An Introduction…

A bit of background about myself: Since my sophomore year of high school I have been interested in astronomy, physics, and mathematics. I received my Bachelor’s degree in Earth and Planetary Sciences concentrating in astronomy/astrophysics from Western Connecticut State University. My coursework and independent readings ultimately led to minors in mathematics and physics.

My intention for this blog is to serve as a reference in astrophysics and related topics to myself as well as others. I aim to share my own research interests and consider selected problems that have fascinated me. I also hope to communicate recent news in the fields of physics and astronomy and discuss the implications of discoveries made.

DISCLAIMER: I am by no means an expert, and as such the posts that I create are of my opinion and my own logic. I may be wrong sometimes, and I hope that the people who see this (assuming that anyone sees this) will respect that.

That being said… Enjoy!

Moon_Astronomy_Club

(ABOVE: An image of the moon taken with a lunar and planetary imaging camera mounted to a Newtonian 130mm reflecting telescope.)

Basics of Tensor Calculus and General Relativity: An Introduction to Manifolds and Coordinates

SOURCE FOR CONTENT: General Relativity: An Introduction for Physicists, Hobson, M.P., Efsttathiou, G., and Lasenby, A.N., 2006. Cambridge University Press. 

D-Dimensional Hypersphere and Gamma Function: Introduction to Thermal Physics, Schroeder D.V. 2000. Addison-Wesley-Longmann. 

IMAGE CREDIT: NASA/JPL. 

The intended purpose of the post is to introduce the concept of manifolds in the context of physics (mathematicians beware!). Furthermore, I will discuss the concepts of Riemannian and pseudo-Riemannian manifolds before moving on towards tensors. This will be the first post in this topic of the series. In order to properly discuss the concepts of general relativity, I will have to break up this part of the series into smaller posts.



 

Part I. In this part of the series, I will discuss the concept of a tensor, and then discuss the introductory topics of manifolds.

A topic that has long eluded a conceptual understanding on my part is a tensor. In the first post of the series we saw a very technical and quite frustrating definition of a tensor. I have read numerous treatments and watched a number of lectures and videos and based on everything I have encountered, here is my understanding of a tensor as of writing this post…

Tensors are geometric objects that can be viewed in a similar way as one views matrices, whose elements are components of the tensor, and will have an overall value. More specifically, a tensor will take two geometric objects as inputs and will give you a scalar (a real number). Furthermore, under a transformation or in a different reference frame, the scalar that the tensor outputs will remain the same in all frames. The objects that change are, in fact, the components of the tensor. These components must obey specific transformation equations so as to preserve the scalar quantity of produced by the tensor. In more mathematical sense, a tensor is a mapping of a number of vectors (including 1-forms and the like) into the real number ordered field.

(**This is the best definition that I could come up with in order to define in a more satisfactory way a tensor.**)

Manifolds

According to the aforementioned reference, a manifold in its most general sense, is any set that one can describe by specifying parameters continuously. In the context of physics, we deal with differentiable manifolds.

A differentiable manifold is a continuous collection of points where each point is differentiable. This definition isn’t any better than the initial definition of a tensor. So, to elaborate a bit, we shall define the concept of continuity: a manifold is continuous if in the local region of a point n_{1}, there exists points whose difference relative to n_{1} is dn.

From this, we can say that a differentiable manifold is a manifold for which we can ascribe to it a scalar field containing points at which it is possible to take derivatives of all orders.

Some examples include: 3-Dimensional Euclidean Space and Phase Space.

3-Dimensional Space

This differentiable manifold requires 3 coordinates (parameters) to specify a single point in the space. Since it requires three parameters the dimension of this particular manifold is 3. Mathematicians sometimes call this 3-space.

Phase Space

This is a manifold that one encounters more often in physics. I came across this manifold (although I did not refer to it as such) while taking my thermodynamics and statistical mechanics course. I found that this manifold requires 6 parameters in order to specify any point. Typically, these parameters include positions (or one radius vector) and velocities or momenta.

A submanifold or surface that I found applicable to phase space would be the D-dimensional hypersphere whose surface area is given by

\displaystyle A_{d}(r)=\frac{2\pi^{d/2}}{(\frac{d}{2}-1)!}r^{d-1}=\frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}r^{d-1}, (1)

where \Gamma(\frac{d}{2}) is the gamma function given by

\displaystyle \Gamma(d+1)\equiv \int_{0}^{\infty}x^{d}\exp{(-x)}dx. (2)

To be more precise, the surface area (Eq.1) of this hypersphere is technically the volume of momentum space, but I am including to present a more concrete example of a manifold.

The next post will make the concepts mentioned here a bit more quantitative. This post was really just to introduce a more conceptual understanding.

 

Clear skies!



 

Derivation of the Euler-Lagrange Equation for a Function of Several Dependent Variables

IMAGE CREDIT: NASA/JPL

SOURCE FOR CONTENT: Classical Dynamics of Particles and Systems. Thornton and Marion. 5th Edition. 



