Equivalent Width Abundance Analysis In Moog
|
|
- Dora Wells
- 5 years ago
- Views:
Transcription
1 Equivalent Width Abundance Analysis In Moog Eric J. Bubar Department of Physics and Astronomy, Clemson University, Clemson, SC ABSTRACT In this cookbook I give in depth steps on how use the MOOG computer software to perform a standard spectroscopic abundance analysis using the abfind routine. Subject headings: Equivalent Widths, abfind 1. Model Atmospheres The following description requires use of a model atmosphere to establish the relevant thermodynamic properties (temperature structure, electron density, etc.) for calculating abundances in. For late type (F, G and K) stars, an appropriate choice for model atmospheres, commonly utilized in the literature, are the 1D plane-parallel model atmospheres of Robert Kurucz Broadly, these model atmospheres break the photosphere of a star (the outer atmosphere where the majority of visible wavelength spectral lines are formed) into many, subsequent layers. Within a given layer the equations of hydrostatic equilibrium hold (pressure balances with gravity) and energy transport is through radiative processes. Perhaps the most essential assumption in these model atmospheres is that of local thermodynamic equilibrium (LTE). In LTE, the transfer equation is solved assuming a blackbody source function and the properties of a small volume of gas are determined by the thermodynamic equilibrium values determined from the local temperatures and pressures. Typically these model atmospheres are created based on the four fundamental physical parameters that are used to describe a star; temperature, surface gravity, metallicity, and microturbulent velocity. 2. Equivalent Widths The abfind package of MOOG uses equivalent widths to force fit abundances using a curve of growth method. Consequently, you need to measure equivalent widths for lines of the elements that you want abundances for. The best choice is going to be lower EXCITATION POTENTIAL lines (χ nu 5-ish ev). You want to find nice clean lines that don t appear to be blended and have shapes that you can fit nicely using SPECTRE. Once you have a measured a bunch of lines for your given element, you need to create a linelist in the MOOG abfind package format.
2 2 3. Linelist You will need to put all of your line equivalent widths into a MOOG format linelist. MOOG is kind of picky about spacing so your list must look the same as the sample list found in the MOOG manual. Make sure that the first line in your file is a commented piece of text. I like to at least put the number of total lines that are in the file on this top line. Starting on the second line of the linelist file, the various columns each give different parameters. COLUMN 1: Wavelength of the line that you measured. COLUMN 2: The ATOMIC NUMBER and IONIZATION STATE of the line that you measured in decimal form. For example, a line of neutral iron would be denoted as The 26 gives the atomic number (in this case iron). The decimal tells MOOG that the ionization state of the line is coming next. The 000 gives the ionization state, in this case neutral. If I wanted singly ionized iron, I would enter in , where the.100 tells MOOG this is a singly ionized state. COLUMN 3: This is the lower EXCITATION POTENTIAL of the line. This tells you the energy you need to give the atom to ionize to the desired state. COLUMN 4: This is the log(gf) or gf value. This basically just tells you what the probability is that a specified transition will occur for a given line. This is the important part for MOOG and is most likely the largest source of uncertainty. Getting lines with good, reliable log(gf) s or gf s is half the battle for the stellar spectroscopist. COLUMN 5: This is the dissociation energy. For my purposes the only number i ve used thus far is 2.2. I believe it becomes much more important for molecular lines, but I like atoms! COLUMN 6: The final column is the equivalent width of the line that you measured in SPECTRE. This will be used for the curve of growth. 4. Check For Trends: Plot of EXCITATION POTENTIAL versus REDUCED EQUIVALENT WIDTH This next step, i m told, is not really performed by a lot of spectroscopists out there, but I trust my mentor, and do what he tells me. I m going for QUALITY here, not QUANTITY, so lets be rigorous so that people will learn to trust our word. It is important to check and see if the lines you measured have any trends that connect the EXCITATION POTENTIAL to the REDUCED EQUIVALENT WIDTH. If they do, there could be problems later on when you ll be guessing, checking and iterating to converge on a solution. The first step is to create a plot of the EXCITATION POTENTIAL versus the log of the REDUCED EQUIVALENT WIDTH. You say, but Eric, I only have an equivalent width!. No problem, the reduced equivalent with is simply the Equivalent Width you measured, divided by the wavelength of the line that you measured that width at. To give a little bit of physical explanation for why we re doing this lets think about the quan-
