Using astroquery

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If you're like me, you've spent a lot of time downloading a table from VizieR, and then trying to insert it into whatever database or table you are currently using. It's annoying, particularly because VizieR either won't output information in a machine-readable format or won't put an empty line into a massive list of results when there's no match in the catalog.
Astroquery is better. It's a python package affiliated with astropy that can grab data from all your favorite catalog servers (Vizier, SIMBAD, IRSA, NED, etc) directly, and use it; combine that with Astropy's ability (or Python's capacity) to print out tables to files, and it's a lot easier to write a script to do it all for you.

Astroquery is in the Python Package Index, so you can install and update it with
pip install astroquery

You also need the 'requests' module to do the web-interfacing (and Astropy v3); it's available on the Python Package Index and comes preloaded in Anaconda. Using it is simple (this part is almost entirely taken from their pages on using VizieR and IRSA):
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Uncertainty Propagation

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All observations have associated uncertainties which must be propagated through your analysis to a an uncertainty on a final result. The principle of uncertainty propagation is fairly simple:

\sigma_x^2 = \sigma_u^2\left(\frac{\partial x}{\partial u}\right)^2 + \sigma_v^2\left(\frac{\partial x}{\partial v}\right)^2+\dots +2\sigma_{uv}^2\left(\frac{\partial x}{\partial u}\right)\left(\frac{\partial x}{\partial v}\right)+\dots

The first two terms on the right are the averages of the squares of the deviations in x produced by the uncertainties in observables u and v respectively. The third term on the right is the average of the cross terms, which cancel out if u and v are uncorrelated. Thus in most situations, a reasonable approximation is:

\sigma_x^2 = \sum\limits_u\sigma_u^2\left(\frac{\partial x}{\partial u}\right)^2

Examples

A typical situation is the sum of two observables each with a multiplicative factor:

x=au+bv

\left(\frac{\partial x}{\partial u}\right)=a,\hspace{10pt}\left(\frac{\partial x}{\partial v}\right)=b

\sigma_x = \sqrt{\sigma_u^2a^2 + \sigma_v^2b^2}

which is the oft used "summing in quadrature."

Perhaps you have the product of two observables:

x=auv+b

\left(\frac{\partial x}{\partial u}\right)=av,\hspace{10pt}\left(\frac{\partial x}{\partial v}\right)=au

\sigma_x = \sqrt{\sigma_u^2(av)^2 + \sigma_v^2(au)^2}=a\sqrt{\sigma_u^2v^2 + \sigma_v^2u^2}

Finally, perhaps you have the inverse of one observable times the exponential of another:

x=\frac{a}{u}e^{bv}

\left(\frac{\partial x}{\partial u}\right)=\left(-\frac{a}{u^2}\right)e^{bv},\hspace{10pt}\left(\frac{\partial x}{\partial v}\right)=\frac{a}{u}be^{bv}

\sigma_x = \sqrt{\sigma_u^2\left(\frac{a}{u^2}e^{bv}\right)^2 + \sigma_v^2\left(\frac{a}{u}be^{bv}\right)^2}=\frac{a}{u}e^{bv}\sqrt{\frac{\sigma_u^2}{u^2}+\sigma_v^2b^2}

Easy!

Setting Up Your BDNYC Astro Python Environment

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Here's a step-by-step tutorial on how to set up your Python development environment.

Distribution and Compiler

The first step is to install Anaconda. Click here, select the appropriate version for your machine and operating system, download the .dmg file, open it and double-click the installer.

What's nice about this distribution is that it includes common plotting, computing, and astronomy packages such as numpy and scipy, astropy, matplotlib, and many others.

Next you'll need a C compiler. If you're using MacOSX, I recommend using Xcode which you can download free here. (This step may take a while so go get a beverage.) Once the download is complete, run the installer.