 

Consider a functional

\displaystyle \phi = \phi(y_{\mu},y_{\mu}^{\prime}; x), (1)

where \mu = 1,2,...,n. By the method used in a previous section of the aforementioned text, we may write

\displaystyle y_{\mu}(\alpha, x) = y_{\mu}(0,x) +\alpha \eta_{\mu}(x). (2)

Additionally, we will find it useful to define

\displaystyle y_{\mu}^{\prime}(\alpha,x) = y_{\mu}^{\prime}(0,x)+\alpha \eta_{\mu}^{\prime}(x). (3)

Further we may also define an integral functional by way of integrating Eq.(1) over the interval x_{1}\leq x \leq x_{2}, and introducing a variational parameter \alpha we have

\displaystyle J(\alpha) = \int_{x_{1}}^{x_{2}} \phi(y_{\mu},y_{\mu};x)dx. (4)

Two necessary conditions that are used to derive the Euler-Lagrange equation include

\displaystyle \frac{\partial J(\alpha)}{\partial \alpha}\bigg\|_{\alpha=0}=0, (5)

and

\displaystyle \eta_{\mu}(x_{1})=\eta_{\mu}(x_{2})=0. (6)

Let us take the derivative of J(\alpha) with respect to \alpha yielding

\displaystyle \frac{\partial J}{\partial \alpha}\bigg\|_{\alpha=0}=\frac{\partial}{\partial \alpha}\int_{x_{1}}^{x_{2}}\phi(y_{\mu},y_{\mu}^{\prime};x)dx. (7)

Carrying out the derivative operator on the right-hand-side of Eq.(7) we get

\displaystyle \frac{\partial J}{\partial \alpha}=\int_{x_{1}}^{x_{2}}\sum_{\mu}\bigg\{\partial_{y_{\mu}}\phi \partial_{\alpha}y_{\mu}+\partial_{y_{\mu}^{\prime}}\phi \partial_{\alpha}y_{\mu}^{\prime}\bigg\}dx. (8)

From Eqs.(2) and (3) we see that

\displaystyle \partial_{\alpha}y_{\mu}= \eta_{\mu}(x), (9)

and

\displaystyle \partial_{\alpha}y_{\mu}^{\prime} = \eta_{\mu}^{\prime}(x). (10)

Thus Eq.(8) becomes…

\displaystyle \frac{\partial J}{\partial \alpha}=\int_{x_{1}}^{x_{2}}\sum_{\mu}\bigg\{\partial_{y_{\mu}}\phi \eta_{\mu}(x)+\partial_{y_{\mu}^{\prime}}\phi \eta_{\mu}^{\prime}(x)\bigg\}dx. (11)

Consider the second term under the summation. We may make use of integration by parts to obtain the following

\displaystyle \frac{\partial J}{\partial \alpha}=\int_{x_{1}}^{x_{2}}\sum_{\mu}\bigg\{\partial_{y_{\mu}}\phi +\frac{d}{dx}(\partial_{y_{\mu}^{\prime}}\phi) \bigg\}\eta_{\mu}(x)dx. (12)

By the necessary condition (Eq.(5)), it follows that

\displaystyle 0 = \sum_{\mu}\bigg\{\partial_{y_{\mu}}\phi +\frac{d}{dx}(\partial_{y_{\mu}^{\prime}}\phi) \bigg\}. (13)

Additionally, in Eq.(13) above, we have also made use of the condition that \eta_{\mu}(x_{1})=\eta_{\mu}(x_{2})=0. Since \eta_{\mu}(x) \neq 0 for any x_{1}\leq x \leq x_{2}, then the terms in the brackets must vanish, yielding the Euler-Lagrange Equation for several dependent variables.



Update: the next posts will be those discussed on my Facebook page. Namely, I intend to continue with my Research Series and my series in Tensor Calculus and General Relativity with various ancillary posts in my Astrophysics Series.

 

Clear Skies!

Astrophysics Series: Derivation of the Total Energy of a Binary Orbit

SOURCE FOR CONTENT: An Introduction to Modern Astrophysics, Carroll & Ostlie, Cambridge University Press. Ch.2 Celestial Mechanics

Here is my solution to one of the problems in the aforementioned text. I derive the total energy of a binary system making use of center-of-mass coordinates. In order to conceptualize it I have used the binary Alpha Centauri A and Alpha Centauri B. While writing this I stumbled upon the Kepler problem, the two-body problem, and the N-body problem. Leave a comment if you would like me to consider that in another post.

Clear Skies!