3 3 tities we re plotting. Okay, the EXCITATION POTENTIAL tells how much energy you need to ionize (or EXCITE) an electron to a given state from the ground state. In general, the HIGH EX- CITATION POTENTIAL LINES form deeper in the atmosphere of the star, where temperatures are high enough to excite atoms to the higher EXCITATION POTENTIAL states. Now, if the line forms deeper, its likely that it might be a little bit weaker just from having to travel through more stuff in the stellar atmosphere before it reaches us. Conversely, a LOWER EXCITATION POTENTIAL LINE doesn t need as much energy to form, so it can form in lower temperature regions, i.e. closer to the stellar surface. Consequently it travels through less stuff so it may be a stronger line. For these reasons you may naturally expect: 1) Lower EXCITATION POTENTIAL lines may form closer to the stellar surface and thus be stronger. - and - 2) Higher EXCITATION POTENTIAL lines may form deeper in the bowels of the stellar atmosphere and thus may be intrinsically weaker. Now, hopefully you can see that there may be a slightly intrinsic trend for lower EXCITATION POTENTIAL lines to give larger equivalent widths, and vice versa, so you need to see if your line choices embellish this effect. If this is the case, you may have problems down the road when fiddling with your spectroscopic parameters. The goal is to get a list of lines that show no such trend so that down the line you will be able to figure out your temperatures and your mictroturbulent velocities separately. You will see down the line what I m talking about, but if you must know now: We will be creating plots of [Fe/H] versus EXCITATION POTENTIAL and [Fe/H] versus log of the REDUCED EQUIVALENT WIDTH. MOOG basically gives us our abundances, (from which we find [Fe/H]) and the EXCITATION POTENTIALS and REDUCED EQUIVALENT WIDTHS are known values from our linelist. The plot of [Fe/H] versus EXCITATION POTENTIAL will be used to determine TEMPERATURE and the plot of [Fe/H] versus log of REDUCED EQUIVALENT WIDTH will be used to find MICROTURBULENT VELOCITY. If there is a trend between these two different abscissa (i.e. x-coordinates, EXCITATION POTENTIAL and REDUCED EQUIVA- LENT WIDTH) then changes in temperature will effect the plot for finding MICROTURBULENCE and vice versa and we have a big mess/headache when trying to spectroscopically determine our best fit solutions. INSERT A PLOT OF EXCITATION POTENTIAL VERSUS REDUCED EQUIV- ALENT WIDTH
4 4 5. Solar Differential Analysis If you are calculating exact abundances, then you don t need this section. However, its most likely that you want to do a differential analysis with respect to the Sun. What you want to do then, is measure a bunch of lines for a given element in your star s spectrum. Then, get a good solar spectrum that has similar resolution and S/N as your spectrum and measure the same lines as you measured in your star. Then, first run MOOG on the linelist you create of your measured SOLAR lines. Use a model atmosphere with the known parameters of the Sun (T= 5777,[g]=4.44,[Fe/H]=0.00,vt=1.10) in your MOOG run. The results of this run will be kept constant, because we KNOW that these are the basic physical parameters of the Sun and we don t want to fiddle with these. We will now do some MOOG running and a little bit of country folk music (i.e. fiddling). 6. Running MOOG Okay, we ve either removed lines that give a trend in EXCITATION POTENTIAL versus REDUCED EQUIVALENT WIDTH or we have found that there are no such trends. This means we re ready to run MOOG! Take your super-terrific pure linelist and we ll make a parameter file for running the ABFIND routine to get abundances.
5 5 Sample Parameter File for Running ABFIND abfind standard out /local/ebubar/moogout/test std.out summary out /local/ebubar/moogout/test sum.out model in /local/ebubar/models/pttmodels/t4748 g4.45 m0.20 vt1.01.atm lines in /local/ebubar/moogin/linelist/bd fe1.lst terminal x11 atmosphere 1 molecules 2 lines 3 flux/int 0 damping 0 units 0 obspectrum 0 plot 4 strong 0
6 6 You can read about what these things mean in the MOOG manual, but basically i m taking a model atmosphere of my star, i m inputting my good linelist, and will output a plot of the results. Once you run moog, you will utilize the second output file for determining what parameters you need to fiddle with. The file that is called summary out is what you will utilize to create your plots of [Fe/H] versus EXCITATION POTENTIAL and [Fe/H] versus log of the REDUCED EQUIVALENT WIDTH. 7. Fiddler on the Roof Now is the fun part. You get to fiddle with your model atmosphere parameters until you start to remove the correlations The idea is to remove the correlations in both the plots of [Fe/H] versus EXCITATION PO- TENTIAL and [Fe/H] versus log of REDUCED EQUIVALENT WIDTH. When I do this, I run MOOG with my model atmosphere parameters for my star. Then, I take that output file and use SUPERMONGO to read in both the abundances for the star and for the Sun. Subtract the Sun abundance from the star abundance and you have the metallicity! In this same SUPERMONGO file I create my relevant plots [Fe/H] versus EXCITATION POTENTIAL If this plot has some slope to it, you need to change your TEMPERATURE. As you fiddle with the temperature in your model atmosphere your least squares fit will either get better or worse. You will keep adjusting the temperature until you flatten the trend. You should aim for an RMS of 0.05-ish and a Correlation coefficient of better than ish [Fe/H] versus log of REDUCED EQUIVALENT WIDTH If this plot has some slope to it, you need to change your MICROTURBULENT VELOCITY. As you fiddle with the microturbulent velocity in your model atmosphere your least squares fit will either get better or worse. Keep adjusting the MICROTURBULENCE until you flatten the trend. You should again aim for an RMS of 0.05-ish and a Correlation coefficient of better than ish.
