Flux, Magnitudes, Coordinates, Time, and FITS

Flux, Magnitudes, Coordinates, Time, and FITS

This companion turns the observational foundations of BIN02 into calculations and a reproducible public-data workflow.

The first part contains worked problems.

The second part uses:

• a frozen SIMBAD-based bright-binary photometry extract,
• a frozen metadata record for the TESS observations of AI Phoenicis,
• a live MAST/TESS workflow using the public archive.

The live script supplied with this bundle is BIN02E1_observational_foundations_lab.py

(1)

Part I: Problems

Problem 1: magnitude difference to flux ratio

Two binary components differ by

(2)

magnitudes in the same passband.

Find

(3)

Problem 2: combined magnitude

Two unresolved components have magnitudes

(4)

Find the combined magnitude.

Problem 3: equal-component binary

Show that an unresolved pair of equal-flux stars is approximately 0.7526 magnitude brighter than either component alone.

Problem 4: eclipse depth

A normalized light curve falls from

(5)

to

(6)

Find:

1. the fractional eclipse depth,
2. the differential magnitude at minimum.

Problem 5: color index

A catalog gives

GBP	= 10.72,	(7)
GRP	= 9.91.	(8)

Find the Gaia color

(9)

Problem 6: small-angle separation

Two resolved components have coordinate differences

Δα	= 0.80 arcsec,	(10)
Δδ	= 0.30 arcsec,	(11)

at mean declination

(12)

Estimate their angular separation.

Problem 7: position angle

Using the offsets from Problem 6, find the approximate position angle measured from north through east.

Problem 8: MJD conversion

Convert

(13)

to JD.

Problem 9: TESS time conversion

A TESS sample has

(14)

Convert to full

(15)

Problem 10: magnitude uncertainty

A flux measurement has

(16)

Estimate the magnitude uncertainty in the small-error approximation.

Problem 11: spectral resolution

A spectrograph operates at

(17)

Estimate:

1. Δλ at 600 nm,
2. the velocity scale c∕R.

Problem 12: third-light dilution

A binary would have an intrinsic fractional eclipse depth

(18)

relative to the binary’s own out-of-eclipse flux.

An unrelated source contributes

(19)

of the binary’s out-of-eclipse flux.

Find the observed eclipse depth.

Part II: Complete worked solutions

Solution 1

Use

(20)

Therefore

	= 10−0.4(2.5)	(21)
	= 10−1	(22)
	= 0.10 .	(23)

The brighter star provides ten times the flux of the fainter star in that passband.

Solution 2

Convert each magnitude to relative flux:

f1	= 10−0.4(8.0),	(24)
f2	= 10−0.4(9.0).	(25)

The total relative flux is

(26)

Factor out 10^−3.2:

ftot

= 10−3.2.

(27)

Thus

mtot	= 8.0 − 2.5 log 10	(28)
	≈ 7.636 .	(29)

The unresolved system is brighter than either component.

Solution 3

For equal component flux F,

(30)

Therefore

mtot − m	= −2.5 log 10(2)	(31)
	≈−0.7526.	(32)

Hence

(33)

Solution 4

The fractional depth is

δ	= 1 − 0.780	(34)
	= 0.220 .	(35)

The differential magnitude is

Δm	= −2.5 log 10(0.780)	(36)
	≈ 0.270 mag .	(37)

Solution 5

GBP − GRP	= 10.72 − 9.91	(38)
	= 0.81 mag .	(39)

Solution 6

The east-west tangent-plane offset is

Δx	= Δα cos δ	(40)
	= 0.80 cos 60∘	(41)
	= 0.40 arcsec.	(42)

The north-south offset is

(43)

Therefore

ρ	=	(44)
	= 0.50 arcsec .	(45)

Solution 7

Measured from north through east,

𝜃	= atan2(Δx, Δy)	(46)
	= atan2(0.40, 0.30)	(47)
	≈ 53.1∘ .	(48)

Solution 8

By definition,

(49)

Therefore

JD	= 60000 + 2400000.5	(50)
	= 2460000.5 .	(51)

Solution 9

Using

(52)

BJDTDB	= 2457000 + 1362.513789	(53)
	= 2458362.513789 .	(54)

Solution 10

σm	≈	(55)
	= 0.0021714 mag.	(56)

Therefore

(57)

Solution 11

The wavelength resolution is

Δλ	=	(58)
	=	(59)
	= 0.0200 nm .	(60)

Using

(61)

	=	(62)
	≈ 9.99 km∕s .	(63)

Solution 12

Let the binary out-of-eclipse flux be

(64)

The contaminating flux is

(65)

The intrinsic eclipse removes

(66)

The observed out-of-eclipse flux is

(67)

Therefore

δobs	=	(68)
	= 0.333 .	(69)

Third light has diluted a 40% intrinsic eclipse to about 33.3% in the observed aperture.