Derivation of the Total Energy of a Binary Orbit:

Setup: Consider the nearest binary star system to our solar system: Alpha Centauri A and Alpha Centauri B. These two stars orbit each other about a common center of mass; a point called a barycenter. The orbital radius vector of Alpha Centauri A is \textbf{r}_{1} and the orbital radius vector of Alpha Centauri B is \textbf{r}_{2}. The masses of Alpha Centauri A and B are m_{1}, and m_{2}, respectively. The total mass of the binary orbit M is the sum of the individual masses of each component. In the context of this system, we encounter what is called the two-body problem of which there exists a special case known as the Kepler Problem (by the way let me know if that would be something that you guys would want to see…). We can simplify this two-body problem by making use of center-of-mass coordinates wherein we define the reduced mass \mu. Therefore, the derivation of the total energy of the binary system of Alpha Centauri A and B will be carried out in such a coordinate system.

To derive this energy equation, one would typically make use of center-of-mass coordinates in which

\displaystyle \textbf{r}_{1}=-\frac{\mu}{m_{1}}r,  (0.1)

and

\displaystyle \textbf{r}_{2}=\frac{\mu}{m_{2}}r, (0.2)

where \mu represents the reduced mass given by

\displaystyle \mu\equiv \frac{m_{1}m_{2}}{m_{1}+m_{2}}=\frac{m_{1}m_{2}}{M}. (0.3)

Recall from conservation of energy that

\displaystyle E = \frac{1}{2}m_{1}\dot{r}_{1}^{2}+\frac{1}{2}m_{2}\dot{r}_{2}^{2}-G\frac{m_{1}m_{2}}{|\mathcal{R}|}, (1)

where |\mathcal{R}| represents the separation distance between the two components. Let us take the derivative of Eqs.(0.1) and (0.2) to get

\displaystyle \dot{r}_{1}=-\frac{\mu}{m_{1}}v, (2.1)

and

\displaystyle \dot{r}_{2}= \frac{\mu}{m_{2}}v. (2.2)

Substitution yields

\displaystyle E = \frac{1}{2}\frac{\mu^{2}}{m_{1}}v^{2}+\frac{1}{2}\frac{\mu^{2}}{m_{2}}v^{2}-G\frac{m_{1}m_{2}}{|\mathcal{R}|}. (3)

Upon making use of the definition of the reduced mass (Eq. (0.3)) we arrive at

\displaystyle E = \frac{1}{2}\mu v^{2}-G\frac{M \mu}{|\mathcal{R}|}. (4)

If we solve for m_{1}m_{2} in Eq.(0.3) we get the total energy of the binary Alpha Centauri A and B. This is true for any binary system assuming center-of-mass coordinates.



 

A Problem in Thermodynamics and Statistical Mechanics: Analytical and Numerical Study of an Einstein Solid

Every physics major at some point in their undergraduate career takes a course in thermodynamics and statistical mechanics. One of my problem sets included a problem that considers an Einstein solid with 50 oscillators and 100 units of energy and then increases the number of oscillators to 5000. I will be presenting my solution to the numerical side of the problem. An Einstein solid can be regarded as

“… a collection of microscopic systems which can store any number of energy ‘units’ of equal size which occur for any quantum-mechanical harmonic oscillator whose potential energy function has the form \displaystyle \frac{1}{2}k_{s}x^{2}…The model of a solid as a collection of identical oscillators with quantized energy units…”

described (defined) by Schroeder in his text Introduction to Thermal Physics. Figure 1 represents the Einstein solid as a whole (in a lattice) and Figure 2 depicts the quantum-mechanical harmonic oscillator interpretation of an Einstein solid.

The problem statement is:

“Use a computer to study the entropy, temperature, and heat capacity of an Einstein solid, as follows. Let the solid contain 50 oscillators (initially), and from 0 to 100 units of energy. Make a table, analogous to Table 3.2, in which each row represents a different value for the energy…Make a graph of entropy vs. energy, and a graph of the heat capacity vs. temperature. Then change the number of oscillators to 5000, and again make a graph of the heat capacity and temperature and entropy and energy, and discuss the predictions and compare it to the predictions to the data for lead, aluminum, and diamond. Estimate the numerical value of \displaystyle \epsilon for each of those solids.”

This problem can be found in the aforementioned text.

 

 

EINSTEIN_SOLID

Figure 1. Einstein Solid (Lattice); Image Credit/Obtained from https://mappingignorance.org/2015/12/17/einstein-and-quantum-solids/

einsol2

Figure 2. Quantum-Mechanical Harmonic Oscillator interpretation of an Einstein solid as a collection of these oscillators. Image Credit: http://hyperphysics.phy-astr.gsu.edu/hbase/Therm/einsol.html



Part I: Let q = 100 units, and let $N = 50$. The corresponding data table for this Einstein solid follows. The following set of equations were used to determine the multiplicity and entropy.