Required Packages

For future installation of packages, I recommend using Pip. To install this, at your Terminal command line type sudo easy_install pip. Then whenever you want to install a package you might need, you just open Terminal and do pip install package_name.

Not every package is this easy (though most are). One package we need is called AstroAsciiData and we can't get it through Pip. Instead go here, click on the link to download, unzip it, and put the folder with the rest of your packages in the directory /anaconda/pkgs/.

Then in Terminal, navigate to that directory with cd /anaconda/pkgs/asciidata-1.1.1 and do python setup.py install. If there are any other packages that Pip can't find, install them this way.

Development Tools

MacOSX comes with the text editing application TextEdit but it is not good for editing code. I strongly recommend using TextMate though it is not free so you should ask your advisor to buy a license for you! Otherwise, some folks find the free TextWrangler to be pretty good.

Next, you'll want to get access to the BDNYC database. Detailed instructions are here on how to setup Dropbox and Github on your machine in order to interact with the database.

Launching Python

Now to use Python, in Terminal just type ipython --pylab. I recommend always launching Python this way to have the plotting library preloaded.

Enjoy!

BDNYC Database Setup

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Read this post first if your development environment needs setting up. Then...

Setting Up the Database

  1. Get someone who has access to the database to share the BDNYCdb Dropbox folder with you.
  2. Go to the BDNYC Github page and click on the Clone in Desktop button in the lower right. Then navigate to the desired directory and click Clone.
  3. Open a Terminal window and type open ~/.bash_profile (or open ~/.tcshrc or where ever your PYTHONPATH variable is) to open your profile in a text editor. Then append the path of your BDNYCdb directory to your PYTHONPATH with a colon (:) and save it.

    In .bash_profile it looks like (for example):

    PYTHONPATH="{$PYTHONPATH}:/Users/Joe/Documents/Python/Modules:/Users/Joe/Documents/Python/Modules/BDNYCdb"

  4. From Terminal, reload your profile with source ~/.bash_profile or the like.

Now the SQLite database is in the shared Dropbox directory as BDNYC.db and the module to access it is in your directory cloned from Github. This way, the module can be version controlled and the database is updated automatically.

Accessing the Database

Now you can load the entire database into a Python variable simply by launching the Python interpreter and pointing the get_db() function to the database file by doing:

In [1]: import BDdb
In [2]: db = BDdb.get_db('/path/to/your/Dropbox/BDNYCdb/BDNYC.db')

Voila! To see an inventory of all database sources, just do:

In [3]: db.inventory()

You can see details on a specific source by entering the id from the inventory list. This will also plot all available spectra for that source for visual inspection.

In [4]: db.inventory(ID=767)

Now that you have the database at your fingertips, you’ll want to get some info out of it. To do this, you can use SQL queries.

Here is a detailed post about how to write an SQL query.

Further documentation for sqlite3 can be found here. Most commands involve wrapping SQL language inside python functions. The main (really only) one we will use to fetch data from the database is the execute() command:

In [5]: data = db.query.execute( "SQL_query_goes_here" ).fetchall()

Example Queries

Some SQL query examples to put in the command above (wrapped in quotes of course):

  1. SELECT shortname, ra, dec FROM sources WHERE (222<ra AND ra<232) AND (5<dec AND dec<15)
  2. SELECT band, magnitude, magnitude_unc FROM photometry WHERE source_id=58
  3. SELECT source_id, band, magnitude FROM photometry WHERE band='z' AND magnitude<15
  4. SELECT wavelength, flux, unc FROM spectra WHERE observation_id=75”

As you hopefully gathered:

  1. Returns the shortname, ra and dec of all objects in a 10 square degree patch of sky centered at RA = 227, DEC = 10
  2. Returns all the photometry and uncertainties available for object 58
  3. Returns all objects and z magnitudes with z less than 15
  4. Returns the wavelength, flux and uncertainty arrays for all spectra of object 75

The above examples are for querying individual tables only. We can query from multiple tables at the same time with the JOIN command like so:

  1. SELECT t.name, p.band, p.magnitude, p.magnitude_unc FROM telescopes as t JOIN photometry AS p ON p.telescope_id=t.id WHERE p.source_id=58
  2. SELECT p1.magnitude-p2.magnitude FROM photometry AS p1 JOIN photometry AS p2 ON p1.source_id=p2.source_id WHERE p1.band='J' AND p2.band='H'
  3. SELECT src.designation, src.unum, spt.spectral_type FROM sources AS src JOIN spectral_types AS spt ON spt.source_id=src.id WHERE spt.spectral_type>=10 AND spt.spectral_type<20 AND spt.regime='optical'
  4. SELECT s.unum, p.parallax, p.parallax_unc, p.publication_id FROM sources as s JOIN parallaxes AS p ON p.source_id=s.id

As you may have gathered:

  1. Returns the survey, band and magnitude for all photometry of source 58
  2. Returns the J-H color for every object
  3. Returns the designation, U-number and optical spectral type for all L dwarfs
  4. Returns the parallax measurements and publications for all sources

Alternative Output

As shown above, the result of a SQL query is typically a list of tuples where we can use the indices to print the values. For example, this source's g-band magnitude:

In [9]: data = db.query.execute("SELECT band,magnitude FROM photometry WHERE source_id=58").fetchall()
In [10]: data
Out[10]: [('u', 25.70623),('g', 25.54734),('r', 23.514),('i', 21.20863),('z', 18.0104)]
In [11]: data[1][1]
Out[11]: 25.54734

However we can also query the database a little bit differently so that the fields and records are returned as a dictionary. Instead of db.query.execute() we can do db.dict.execute() like this:

In [12]: data = db.dict.execute("SELECT * FROM photometry WHERE source_id=58").fetchall()
In [13]: data
Out[13]: [<sqlite3.Row at 0x107798450>, <sqlite3.Row at 0x107798410>, <sqlite3.Row at 0x107798430>, <sqlite3.Row at 0x1077983d0>, <sqlite3.Row at 0x1077982f0>]
In [14]: data[1]['magnitude']
Out[14]: 25.54734

Database Schema and Browsing

In order to write the SQL queries above you of course need to know what the names of the tables and fields in the database are. One way to do this is:

In [15]: db.query.execute("SELECT sql FROM sqlite_master").fetchall()

This will print a list of each table, the possible fields, and the data type (e.g. TEXT, INTEGER, ARRAY) for that field.

SQL browserEven easier is to use the SQLite Database Browser pictured at left which lets you expand and collapse each table.

It even allows you to manually create/edit/destroy records with a very nice GUI.

IMPORTANT: Keep in mind that if you change a database record, you immediately change it for everyone since we share the same database file on Dropbox. Be careful!

Always check and double-check that you are entering the correct data for the correct source before you save any changes with the SQLite Database Browser.

SQL Queries

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An SQL database is comprised of a bunch of tables (kind of like a spreadsheet) that have fields (column names) and records (rows of data). For example, our database might have a table called students that looks like this:

id first last grade GPA
1 Al Einstein 6 2.7
2 Annie Cannon 6 3.8
3 Artie Eddington 8 3.2
4 Carlie Herschel 8 3.2

So in our students table, the fields are [id, first, last, grade, GPA], and there are a total of four records, each with a required yet arbitrary id in the first column.

To pull these records out, we tell SQL to SELECT values for the following fields FROM a certain table. In SQL this looks like:

In [1]: db.execute("SELECT id, first, last, grade, GPA FROM students").fetchall()
Out[1]: [(1,'Al','Einstein',6,2.7),(2,'Annie','Cannon',6,3.8),(3,'Artie','Eddington',8,3.2),(4,'Carlie','Herschel',8,3.2)]

Or equivalently, we can just use a wildcard "*" if we want to return all fields with the SQL query "SELECT * FROM students".