7 Fe II for MICROTURBULENCE Lets now say you were able to measure lots of Fe II lines (bully for you!). You can use these lines to determine the microturbulent velocity parameter. Again, make plots of EXCITATION POTENTIAL versus log(reduced EQUIVALENT WIDTH). Check for trends and strange lines. Of course, you may have been just fine with getting a great fit with just the Fe I lines. That is quite excellent as well. Therefore, this step may not even be a necessary one. In that case, you ll be jumping ahead to find logg. But if you want to use Fe II to find microturbulence and there are strange trends in existence, you may have to deal with strange lines in the method described below Dealing with Strange Lines Do you see some line that looks odd in one of your trend plots? This is entirely possible (and likely). The first step is to return to your measurements and make sure that they are correct. Perhaps you transposed wrong or were having a bad measuring day. If the line measurement still looks fine, then its time to get a little more involved. Look to see if there are any other lines of significant strength around your alleged Fe II line. Its possible you have measured a blended line feature. You can use MOOG to create synthetic spectra of the lines in your region of interest. A great way to do this is to get all the lines from VALD for your region of interest. Then put these in a MOOG format linelist and plug it in to create a synthetic spectrum. Once you have a synthetic spectrum, check the FIRST output file that you get from running that MOOG synthesis. Look a good ways down and you should see a listing of all the wavelengths from your linelist along with various columns. One of these columns is labeled strength. This column gives a magical number that gives the relative strength of each of the lines that are in your linelist. If you see a line near your Fe II line that is either stronger or comparable in strength when using this magical line strength column, then your line is likely blended and perhaps isn t actually an Fe II line at all! This spectral synthesis method is a good way to check lines that may otherwise have escaped your notice as being blended. Of course its also always possible that in your fervor to find as many good lines as possible you included blended lines by mistake, with the good intentions of improving your results. It happens. You just want to check lines that appear to be giving wonky/inconsistent results because there really is physics underlying all of this black magic and it should be consistent. Of course, you may have inconsistencies, and thats perfectly fine. In fact, thats when you ll most likely discover something new and exciting! 7.4. Fe II for LOG(G) Okay, so you got a beautiful fit by fiddling with Temperature and Microturbulence (correlation coefficients are less than 0.005). Now, we re going to take these pristine results and fiddle with
8 8 log(g) and wipe out all that hard work. You need to get the metallicity of your star based on your Fe II line equivalent widths ([Fe II/H]). You should just take an average of all your Fe II abundances (you won t have enough Fe II lines measured to try and remove correlations with EXCITATION POTENTIAL or REDUCED EQUIVALENT WIDTH) to get some FINAL abundance [Fe II/H]. Now, for solar-type stars (i.e. F, G, and K), singly ionized Fe lines are sensitive to changes in the pressure (and thus gravity), while neutral lines are essentially independent of surface gravity effects (not really true in a strict sense, but approximately its good enough). So, we ll assume you have determined a temperature from removing the correlation in [Fe I/H] versus EXCITATION POTENTIAL (aka EXCITATION BALANCE) and a microturbulence by removing the correlation in [Fe I/H] versus REDUCED EQUIVALENT WIDTH (equivalent width balance). In doing this, you should have converged to a single abundance for the star (i.e. an average [Fe I/H]. Now, you need to adjust the surface gravity (logg) until the abundance you derive from Fe II lines ([Fe II/H]) matches with that from Fe I lines ([Fe I/H]). The pain in this comes from the parameters all being interdependent. As you change the gravity, your temperature, microturbulence and metallicity solutions will likely fall apart (sometimes by a lot, sometimes by a little). The trick now is to iterate on the parameters until you get a solution over all parameters. In summary, you need to do this: 7.5. Summary 1) Tweak the temperature in order to remove the correlation in [Fe I/H] versus EXCITATION POTENTIAL (EXCITATION BALANCE) 2) Tweak the microturbulent velocity in order to remove the correlation in [Fe I/H] versus RE- DUCED EQUIVALENT WIDTH (line strength balance or equivalent width balance) 3) Tweak surface gravity in order to force [Fe II/H] to be equal to [Fe I/H] which you have from steps 1 and 2. Once you satisfy these three steps, you have performed your standard spectroscopic abundance analysis. Congratulations! Now start the whole messy process over again with another star and watch as you rip your hair trying to converge to unique solutions.