\displaystyle \Omega(N,q) = {q+N-1 \choose q} = \frac{(q+N-1)!}{q! (N-1)!},     (1)

and

\displaystyle S = Nk \ln{\Omega},     (2)

where \Omega is the multiplicity. The remaining quantities of temperature were obtained using a simplified form of the central difference equations for the first order derivative. The respective definitions of temperature and heat capacity are

\displaystyle T = \frac{\partial U}{\partial S},       (3)

and

\displaystyle C_{V} = \frac{\partial U}{\partial T},           (4)

where U represents the internal energy of the Einstein solid, and S is the entropy. The generalized from of the first order central difference approximation has the form

\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j+1}-y_{j-1}}{2h} + \mathcal{O}(h^{2}),  (5)

where \mathcal{O}(h^{2}) represents the higher order terms, in this case, the quadratic, cubic, quartic, and so on, and h is the step size for each iteration. For the final iteration (when q = 100 units), instead of using the central difference approximation, a backward difference approximation was employed since there does not exist data for q = 101 units of energy. The backward difference approximation has the form

\displaystyle \frac{dy_{j}}{dx}\approx \frac{y_{j}-y_{j-1}}{h}+\mathcal{O}(h).     (6)

Table I (Dimensionless Parameters):

Energy q Ω S/k kT/ε C/Nk
0 1 0 0 N/A
1 50 3.912023005 0.27969284 0.121826198
2 1275 7.150701458 0.328336604 0.453606383
3 22100 10.00333289 0.367875021 0.536183525
4 292825 12.58733044 0.402937926 0.593741905
5 3162510 14.96687657 0.43524436 0.637773801
6 28989675 17.18245029 0.465656087 0.673043377
7 231917400 19.26189183 0.494675894 0.702124659
8 1652411475 21.22550156 0.522626028 0.72660015
9 10648873950 23.08871999 0.549726805 0.747522024
10 62828356305 24.86367234 0.576136157 0.765628174
11 3.427E+11 26.56012163 0.601971486 0.781456694
12 1.74206E+12 28.18608885 0.627322615 0.795411957
13 8.30828E+12 29.74827387 0.652259893 0.807805226
14 3.73873E+13 31.25235127 0.676839501 0.818880855
15 1.59519E+14 32.70318415 0.701107048 0.828833859
16 6.48046E+14 34.1049827 0.725100078 0.837822083
17 2.51594E+15 35.4614241 0.748849881 0.845974847
18 9.3649E+15 36.77574496 0.772382808 0.853399232
19 3.35165E+16 38.05081369 0.795721261 0.860184741
20 1.15632E+17 39.28918792 0.818884446 0.866406816
21 3.8544E+17 40.49316072 0.84188895 0.872129523
22 1.24392E+18 41.66479814 0.864749193 0.877407641
23 3.89401E+18 42.80597005 0.887477794 0.882288296
24 1.18443E+19 43.91837566 0.910085848 0.88681226
25 3.5059E+19 45.00356493 0.932583169 0.891014994
26 1.01132E+20 46.0629565 0.954978471 0.89492748
27 2.84667E+20 47.09785298 0.977279528 0.898576916
28 7.82835E+20 48.10945389 0.999493303 0.901987268
29 2.10556E+21 49.09886689 1.021626052 0.905179739
30 5.54463E+21 50.06711736 1.043683421 0.908173155
31 1.43087E+22 51.01515679 1.065670516 0.910984284
32 3.6219E+22 51.94387004 1.087591972 0.913628113
33 8.99987E+22 52.85408172 1.109452006 0.916118071
34 2.19703E+23 53.74656181 1.131254467 0.918466228
35 5.27286E+23 54.62203054 1.153002873 0.920683463
36 1.24498E+24 55.48116286 1.174700452 0.9227796
37 2.89374E+24 56.32459225 1.196350168 0.924763536