We can modify our SQL query to change the order of fields or only return certain ones as well. For example:

In [2]: db.execute("SELECT last, first, GPA FROM students").fetchall()
Out[1]: [('Einstein','Al',2.7),('Cannon','Annie',3.8),('Eddington','Artie',3.2),('Herschel','Carlie',3.2)]

Now that we know how to get records from tables, we can restrict which records it returns with the WHERE statement:

In [3]: db.execute("SELECT last, first, GPA FROM students WHERE GPA>3.1").fetchall()
Out[3]: [('Cannon','Annie',3.8),('Eddington','Artie',3.2),('Herschel','Carlie',3.2)]

Notice the first student had a GPA less than 3.1 so he was omitted from the result.

Now let's say we have a second table called quizzes which is a table of every quiz grade for all students that looks like this:

id student_id quiz_number score
1 1 3 89
2 2 3 96
3 3 3 94
4 4 3 92
5 1 4 78
6 3 4 88
7 4 4 91

Now if we want to see only Al's grades, we have to JOIN the tables ON some condition. In this case, we want to tell SQL that the student_id (not the id) in the quizzes table should match the id in the students table (since only those grades are Al's). This looks like:

In [4]: db.execute("SELECT quizzes.quiz_number, quizzes.score FROM quizzes JOIN students ON students.id=quizzes.student_id WHERE students.last='Einstein'").fetchall()
Out[4]: [(3,89),(4,78)]

So students.id=quizzes.student_id associates each quiz with a student from the students table and students.last='Einstein' specifies that we only want the grades from the student with last name Einstein.

Similarly, we can see who scored 90 or greater on which quiz with:

In [5]: db.execute("SELECT students.last, quizzes.quiz_number, quizzes.score FROM quizzes JOIN students ON students.id=quizzes.student_id WHERE quizzes.score>=90").fetchall()
Out[5]: [('Cannon',3,96),('Eddington',3,94),('Herschel',3,92),('Herschel',4,91)]

That's it! We can JOIN as many tables as we want with as many restrictions we need to pull out data in the desired form.

This is powerful, but the queries can become lengthy. A slight shortcut is to use the AS statement to assign a table to a variable (e.g. students => s, quizzes => q) like such:

In [6]: db.execute("SELECT s.last, q.quiz_number, q.score FROM quizzes AS q JOIN students AS s ON s.id=q.student_id WHERE q.score>=90").fetchall()
Out[6]: [('Cannon',3,96),('Eddington',3,94),('Herschel',3,92),('Herschel',4,91)]

Ground Based Photometry

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Here I want to calculate some photometric points from spectra for comparison with published values for a bunch of known brown dwarfs.

In order to get the true magnitude m for an object, I first need to calculate the instrumental magnitude m_\text{inst} and then correct for a number of effects. That is, I calculate the apparent magnitude from a particular place on the Earth and then add corrections to determine what it would be if we measured from space.

After all our corrections are made, the magnitude is given by:

m=m_\text{inst}-ZP_m-k_m\cdot X

Instrumental Magnitude

The first term on the right in the equation above is the magnitude measured by the instrument on the ground, given by:

m_\text{inst}=-2.5\log\left(\int f_\lambda (\lambda)\left(\frac{\lambda}{hc}\right)S_m(\lambda)d\lambda\right)

Where f_\lambda (\lambda) is the energy flux density of the source in units of [erg s-1 cm-2 A-1] and S_m(\lambda) is the scalar filter throughput for the band of interest.

Since I will be comparing my calculated magnitudes to photometry taken with photon counting devices, the factor of \frac{\lambda}{hc} converts f_\lambda (\lambda) to a photon flux density in units [photons s-1 cm-2 A-1].