ARES+MOOG From EWs to stellar parameters
ARES+MOOG From EWs to stellar parameters Kepler 10 Artistic View Sérgio Sousa (CAUP) ExoEarths Team (http://www.astro.up.pt/exoearths/) Wroclaw Poland 2013 Homogeneous Analysis Spectroscopic Stellar Parameters
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More informationLesson 1: Forces. Fascinating Education Script Fascinating Intro to Chemistry Lessons. Slide 1: Introduction. Slide 2: Forces
Fascinating Education Script Fascinating Intro to Chemistry Lessons Lesson 1: Forces Slide 1: Introduction Slide 2: Forces Hi. My name is Sheldon Margulies, and we re about to learn what things are made
More informationφ(ν)dν = 1. (1) We can define an average intensity over this profile, J =
Ask about final Saturday, December 14 (avoids day of ASTR 100 final, Andy Harris final). Decided: final is 1 PM, Dec 14. Rate Equations and Detailed Balance Blackbodies arise if the optical depth is big
More informationEM Waves in Media. What happens when an EM wave travels through our model material?
EM Waves in Media We can model a material as made of atoms which have a charged electron bound to a nucleus by a spring. We model the nuclei as being fixed to a grid (because they are heavy, they don t
More informationHypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =
Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,
More informationSpectral synthesis codes and methods of analysis : what do we have on the market Tatiana Ryabchikova Institute of Astronomy RAS
Spectral synthesis codes and methods of analysis : what do we have on the market Tatiana Ryabchikova Institute of Astronomy RAS Spring school of spectroscopic data analysis 8-12 April, Wroclaw, Poland
More informationRegression, part II. I. What does it all mean? A) Notice that so far all we ve done is math.
Regression, part II I. What does it all mean? A) Notice that so far all we ve done is math. 1) One can calculate the Least Squares Regression Line for anything, regardless of any assumptions. 2) But, if
More informationImplicit Differentiation Applying Implicit Differentiation Applying Implicit Differentiation Page [1 of 5]
Page [1 of 5] The final frontier. This is it. This is our last chance to work together on doing some of these implicit differentiation questions. So, really this is the opportunity to really try these
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationSISD Training Lectures in Spectroscopy
SISD Training Lectures in Spectroscopy Anatomy of a Spectrum Visual Spectrum of the Sun Blue Spectrum of the Sun Morphological Features in Spectra λ 2 Line Flux = Fλ dλ λ1 (Units: erg s -1 cm -2 ) Continuum
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationarxiv: v1 [astro-ph.im] 6 Nov 2017
Measuring Stellar Atmospheric Parameters with ARES+MOOG S. G. Sousa and D. T. Andreasen arxiv:1711.01839v1 [astro-ph.im] 6 Nov 2017 Abstract The technical aspects in the use of an Equivalent Width (EW)
More informationFIA0221: Taller de Astronomía II. Lecture 14 Spectral Classification of Stars
FIA0221: Taller de Astronomía II Lecture 14 Spectral Classification of Stars Spectral types along the stellar CMD. Oh, Be A Fine Girl Kiss Me! Classification of Stellar spectra: The MK system: strong He+
More informationShort/Simple Definitions:
Eric Joseph Bubar Stellar Atmosphere/Interiors Portfolio CHAPTER : CURVES OF GROWTH Short/Simple Definitions: Curve of Growth: Plot of equivalent widths versus number of absorbing atoms that created that
More information12. Physical Parameters from Stellar Spectra. Fundamental effective temperature calibrations Surface gravity indicators Chemical abundances
12. Physical Parameters from Stellar Spectra Fundamental effective temperature calibrations Surface gravity indicators Chemical abundances 1 Fundamental Properties of Stars Temperature (T) Radius (R) Chemical
More informationMATH 308 COURSE SUMMARY
MATH 308 COURSE SUMMARY Approximately a third of the exam cover the material from the first two midterms, that is, chapter 6 and the first six sections of chapter 7. The rest of the exam will cover the
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More information5.2 Infinite Series Brian E. Veitch
5. Infinite Series Since many quantities show up that cannot be computed exactly, we need some way of representing it (or approximating it). One way is to sum an infinite series. Recall that a n is the
More informationOrientation: what is physical chemistry about?