38 6.62514E+24 57.1529142 1.217954752 0.926643346
39 1.4949E+25 57.96668937 1.239516722 0.928426373
40 3.32616E+25 58.76644629 1.261038406 0.930119309
41 7.30133E+25 59.55268389 1.282521958 0.931728264
42 1.58195E+26 60.32587378 1.303969378 0.933258829
43 3.38465E+26 61.08646224 1.325382522 0.934716125
44 7.15391E+26 61.8348721 1.346763119 0.936104855
45 1.49437E+27 62.57150439 1.368112778 0.937429341
46 3.08621E+27 63.29673989 1.389433002 0.938693566
47 6.30374E+27 64.01094048 1.410725193 0.9399012
48 1.27388E+28 64.71445045 1.431990666 0.941055635
49 2.54776E+28 65.40759763 1.45323065 0.942160007
50 5.04457E+28 66.09069447 1.4744463 0.943217222
51 9.89131E+28 66.76403902 1.495638697 0.944229971
52 1.9212E+29 67.42791582 1.516808861 0.945200757
53 3.6974E+29 68.08259672 1.537957749 0.946131903
54 7.05244E+29 68.72834166 1.559086264 0.947025573
55 1.33355E+30 69.36539938 1.580195257 0.947883783
56 2.50041E+30 69.99400804 1.60128553 0.948708411
57 4.64989E+30 70.61439586 1.622357843 0.949501213
58 8.57824E+30 71.22678169 1.643412912 0.95026383
59 1.57025E+31 71.83137547 1.664451416 0.950997796
60 2.85263E+31 72.42837879 1.685473998 0.951704548
61 5.14408E+31 73.01798529 1.706481267 0.952385432
62 9.20957E+31 73.60038111 1.7274738 0.953041713
63 1.63726E+32 74.17574525 1.748452147 0.953674575
64 2.89078E+32 74.74424999 1.769416829 0.954285135
65 5.06999E+32 75.30606117 1.79036834 0.95487444
66 8.83407E+32 75.86133855 1.811307153 0.955443478
67 1.52948E+33 76.41023613 1.832233716 0.955993177
68 2.63161E+33 76.95290236 1.853148456 0.956524415
69 4.50043E+33 77.48948048 1.874051781 0.957038017
70 7.65073E+33 78.02010873 1.894944079 0.957534764
71 1.29308E+34 78.54492059 1.915825721 0.958015394
72 2.17309E+34 79.06404502 1.936697061 0.958480604
73 3.63175E+34 79.57760662 1.957558438 0.958931052
74 6.03655E+34 80.08572588 1.978410175 0.959367363
75 9.98043E+34 80.58851934 1.999252581 0.959790129
76 1.64152E+35 81.08609973 2.020085953 0.960199908
77 2.68612E+35 81.57857622 2.040910573 0.960597234
78 4.37356E+35 82.06605448 2.061726714 0.960982609
79 7.08627E+35 82.54863689 2.082534635 0.961356514
80 1.14266E+36 83.02642266 2.103334587 0.961719401
81 1.8339E+36 83.49950796 2.124126809 0.962071705
82 2.92977E+36 83.96798603 2.14491153 0.962413835
83 4.65939E+36 84.43194735 2.165688971 0.962746183
84 7.37736E+36 84.89147968 2.186459344 0.963069122
85 1.16302E+37 85.34666822 2.207222854 0.963383006
86 1.82567E+37 85.7975957 2.227979695 0.963688173
87 2.85392E+37 86.24434247 2.248730056 0.963984945
88 4.44304E+37 86.68698658 2.269474119 0.964273631
89 6.8892E+37 87.12560389 2.290212057 0.964554522
90 1.064E+38 87.56026816 2.31094404 0.9648279
91 1.63692E+38 87.99105107 2.331670228 0.965094032
92 2.50876E+38 88.41802239 2.352390779 0.965353173
93 3.83058E+38 88.84124995 2.373105841 0.965605568
94 5.82737E+38 89.2607998 2.39381556 0.965851451
95 8.83307E+38 89.67673621 2.414520077 0.966091045
96 1.33416E+39 90.08912176 2.435219526 0.966324564
97 2.00812E+39 90.4980174 2.455914037 0.966552213
98 3.01218E+39 90.90348251 2.476603736 0.966774189
99 4.50306E+39 91.30557493 2.497288745 1.287457337
100 6.70955E+39 91.70435105 2.507672727 1.926043463