Zero Point Correction

The second term in our magnitude equation is a first order correction to compare m_\text{inst} to some standard we define as zero. I will use a flux calibrated spectrum of the A0 star Vega to calculate the zero point magnitude for the band:

ZP_m=-2.5\log\left(\int f_{\lambda\text{ Vega}}(\lambda)\left(\frac{\lambda}{hc}\right)S_m(\lambda)d\lambda\right)

Just as we obtained our instrumental magnitude above.

Extinction Correction

The third term is to correct for the extinction of the source flux due to atmospheric absorption. We can get closer to the true apparent magnitude (above the atmosphere) by adding an extinction term:

k_m\cdot\sec (z)=k_m\cdot X

Where k_m is the extinction coefficient for the band of interest and \sec(z)=X is the airmass.

The airmass is the optical path length of the atmosphere, which attenuates the source flux depending on its angle from the zenith z . Approximating the truly spherical atmosphere as plane-parallel, the airmass goes from X=1 at z=0^\circ to X=2 at z=60^\circ . At zenith angles greater than that, the plane-parallel approximation falls apart and the airmass term gets complicated.

Where the airmass is the amount of atmosphere in the line of sight, the extinction coefficient is the amount by which the incident light is attenuated as it travels through the airmass. The extinction coefficient is related to the optical depth \tau of the atmosphere as:

m-m_0=-2.5\log\left(\frac{I}{I_0}\right)=-2.5\log (e^{-\tau X})=1.086\cdot\tau\cdot X=k_m\cdot X

Where m and m_0 are the magnitudes below and above the atmosphere respectively.

Example: J21512543-2441000

As an example, I'd like to calculate the 2MASS J-band magnitude of the brown dwarf at 21h51m25.43s -24d41m00s given a low resolution NIR energy flux density from the SpeX Prism instrument on the 3m NASA Infrared Telescope Facility.

J21512543-2441000

Interpolating the filter throughput to the object spectrum and then integrating as in the equation above, I get J_\text{inst}=10.046 as my instrumental magnitude in the J-band.

Performing the same procedure on the flux calibrated spectrum of Vega, I get ZP_J=-5.721 for my J-band zero point magnitude.

Checking the FITS file header, I will use X=1.444625 for the airmass. The mean extinction coefficient for the MKO system J-band is given as k_J=0.0153 in Tokunaga & Vacca (2007), making the atmospheric correction term k_J\cdot X=0.0221 .

The corrected magnitude is then:

J=J_\text{inst}-ZP_J-k_J\cdot X=10.046-(-5.721)-0.0221=15.745

Which is only 0.007 magnitudes off from the value of J=15.752 from the 2MASS catalog.

Remaining Problems

As shown in the example above, this works... but not for every object.

2MASS apparent J magnitudes vs. my calculated apparent j magnitudes for 67 brown dwarfs. The solid black line is for perfect agreement and the dashed line is a best fit of the data.

2MASS apparent J magnitudes vs. my calculated apparent j magnitudes for 67 brown dwarfs. The solid black line is for perfect agreement and the dashed line is a best fit of the data.

2MASS apparent H magnitudes vs. my calculated apparent h magnitudes for 67 brown dwarfs. The solid black line is for perfect agreement and the dashed line is a best fit of the data.

2MASS apparent H magnitudes vs. my calculated apparent h magnitudes for 67 brown dwarfs. The solid black line is for perfect agreement and the dashed line is a best fit of the data.

I whittled down my sample of 875 to only those objects with flux units and airmass values taken at Mauna Kea so that I could use the same extinction coefficient and make sure they are all in the same units of [erg s-1 cm-2 A-1].

Then I pulled the 2MASS catalog J and H magnitudes with uncertainties for these remaining objects and plotted them against my calculated values with uncertainties.

To the left are the plots of the 67 objects that fit the selection criteria in J-band (above) and H-band (below).

Though it's not the biggest sample, the deviation of the best fit line from unity suggests I'm off by a factor of 0.9 from the 2MASS catalog value across the board.