1 Orientation: what is physical chemistry about? Chemistry is traditionally divided into a small number of subfields, namely organic, inorganic, analytical and physical chemistry. It s fairly easy to say
More informationAstronomy 102 Math Review
Astronomy 102 Math Review 2003-August-06 Prof. Robert Knop r.knop@vanderbilt.edu) For Astronomy 102, you will not need to do any math beyond the high-school alegbra that is part of the admissions requirements
More informationAddition of Opacities and Absorption
Addition of Opacities and Absorption If the only way photons could interact was via simple scattering, there would be no blackbodies. We ll go into that in much more detail in the next lecture, but the
More informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationCHAPTER 7: TECHNIQUES OF INTEGRATION
CHAPTER 7: TECHNIQUES OF INTEGRATION DAVID GLICKENSTEIN. Introduction This semester we will be looking deep into the recesses of calculus. Some of the main topics will be: Integration: we will learn how
More informationIntroduction to Algebra: The First Week
Introduction to Algebra: The First Week Background: According to the thermostat on the wall, the temperature in the classroom right now is 72 degrees Fahrenheit. I want to write to my friend in Europe,
More informationLECTURE 15: SIMPLE LINEAR REGRESSION I
David Youngberg BSAD 20 Montgomery College LECTURE 5: SIMPLE LINEAR REGRESSION I I. From Correlation to Regression a. Recall last class when we discussed two basic types of correlation (positive and negative).
More informationBig Bang, Black Holes, No Math
ASTR/PHYS 109 Dr. David Toback Lectures 8 & 9 1 Prep For Today (is now due) L9 Reading: BBBHNM Unit 2 (already due) Pre-Lecture Reading Questions (PLRQ) Unit 2 Revision (if desired), Stage 2: Was due today
More informationConfidence intervals
Confidence intervals We now want to take what we ve learned about sampling distributions and standard errors and construct confidence intervals. What are confidence intervals? Simply an interval for which
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More informationMeasuring the Properties of Stars (ch. 17) [Material in smaller font on this page will not be present on the exam]
Measuring the Properties of Stars (ch. 17) [Material in smaller font on this page will not be present on the exam] Although we can be certain that other stars are as complex as the Sun, we will try to
More information1 Measurement Uncertainties
1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise.
More informationThe Hydrogen Atom According to Bohr
The Hydrogen Atom According to Bohr The atom We ve already talked about how tiny systems behave in strange ways. Now let s s talk about how a more complicated system behaves. The atom! Physics 9 4 Early
More informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More information4 HOW DID THE EARTH FORM?
4 HOW DID THE EARTH FORM? New stars and space debris spinning like pizza dough are a couple of the things that explain the formation of solar systems like ours. In this three-part lecture, David Christian
More informationV. Stars.
V. Stars http://sgoodwin.staff.shef.ac.uk/phy111.html 0. The local HR diagram We saw that locally we can make an HR diagram of absolute luminosity against temperature. We find a main sequence, giants and
More informationGravity and Orbits. 1. Choose the picture you think shows the gravity forces on the Earth and the Sun.
Name: Grade: Gravity and Orbits Pre-lab 1. Choose the picture you think shows the gravity forces on the Earth and the Sun. (a longer arrow to represents a big force, and a shorter arrow represent a smaller
More informationMITOCW ocw f99-lec30_300k
MITOCW ocw-18.06-f99-lec30_300k OK, this is the lecture on linear transformations. Actually, linear algebra courses used to begin with this lecture, so you could say I'm beginning this course again by
More informationThe Basics of Light. Sunrise from the Space Shuttle, STS-47 mission. The Basics of Light
The Basics of Light The sun as it appears in X-ray light (left) and extreme ultraviolet light (right). Light as energy Light is remarkable. It is something we take for granted every day, but it's not something
More information#29: Logarithm review May 16, 2009
#29: Logarithm review May 16, 2009 This week we re going to spend some time reviewing. I say re- view since you ve probably seen them before in theory, but if my experience is any guide, it s quite likely
More information- a value calculated or derived from the data.
Descriptive statistics: Note: I'm assuming you know some basics. If you don't, please read chapter 1 on your own. It's pretty easy material, and it gives you a good background as to why we need statistics.
More informationAxiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:
Axiomatic systems Revisiting the rules of inference Material for this section references College Geometry: A Discovery Approach, 2/e, David C. Kay, Addison Wesley, 2001. In particular, see section 2.1,
More informationMITOCW watch?v=ko0vmalkgj8
MITOCW watch?v=ko0vmalkgj8 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationBig-oh stuff. You should know this definition by heart and be able to give it,
Big-oh stuff Definition. if asked. You should know this definition by heart and be able to give it, Let f and g both be functions from R + to R +. Then f is O(g) (pronounced big-oh ) if and only if there
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra, Spring 2005 Please use the following citation format: Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology:
More informationLine Integrals and Path Independence
Line Integrals and Path Independence We get to talk about integrals that are the areas under a line in three (or more) dimensional space. These are called, strangely enough, line integrals. Figure 11.1
More informationChapter 3 ALGEBRA. Overview. Algebra. 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents. Section 3.
4 Chapter 3 ALGEBRA Overview Algebra 3.1 Linear Equations and Applications 3.2 More Linear Equations 3.3 Equations with Exponents 5 LinearEquations 3+ what = 7? If you have come through arithmetic, the
More informationPHY 101L - Experiments in Mechanics
PHY 101L - Experiments in Mechanics introduction to error analysis What is Error? In everyday usage, the word error usually refers to a mistake of some kind. However, within the laboratory, error takes
More informationAppendix A Powers of Ten
Conclusion This has been a theory book for observational amateur astronomers. This is perhaps a bit unusual because most astronomy theory books tend to be written for armchair astronomers and they tend
More informationASTRO 114 Lecture Okay. We re going to continue now with our discussion of stars. We re moving
ASTRO 114 Lecture 40 1 Okay. We re going to continue now with our discussion of stars. We re moving into the next chapter and the next chapter not only talks about the properties of stars but the entire
More informationOne sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:
One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get
More informationCOLLEGE ALGEBRA. Solving Equations and Inequalities. Paul Dawkins
COLLEGE ALGEBRA Solving Equations and Inequalities Paul Dawkins Table of Contents Preface... ii Solving Equations and Inequalities... 3 Introduction... 3 s and Sets... 4 Linear Equations... 8 Application
More informationLecture Outline: Spectroscopy (Ch. 4)
Lecture Outline: Spectroscopy (Ch. 4) NOTE: These are just an outline of the lectures and a guide to the textbook. The material will be covered in more detail in class. We will cover nearly all of the
More informationPhysics Motion Math. (Read objectives on screen.)
Physics 302 - Motion Math (Read objectives on screen.) Welcome back. When we ended the last program, your teacher gave you some motion graphs to interpret. For each section, you were to describe the motion
More informationBig Bang, Black Holes, No Math
ASTR/PHYS 109 Dr. David Toback Lectures 10, 11 & 12 1 Prep For Today (is now due) L12 Reading: (BBBHNM Unit 2) Pre-Lecture Reading Questions: If you were misgraded, need help or an extension let me know
More informationCHM 105 & 106 UNIT TWO, LECTURE EIGHT 1 IN OUR PREVIOUS LECTURE WE WERE LOOKING AT CONCENTRATION UNITS FOR SOLUTIONS
CHM 105 & 106 UNIT TWO, LECTURE EIGHT 1 CHM 105/106 Program 15: Unit 2 Lecture 8 IN OUR PREVIOUS LECTURE WE WERE LOOKING AT CONCENTRATION UNITS FOR SOLUTIONS AND WE HAD LOOKED AT PERCENT BY MASS AND PERCENT
More informationASTRO 114 Lecture Okay. We re now gonna continue discussing and conclude discussing the entire
ASTRO 114 Lecture 55 1 Okay. We re now gonna continue discussing and conclude discussing the entire universe. So today we re gonna learn about everything, everything that we know of. There s still a lot
More informationFundamentals of Semiconductor Devices Prof. Digbijoy N. Nath Centre for Nano Science and Engineering Indian Institute of Science, Bangalore
Fundamentals of Semiconductor Devices Prof. Digbijoy N. Nath Centre for Nano Science and Engineering Indian Institute of Science, Bangalore Lecture - 05 Density of states Welcome back. So, today is the
More informationLecture 10: Powers of Matrices, Difference Equations
Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each
More informationc 1 v 1 + c 2 v 2 = 0 c 1 λ 1 v 1 + c 2 λ 1 v 2 = 0
LECTURE LECTURE 2 0. Distinct eigenvalues I haven t gotten around to stating the following important theorem: Theorem: A matrix with n distinct eigenvalues is diagonalizable. Proof (Sketch) Suppose n =
More informationof 8 28/11/ :25
Paul's Online Math Notes Home Content Chapter/Section Downloads Misc Links Site Help Contact Me Differential Equations (Notes) / First Order DE`s / Modeling with First Order DE's [Notes] Differential Equations
More informationN H 2 2 NH 3 and 2 NH 3 N H 2
Chemical Equilibrium Notes (Chapter 18) So far, we ve talked about all chemical reactions as if they go only in one direction. However, as with many things in life, chemical reactions can go both in the
More informationThe following are generally referred to as the laws or rules of exponents. x a x b = x a+b (5.1) 1 x b a (5.2) (x a ) b = x ab (5.