 Graphing the entropy vs. energy, and the heat capacity vs. temperature gives the following:

Graphs I & II

EinsteinSolid_Graphs_1



 

Part II: Let q = 100 units and let N = 5000. Using this in the calculation yields the following table for this Einstein solid. This “dilutes” the system and lowers the temperature:

Table II (Dimensionless  Parameters):

Energy q Ω S/k kT/ε C/Nk
0 1 0 0 N/A
1 5000 8.517193 0.122388 0.003049
2 12502500 16.34144 0.131206 0.026553
3 2.08E+10 23.76042 0.137453 0.035575
4 2.61E+13 30.89192 0.14245 0.043342
5 2.61E+16 37.80047 0.146681 0.050387
6 2.18E+19 44.52691 0.150388 0.056922
7 1.56E+22 51.09939 0.153709 0.063064
8 9.74E+24 57.53854 0.156731 0.068885
9 5.42E+27 63.86011 0.159515 0.074436
10 2.72E+30 70.07651 0.162105 0.079755
11 1.24E+33 76.19781 0.164531 0.08487
12 5.16E+35 82.23229 0.166818 0.089804
13 1.99E+38 88.18693 0.168985 0.094575
14 7.13E+40 94.06767 0.171047 0.099198
15 2.38E+43 99.87961 0.173017 0.103687
16 7.47E+45 105.6272 0.174905 0.108052
17 2.2E+48 111.3144 0.176719 0.112303
18 6.14E+50 116.9446 0.178467 0.116448
19 1.62E+53 122.5209 0.180154 0.120494
20 4.07E+55 128.0462 0.181787 0.124447
21 9.73E+57 133.5229 0.183368 0.128314
22 2.22E+60 138.9532 0.184904 0.132098
23 4.85E+62 144.3393 0.186396 0.135806
24 1.02E+65 149.683 0.187849 0.13944
25 2.04E+67 154.9861 0.189265 0.143004
26 3.94E+69 160.2502 0.190646 0.146503
27 7.34E+71 165.4768 0.191995 0.149939
28 1.32E+74 170.6671 0.193314 0.153316
29 2.28E+76 175.8226 0.194604 0.156635
30 3.83E+78 180.9444 0.195868 0.159899
31 6.21E+80 186.0336 0.197106 0.16311
32 9.77E+82 191.0912 0.19832 0.166272
33 1.49E+85 196.1183 0.199512 0.169384
34 2.21E+87 201.1157 0.200682 0.172451
35 3.17E+89 206.0843 0.201831 0.175472
36 4.44E+91 211.025 0.202961 0.17845
37 6.04E+93 215.9384 0.204073 0.181387
38 8E+95 220.8254 0.205166 0.184283
39 1.03E+98 225.6866 0.206243 0.18714
40 1.3E+100 230.5227 0.207304 0.18996
41 1.6E+102 235.3343 0.208349 0.192743
42 1.9E+104 240.122 0.209379 0.195491
43 2.3E+106 244.8863 0.210395 0.198205
44 2.6E+108 249.6279 0.211397 0.200885
45 2.9E+110 254.3472 0.212386 0.203533
46 3.2E+112 259.0447 0.213363 0.20615
47 3.4E+114 263.7209 0.214327 0.208736
48 3.6E+116 268.3762 0.215279 0.211293
49 3.7E+118 273.0112 0.21622 0.213821
50 3.7E+120 277.6261 0.21715 0.21632
51 3.7E+122 282.2214 0.218069 0.218793
52 3.6E+124 286.7975 0.218978 0.221238
53 3.4E+126 291.3548 0.219877 0.223658
54 3.2E+128 295.8935 0.220766 0.226052
55 2.9E+130 300.4141 0.221646 0.228422
56 2.7E+132 304.9169 0.222517 0.230768
57 2.4E+134 309.4022 0.22338 0.23309
58 2.1E+136 313.8703 0.224233 0.235388
59 1.8E+138 318.3214 0.225079 0.237665
60 1.5E+140 322.756 0.225917 0.239919
61 1.2E+142 327.1743 0.226746 0.242152
62 1E+144 331.5765 0.227568 0.244363
63 8.1E+145 335.9628 0.228383 0.246555
64 6.4E+147 340.3337 0.229191 0.248725
65 5E+149 344.6892 0.229991 0.250876
66 3.8E+151 349.0296 0.230785 0.253008
67 2.9E+153 353.3553 0.231572 0.255121
68 2.2E+155 357.6663 0.232353 0.257215
69 1.6E+157 361.9629 0.233127 0.259291
70 1.1E+159 366.2453 0.233896 0.261349
71 8.2E+160 370.5137 0.234658 0.263389
72 5.8E+162 374.7683 0.235414 0.265413
73 4E+164 379.0093 0.236165 0.267419
74 2.7E+166 383.237 0.23691 0.269409
75 1.9E+168 387.4514 0.23765 0.271382
76 1.2E+170 391.6527 0.238384 0.273339
77 8.2E+171 395.8412 0.239113 0.275281
78 5.3E+173 400.0169 0.239837 0.277207
79 3.4E+175 404.1802 0.240556 0.279118
80 2.2E+177 408.331 0.24127 0.281015
81 1.4E+179 412.4696 0.24198 0.282896
82 8.4E+180 416.5962 0.242684 0.284763
83 5.2E+182 420.7108 0.243384 0.286616
84 3.1E+184 424.8136 0.24408 0.288455
85 1.9E+186 428.9048 0.244771 0.29028
86 1.1E+188 432.9845 0.245458 0.292092
87 6.5E+189 437.0529 0.24614 0.29389
88 3.7E+191 441.11 0.246819 0.295676
89 2.1E+193 445.156 0.247493 0.297448
90 1.2E+195 449.191 0.248164 0.299208
91 6.7E+196 453.2152 0.24883 0.300955
92 3.7E+198 457.2286 0.249493 0.30269
93 2E+200 461.2315 0.250152 0.304413
94 1.1E+202 465.2238 0.250807 0.306124
95 5.9E+203 469.2057 0.251458 0.307823
96 3.2E+205 473.1774 0.252106 0.30951
97 1.7E+207 477.1389 0.252751 0.311187
98 8.6E+208 481.0903 0.253392 0.312851
99 4.4E+210 485.0318 0.254029 0.418439
100 2.3E+212 488.9634 0.254345 0.628243

 

Thus the graphs of the entropy vs. energy and heat capacity vs. temperature follow:

 EinsteinSolid_Graphs_2

Figure 2. Graphs III and IV.