But more worrisome is the fact that most of the calculated magnitudes are not within the errors of the 2MASS magnitudes. This deviation ranges from very good agreement of a few thousandths of a magnitude up to the worst offenders of about 0.8 mags.

Filter Effective Wavelength(s)

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The effective wavelength of a filter for narrow band photometry can easily be approximated by a constant and just looked up when needed. For broad band photometry, however, the width of the filter and the amount of flux in the band being measured actually come into play.

The effective wavelength of a filter is given by:

\lambda_\text{eff}=\frac{\int \lambda \text{ }f_\lambda (\lambda)\text{ }S(\lambda)\text{ } d\lambda}{\int f_\lambda (\lambda)\text{ } S(\lambda)\text{ } d\lambda}

Where S(\lambda) is the scalar filter throughput and f_\lambda is the flux density in units of [erg s-1 cm-2 A-1] or [photons s-1 cm-2 A-1] depending upon whether you are using an energy measuring or a photon counting detector, respectively.

Here are the results for 67 brown dwarfs with complete spectrum coverage of the 2MASS J-band:

Effective wavelength values for the 2MASS J-band filter. Blue and green circles indicate lambda calculated using photon flux densities (PFD) and energy flux densities (EFD) respectively. Filled circles are for 67 confirmed brown dwarfs. Open circles are for Vega.

Effective wavelength values for the 2MASS J-band filter. Blue and green circles indicate lambda calculated using photon flux densities (PFD) and energy flux densities (EFD) respectively. Filled circles are for 67 confirmed brown dwarfs. Open circles are for Vega.

The red line on the plot shows the specified value given by 2MASS. For fainter objects like brown dwarfs (filled circles), the calculated effective wavelength of the J-band filter can shift redward by as much as 150 angstroms. Vega (open circles) shifts it blueward by about 30 angstroms.

The difference is small but measurable and demonstrates the dependence of the filter width, source spectrum, and detector type on the effective wavelength \lambda_\text{eff} while doing broad band photometry.

Visualizing results from low-res NIR spectral fits

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In my first serious foray into Python and github I adapted some plotting code from Dan Foreman_Mackey with the help of Adrian Price-Whelan  and Joe Filippazzo to create contour plots and histograms of my fitting results! These are histograms MCMC results for model fits to a low-resolution near-infrared spectrum of a young L5 brown dwarf, in temperature and gravity atmospheric parameters. The colors represent different segments of the spectrum - purple is YJH, blue is YJ, red is H. The take-away is that different segments of the spectrum results in different temperatures, and all parts of the spectrum make it look old (high gravity). This is probably because the overly-simplistic dust treatments in the models are not sufficient for young, low-mass objects.

L5_young_JH_J_H_rotate

The baseline for comparison is the result for a field L5 dwarf, shown below. The temperatures are much more consistent with one another and what you would expect for an L5 spectral type from other methods, and the gravities are similarly high (although too high for comfort for J band). This is reassuring for the method in general and probably means that we need most sophisticated dust treatments in the models to handle giant-exoplanet-like young brown dwarfs. Paper will be submitted soon!

L5_JH_J_H_rotate

Photon Flux Density vs. Energy Flux Density

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One of the subtleties of photometry is the difference between magnitudes and colors calculated using energy flux densities (EFD) and photon flux densities (PFD).

The complication arises since the photometry presented by many surveys is calculated using PFD but spectra (specifically the synthetic variety) is given as EFD. The difference is small but measurable so let's do it right.

The following is the process I used to remedy the situation by switching my models to PFD so they could be directly compared to the photometry from the surveys. Thanks to Mike Cushing for the guidance.

Filter Zero Points

Before we can calculate the magnitudes, we need filter zero points calculated from PFD. To do this, I started with a spectrum of Vega in units of [erg s-1 cm-2 A-1] snatched from STSci.