Chapter 5 Exponents 5. Exponent Concepts An exponent means repeated multiplication. For instance, 0 6 means 0 0 0 0 0 0, or,000,000. You ve probably noticed that there is a logical progression of operations.
More informationUncertainty: A Reading Guide and Self-Paced Tutorial
Uncertainty: A Reading Guide and Self-Paced Tutorial First, read the description of uncertainty at the Experimental Uncertainty Review link on the Physics 108 web page, up to and including Rule 6, making
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More informationMath 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions
Math 111, Introduction to the Calculus, Fall 2011 Midterm I Practice Exam 1 Solutions For each question, there is a model solution (showing you the level of detail I expect on the exam) and then below
More information(c) Sketch the ratio of electron to gas pressure for main sequence stars versus effective temperature. [1.5]
1. (a) The Saha equation may be written in the form N + n e N = C u+ u T 3/2 exp ( ) χ kt where C = 4.83 1 21 m 3. Discuss its importance in the study of stellar atmospheres. Carefully explain the meaning
More informationRING DISCOVERED AROUND DWARF PLANET
RING DISCOVERED AROUND DWARF PLANET Haumea, a dwarf planet in the Kuiper Belt was just found to have a ring. Why? Hint: what causes the Jovian planet rings? Artist's conception, not a real photo RING DISCOVERED
More informationappstats27.notebook April 06, 2017
Chapter 27 Objective Students will conduct inference on regression and analyze data to write a conclusion. Inferences for Regression An Example: Body Fat and Waist Size pg 634 Our chapter example revolves
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationLecture 2 - Length Contraction
Lecture 2 - Length Contraction A Puzzle We are all aware that if you jump to the right, your reflection in the mirror will jump left. But if you raise your hand up, your reflection will also raise its
More informationHOLLOMAN S AP STATISTICS BVD CHAPTER 08, PAGE 1 OF 11. Figure 1 - Variation in the Response Variable
Chapter 08: Linear Regression There are lots of ways to model the relationships between variables. It is important that you not think that what we do is the way. There are many paths to the summit We are
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationMITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4
MITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4 PROFESSOR: OK, this lecture, this day, is differential equations day. I just feel even though these are not on the BC exams, that we've got everything
More informationThe Euler Method for the Initial Value Problem
The Euler Method for the Initial Value Problem http://people.sc.fsu.edu/ jburkardt/isc/week10 lecture 18.pdf... ISC3313: Introduction to Scientific Computing with C++ Summer Semester 2011... John Burkardt
More informationAstronomy 421. Lecture 14: Stellar Atmospheres III
Astronomy 421 Lecture 14: Stellar Atmospheres III 1 Lecture 14 - Key concepts: Spectral line widths and shapes Curve of growth 2 There exists a stronger jump, the Lyman limit, occurring at the wavelength
More informationNote that we are looking at the true mean, μ, not y. The problem for us is that we need to find the endpoints of our interval (a, b).
Confidence Intervals 1) What are confidence intervals? Simply, an interval for which we have a certain confidence. For example, we are 90% certain that an interval contains the true value of something
More informationMetal Poor Stars: A Review for Non-Observers. Charli Sakari
Metal Poor Stars: A Review for Non-Observers Charli Sakari Outline Summary: What we know and have discussed already How should we interpret published stellar abundances? Martin Asplund et al.: NOT observed
More informationGravity and Orbits Activity Page 1. Name: Grade: Gravity and Orbits. Pre-lab. 1. In the picture below, draw how you think Earth moves.