 

phptcs1Nq

Figure 3. (Figure 1.14 of Schroeder’s Thermal Physics) Heat Capacity curves for Lead (Pb), Aluminum (Al), and Diamond, respectively as a function of temperature in Kelvin.

 

Graph II shows the prediction for heat capacity as a function of temperature of an Einstein solid for which there are 100 units of energy and 50 oscillators. The data exhibits a trend that appears to reach an asymptote quickly, then when the temperature reaches T ≈ 2.5, there is a sudden increase in the value of the heat capacity. The approach to determining the final data points was switched from a central difference approximation to a backward difference approximation of the last two entries corresponding to energies q = 99 and q = 100 units. If we ignore the last two, the curve approaches an asymptote at CV = 1. However, the graphs produced are of the dimensionless quantities involved. The overall curve appears to be logarithmic and resembles the heat capacity curve for lead. The initial increase is almost immediate and its slope appears to be slightly less than lead but greater than aluminum.

Graphs III and IV show the prediction for heat capacity in terms of temperature of an Einstein solid for which the energy is the same, but the number of oscillators is now 5000. The temperature has been reduced and the heat capacity vs. temperature yields a graph that shows a trendline that appears linear. Comparing to Figure 3(Fig. 1.14 in the text), this graph resembles the heat capacity curve for diamond. In Figure 3, the diamond curve is linear throughout. The only discrepancies among Graph IV and Figure 3 are the final two data points in Graph IV. Again, a backward difference approximation was used to determine the final data points for this Einstein solid as well.  The value for the constant ε was determined by finding the quotient of the entropy and temperature columns and taking the average value of ε for each energy.

 

This was the numerical analysis of an Einstein solid’s temperature, energy, entropy, and heat capacity. In the next post, I shall discuss the analytical version of this analysis.

A “Proof” of the Sturm-Liouville Theorem/Problem

IMAGE CREDIT: NASA/JPL: This shows Jupiter’s Great Red Spot; a storm that has been occurring for over 300 years now. Quite recently, however, observations show that the Spot appears to be shrinking in size. 

About a week ago, I was looking through my notebooks and came across an unfinished problem posed by one of my professors. Unfortunately, I was not able to solve the problem during the semester. However, I thought it might be something interesting to consider. I did a quick search and found that the problem he gave us was to prove the Sturm-Liouville Theorem.

The main brute-force method to analytically solving a given partial differential equation is the separation of variables. This method is heavily used by physicists and in doing so transforms the initial boundary value problem (IBVP) into a Sturm-Liouville problem in which we have an ordinary differential equation and linear homogeneous boundary conditions.

Continue reading A “Proof” of the Sturm-Liouville Theorem/Problem

A Narrow, Technical Problem in Partial Differential Equations

While I was in school, one of my professors set this problem to me and my classmates and challenged us to solve it over the next few days. I found the challenge intriguing and it fascinated me, so I thought it was worth sharing. The problem was this:


Show that

\displaystyle v(x,t) = \int_{-\infty}^{\infty} f(x-y,t)g(y)dy,    (1.1)

where \displaystyle g(y) has finite support and also satisfies the PDE

\displaystyle \frac{\partial v}{\partial t} = -\kappa \frac{\partial^{2}v}{\partial x^{2}}.   (1.2)



First off, what does finite support mean? Mathematically speaking, a function has support which is characterized by a subset of its domain whose members do not map to zero, and yet are finite. (Just as a quick note: much of the proper definitions require an understanding in mathematical analysis and measure theory, something which I have not studied in detail, so take that explanation with a grain of salt.)

As for the solution, we can rewrite the given PDE as

\displaystyle \frac{\partial v}{\partial t} - \kappa \frac{\partial^{2}v}{\partial x^{2}} = 0.    (2)

The PDE requires a first-order time derivative and a second-order spatial derivative.

\displaystyle \therefore \frac{\partial v}{\partial t} = \frac{\partial}{\partial t}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy,   (3.1)

and

\displaystyle \frac{\partial^{2} v}{\partial x^{2}} = \frac{\partial^{2}}{\partial x^{2}}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy.    (3.2)

Next, we substitute Eqs. (3.1) and (3.2) into Eq.(2), yielding

\displaystyle \frac{\partial}{\partial t}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy -\kappa \frac{\partial^{2}}{\partial x^{2}}\int_{-\infty}^{\infty} f(x-y,t)g(y)dy = 0.    (4)

Note that taking the derivative of a function and then integrating that function is equivalent to integrating the function and differentiating the same function, in conjunction with the fact that the sum or difference of the integrals is the integral of the sum or difference (proofs of these facts are typically covered in a course in real analysis). Taking advantage of these gives

\displaystyle \int_{-\infty}^{\infty} \bigg\{\frac{\partial}{\partial t}f(x-y,t)-\kappa\frac{\partial^{2}}{\partial x^{2}}f(x-y,t)\bigg\}g(y)dy = 0.   (5)

Notice that the terms contained in the brackets equate to \displaystyle 0. This means that

\displaystyle \int_{-\infty}^{\infty} 0 \cdot g(y)dy = 0.   (6)

This implies that the function \displaystyle v(x,t) does satisfy the given PDE (Eq.(2)).