Then the zero point flux density in [photons s-1 cm-2 A-1] is:

F_{zp}=\frac{\int p_V(\lambda) S(\lambda) d\lambda}{\int S(\lambda)d\lambda}=\frac{\int e_V(\lambda)\left( \frac{\lambda}{hc}\right) S(\lambda) d\lambda}{\int S(\lambda)d\lambda}

Where e_V is the given energy flux density in [erg s-1 cm-2 A-1] of Vega, p_V is the photon flux density in [photons s-1 cm-2 A-1], and S(\lambda) is the scalar filter throughput.

Since I'm starting with a spectrum of Vega in EFD units, I need to multiply by \frac{\lambda}{hc} to convert it to PFD units.

In Python, this looks like:

def zp_flux(band):
    from scipy import trapz, interp, log10
    (wave, flux), filt, h, c = vega(), get_filters()[band], 6.6260755E-27 # [erg*s], 2.998E14 # [um/s]
    I = interp(wave, filt['wav'], filt['rsr'], left=0, right=0)
    return trapz(I*flux*wave/(h*c), x=wave)/trapz(I, x=wave))

Calculating Magnitudes

Now that we have the filter zero points, we can calculate the magnitudes using:

m = -2.5\log\left(\frac{F_\lambda}{F_{zp}}\right)

Where m is the apparent magnitude and F_\lambda is the flux from our source given similarly by:

F_{\lambda}=\frac{\int p_\lambda(\lambda) S(\lambda) d\lambda}{\int S(\lambda)d\lambda}=\frac{\int e_\lambda(\lambda)\left( \frac{\lambda}{hc}\right) S(\lambda) d\lambda}{\int S(\lambda)d\lambda}

Since the synthetic spectra I'm using are given in EFD units, I need to multiply by \frac{\lambda}{hc} to convert it to PFD units just as I did with my spectrum of Vega.

In Python the magnitudes are obtained the same way as above but we use the source spectrum in [erg s-1 cm-2 A-1] instead of Vega. Then the magnitude is just:

mag = -2.5*log10(source_flux(band)/zp_flux(band))

Below is an image that shows the discrepancy between using EFD and PFD to calculate colors for comparison with survey photometry.

The circles are colors calculated from synthetic spectra of low surface gravity (large circles) to high surface gravity (small circles). The grey lines are iso-temperature contours. The jumping shows the different results using PFD and EFD. The stationary blue stars, green squares and red triangles are catalog photometric points calculated from PFD.

The circles are colors calculated from synthetic spectra of low surface gravity (large circles) to high surface gravity (small circles). The grey lines are iso-temperature contours. The jumping shows the different results using PFD and EFD. The stationary blue stars, green squares and red triangles are catalog photometric points calculated from PFD.

Other Considerations

The discrepancy I get between the same color calculated from PFD and EFD though is as much as 0.244 mags (in r-W3 at 1050K), which seems excessive. The magnitude calculation reduces to:

m = -2.5\log\left( \frac{\int e_\lambda(\lambda)S(\lambda) \lambda d\lambda}{\int e_V(\lambda) S(\lambda) \lambda d\lambda}\right)

Since the filter profile is interpolated with the spectrum before integration, I thought the discrepancy must be due only to the difference in resolution between the synthetic and Vega spectra. In other words, I have to make sure the wavelength arrays for Vega and the source are identical so the trapezoidal sums have the same width bins.

This reduces the discrepancy in r-W3 at 1050K from -0.244 mags to -0.067 mags, which is better. However, the discrepancy in H-[3.6] goes from 0.071 mags to -0.078 mags.

To Recapitulate

In summary, I had a spectrum of Vega and some synthetic spectra all in energy flux density units of [erg s-1 cm-2 A-1] and some photometric points from the survey catalogs calculated from photon flux density units of [photons s-1 cm-2 A-1].

In order to compare apples to apples, I first converted my spectra to PFD by multiplying by \frac{\lambda}{hc} at each wavelength point before integrating to calculate my zero points and magnitudes.