Name: Grade: Gravity and Orbits Pre-lab 1. In the picture below, draw how you think Earth moves. 2. Draw a picture using arrows to show what you think the forces might be on the Earth and the Sun. You
More informationChapter 27 Summary Inferences for Regression
Chapter 7 Summary Inferences for Regression What have we learned? We have now applied inference to regression models. Like in all inference situations, there are conditions that we must check. We can test
More informationBig Bang, Black Holes, No Math
ASTR/PHYS 109 Dr. David Toback Lecture 5 1 Prep For Today (is now due) L5 Reading: No new reading Unit 2 reading assigned at the end of class Pre-Lecture Reading Questions: Unit 1: Grades have been posted
More informationPart I Electrostatics. 1: Charge and Coulomb s Law July 6, 2008
Part I Electrostatics 1: Charge and Coulomb s Law July 6, 2008 1.1 What is Electric Charge? 1.1.1 History Before 1600CE, very little was known about electric properties of materials, or anything to do
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationLecture 12: Quality Control I: Control of Location
Lecture 12: Quality Control I: Control of Location 10 October 2005 This lecture and the next will be about quality control methods. There are two reasons for this. First, it s intrinsically important for
More informationFingerprinting the Stars Lab (Sarah Hansen & Monica Valluri)
Fingerprinting the Stars Lab (Sarah Hansen & Monica Valluri) Introduction Every element produces a unique fingerprint of spectral lines. By identifying the spectral features in stellar spectra, we can
More information= v = 2πr. = mv2 r. = v2 r. F g. a c. F c. Text: Chapter 12 Chapter 13. Chapter 13. Think and Explain: Think and Solve:
NAME: Chapters 12, 13 & 14: Universal Gravitation Text: Chapter 12 Chapter 13 Think and Explain: Think and Explain: Think and Solve: Think and Solve: Chapter 13 Think and Explain: Think and Solve: Vocabulary:
More informationMITOCW MITRES18_005S10_DiffEqnsGrowth_300k_512kb-mp4
MITOCW MITRES18_005S10_DiffEqnsGrowth_300k_512kb-mp4 GILBERT STRANG: OK, today is about differential equations. That's where calculus really is applied. And these will be equations that describe growth.
More informationPHYSICS 107. Lecture 27 What s Next?
PHYSICS 107 Lecture 27 What s Next? The origin of the elements Apart from the expansion of the universe and the cosmic microwave background radiation, the Big Bang theory makes another important set of
More informationCASE STUDY FOR USE WITH SECTION B
GCE A level 325/0-A PHYSICS PH5 Assessment Unit CASE STUDY FOR USE WITH SECTION B Pre-Release Material To be opened on receipt A new copy of this Case Study will be given out in the examination 325 0A00
More informationclass 21 Astro 16: Astrophysics: Stars, ISM, Galaxies November 20, 2018
Topics: Post-main-sequence stellar evolution, degeneracy pressure, and white dwarfs Summary of reading: Review section 2 of Ch. 17. Read the beginning and first section of Ch. 18 (up through the middle
More informationMITOCW ocw-18_02-f07-lec02_220k
MITOCW ocw-18_02-f07-lec02_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationExperiment 1 Make a Magnet
Magnets Here s a riddle. I stick to some things but not to others. I stick but I m not sticky. I attract some things but push other things away and, if allowed to move, I will always point the same way.
More informationParallax: Space Observatories. Stars, Galaxies & the Universe Announcements. Stars, Galaxies & Universe Lecture #7 Outline
Stars, Galaxies & the Universe Announcements HW#4: posted Thursday; due Monday (9/20) Reading Quiz on Ch. 16.5 Monday (9/20) Exam #1 (Next Wednesday 9/22) In class (50 minutes) first 20 minutes: review
More informationDescriptive Statistics (And a little bit on rounding and significant digits)
Descriptive Statistics (And a little bit on rounding and significant digits) Now that we know what our data look like, we d like to be able to describe it numerically. In other words, how can we represent
More informationMITOCW watch?v=poho4pztw78
MITOCW watch?v=poho4pztw78 GILBERT STRANG: OK. So this is a video in which we go for second-order equations, constant coefficients. We look for the impulse response, the key function in this whole business,
More informationContents of The Universe and Deforming Solids
Skyscrapers in the 2011 Japan Earthquake Contents of The Universe and Deforming Solids For most of this course, we ve talked about physics we ve known about for > 100 years. Today, we ll discuss some physics
More informationWhy is it hard to detect planets around other stars?
Extrasolar planets Why is it hard to detect planets around other stars? Planets are small and low in mass Planets are faint The angular separation between planets and their stars is tiny Why is it hard
More informationSoftware Testing Lecture 2
Software Testing Lecture 2 Justin Pearson September 25, 2014 1 / 1 Test Driven Development Test driven development (TDD) is a way of programming where all your development is driven by tests. Write tests
More informationASTRONOMY 114 Lecture 2 1. Today we re gonna be talking about more than just astronomy. We re gonna be
ASTRONOMY 114 Lecture 2 1 Today we re gonna be talking about more than just astronomy. We re gonna be discussing science in general. Now, I ve already mentioned to you we are gonna be talking about other
More information