References:

Definition of Support in Mathematics: https://en.wikipedia.org/wiki/Support_(mathematics)

Observing the Variable Star W Ursae Majoris

While I was an undergraduate, one of my smaller research projects involved observing the variable star W Ursae Majoris.

In general, there are six types of binary star systems: Optical double, Visual binary, Astrometric binary, Eclipsing binary, Spectrum binary, and Spectroscopic binary.

In this project, my classmate and I were interested in the eclipsing binary (EW) W Ursae Majoris. An eclipsing binary is a binary system in which one of the stars will pass in front of its companion, effectively causing an eclipse. We are able to observe this by way of generating the light curves of the system. An example light curve is shown below:

Image result for eclipsing binary light curve

(Image was obtained at the URL:  https://imagine.gsfc.nasa.gov/educators/hera_college/binary-model.html)

The graph shows a plot of intensity over time (which in this case is an orbital period). Observations of an EW should show dips in the intensity of the two stars. What is really fascinating to me is that we can gain valuable information from this graph. For example, the length of a dip can indicate the masses of the star. If we have a star of mass m_{1} and the other is m_{2} such that m_{2}>m_{1}, and if the duration of the decrease in intensity of the system is significant we can then infer that the mass passing in front of its companion is that of m_{1}. By default, the mass that is being “eclipsed” is m_{2}. Conversely, if the intensity decreases but only for a short while, the positions are reversed, with m_{2} passing in front (relatively speaking) and m_{1} is being “eclipsed”. (I am assuming that the barycenter (i.e. the system’s center of mass) is equidistant from the centers of the two stars.)

Another form of classification of binary stars is whether or not the binary system components are touching or not. More precisely, there are three kinds of close binaries: detached, semi-detached, and contact binary. There are sub-categories of contact binaries: near contact, contact, overcontact,  and double contact.

An equipotential surface map of a system (assuming that the binary system has a mass ratio of 2:1, which may be incorrect as most W UMa binaries have a mass ratio of 10:1) is shown below:

Related image

Image Credit: Fig.1 of Terrell, D., Eclipsing Binary Stars: Past, Present, and Future. JAAVSO Vol. 30, 2001.

To quickly elaborate, each type of contact binary will fill its inner Lagrangian surface (aka Roche lobes) to an extent. In the context of our project, W Ursae Majoris is an overcontact eclipsing binary system.  This type of binary will overfill its inner Lagrangian surface. As a result of this, processes such as mass transfer and accretion can occur. The diagram below shows the orbital evolution of a W UMa EW AC Bootis (in addition to being its own binary system, W UMa is also a class of close binaries)

Image result for overcontact binary roche lobe diagram

Image Credit: Fig. 15 of Alton, K., A Unified Roche-Model Light Curve Solution for the W UMa Binary AC Bootis. JAAVSO. Vol. 38, 2010.

The objective of the project was to image the eclipsing binary, measure the apparent magnitude, to process the images, and to obtain a light curve. To observe this system, a classmate and I made use of the 20″ Ritchey-Chrétien telescope at the university observatory. We made use of the CCD camera attached and set a sequence of images to be taken every two minutes. W UMa has a period of approximately 8 hours, however, due to time constraints (and as much as I would have liked to, the weather was not conducive for observations exceeding two hours), we ended up only taking images for around two hours.

After the session was over, we ended up taking a total of roughly 40-50 images. Additionally, the software used to capture the images simultaneously measured the magnitude of W UMa at the time each image was taken. This allowed us to use Excel (and later on MATLAB) to obtain a partial light curve. However, since this is a partial light curve, we can say that an eclipse (and a short one at that) occurs, yet we cannot determine whether or not the local minimum depicted in the graph below is a primary or a secondary minimum–we simply do not have enough data.

EW UMa light curve

In addition to the partial light curve above, we were able to process the images (using Registax v.6). Below is a stacked image of W UMa. The big blob near the center of the image is the binary. The binary is not able to be resolved by telescopes component-wise.

 

UMA_newprocessed_1

 



References: 

Caroll, B.W., and Ostlie, D. A., Introduction to Modern Astrophysics. 2017. Cambridge University Press. 7.

Catalog and Atlas of Eclipsing Binaries (CALEB): Types of Binary Stars

http://www.daviddarling.info/encyclopedia/C/close_binary.html

American Association of Variable Star Observers (AAVSO) URL: https://www.aavso.org/vsots_wuma

Journal of American Association of Variable Star Observers: Figure References

 

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