, it's hard to say for certain. But one thing is for sure: if we want to truly isolate the effect of semi-major axis, then we must take the age of our binaries into account as well. For future studies, we have two tasks: first, we must acquire a much larger sample size in order to fill in the game in our semi-major axis range so that we can properly fit a polynomial relation to characterize the variation in tidal locking. This will require conquering the magnitude limitations encountered with our current equipment. And secondly: we must find some way to account for the age of our binaries, either by controlling for age or by accounting for age in conjunction with the semi-major axes of our binaries. 7. References
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