Part III: Frozen archive exercises

The file BIN02E1_frozen_photometry.csv

(70)

It preserves a small educational snapshot of public photometric values used in this sequence.

The file includes bright-system values from the SIMBAD snapshot already introduced in BIN01 and a separate AI Phe metadata file for the TESS data lab.

Exercise A: Algol flux ratios

Using the frozen Algol values

B	= 2.07,	(71)
V	= 2.12,	(72)
J	= 2.16,	(73)
H	= 1.95,	(74)
K	= 2.08,	(75)

compute

(76)

and compare the relative B- and V -band fluxes using the magnitude relation.

Do not interpret the result as a pure temperature measurement without considering

• the unresolved multiple-star nature of the system,
• passband calibration,
• orbital phase,
• extinction.

Exercise B: Sirius versus Algol in V

The frozen snapshot gives approximately

V Sirius	= −1.46,	(77)
V Algol	= 2.12.	(78)

The magnitude difference is

(79)

Therefore the ratio of their observed V -band fluxes is

	= 100.4(3.58)	(80)
	≈ 27.0 .	(81)

This is an observed flux ratio at Earth, not a luminosity ratio.

Part IV: TESS/MAST data lab with AI Phoenicis

AI Phoenicis is a bright detached eclipsing binary observed by TESS.

The frozen metadata file BIN02E1_AI_Phe_TESS_metadata.csv

(82)

records the observational facts needed to reproduce the introductory lab.

The important identifiers are

target	= AI Phe,	(83)
TIC	= 102069549,	(84)
sector	= 2.	(85)

Published analyses of these data report that TESS observed AI Phe in Sector 2 with two-minute cadence and that the sector covered both primary and secondary eclipse.

Lab 1: live archive search

The supplied Python script uses lightkurve to search MAST:

search = lk.search_lightcurve(
    "AI Phe",
    mission="TESS",
    sector=2
)

Record:

• the number of returned products,
• author or pipeline,
• cadence,
• sector,
• exposure time if reported,
• target identifiers.

Lab 2: inspect the light curve

Download one standard SPOC light-curve product.

Inspect

• time,
• SAP flux,
• PDCSAP flux,
• flux errors,
• quality flags.

Do not begin by plotting only the default flux column.

First determine what that column represents.

Lab 3: quality masking

Create a strict introductory mask using

(86)

Then compare the number of retained and rejected samples.

This is intentionally simple.

Later methods articles can adopt mission bitmasks tailored to a specific scientific question.

Lab 4: normalized flux

For the retained PDCSAP samples, define

(87)

Plot

(88)

against BTJD.

Identify the primary and secondary eclipse.

Lab 5: differential magnitudes

Convert the normalized flux to

(89)

Verify that

• out-of-eclipse points lie near Δm = 0,
• lower flux corresponds to larger differential magnitude.

Lab 6: SAP versus PDCSAP

Plot normalized SAP and PDCSAP flux on the same time interval.

Ask:

1. Which long-term structures differ?
2. Are the eclipse depths identical?
3. Could detrending affect a physical parameter inferred from the baseline?

The correct lesson is not that one column is universally superior.

The lesson is that pipeline processing choices are part of the measurement model.

Part V: FITS inspection

If the downloaded light curve is available locally, inspect it with

from astropy.io import fits

with fits.open(filename) as hdul:
    hdul.info()
    print(hdul[0].header)
    print(hdul[1].header)
    print(hdul[1].columns)

Record at least:

• number of HDUs,
• extension names,
• time-system keywords,
• target and sector identifiers,
• column units,
• quality column type.

A FITS light curve is not merely an array of numbers.

The header provides the context necessary to interpret those numbers.

Part VI: What should be saved

A reproducible submission for this lab should save:

1. the exact archive search result,
2. the downloaded product identifier,
3. the frozen metadata CSV,
4. the Python script and package versions,
5. the quality-mask rule,
6. one SAP-versus-PDCSAP plot,
7. one normalized eclipse plot,
8. one short text file describing the time convention.

Data-provenance note

The AI Phe metadata shipped with this lesson is a small frozen record assembled from the published TESS analysis of AI Phe and public mission documentation.

The frozen file is not a replacement for the MAST light-curve file.

Students should rerun the live query whenever network access is available and compare the returned metadata with the frozen record.

Summary

This companion establishes five operational habits:

1. convert magnitudes to linear flux before adding sources,
2. treat coordinates and time standards as part of the measurement,
3. carry uncertainty and quality information,
4. inspect FITS metadata before analysis,
5. preserve a live-query path and a frozen reproducibility checkpoint.

These habits will be reused throughout the entire BIN series.

References

References