Working With Astronomical Data: Tables, Units, Plots, and Uncertainty

What “Astronomical Data” Looks Like in Practice

Astronomical data is usually delivered as structured numbers: tables of measurements, time series, images with pixel values, and catalogs that combine many sources. Even when the original observation was complex, most analysis steps reduce to a few recurring tasks: reading a table, understanding the units, plotting relationships, and propagating uncertainty. This chapter focuses on those practical skills so you can reliably interpret and transform datasets without accidentally changing their meaning.

Two habits prevent most mistakes: (1) treat units as part of the data, not as decoration; (2) treat uncertainty as a first-class column, not as an afterthought. When you do that, your plots and derived quantities remain physically meaningful.

Tables and Catalogs: Columns, Metadata, and Common Pitfalls

Most astronomical datasets arrive as catalogs (tables) with one row per object or one row per measurement. You might see formats like CSV, TSV, FITS tables, or VOTable. Regardless of format, the same ideas apply: each column has a name, a meaning, a unit, and often a description in metadata. If you ignore metadata, you risk mixing incompatible quantities (for example, degrees vs radians, or flux vs magnitude).

Typical columns you will encounter

• Identifiers: object IDs, survey IDs, cross-match IDs.
• Coordinates: right ascension (RA) and declination (Dec), often in degrees; sometimes in sexagesimal strings.
• Time: observation time (MJD, JD, BJD), exposure start/stop, cadence.
• Photometry: fluxes, magnitudes, colors, and their uncertainties.
• Spectroscopy: radial velocity, redshift, line equivalent widths, and uncertainties.
• Quality flags: bitmasks or integer codes indicating issues (saturation, blending, poor fit).
• Derived parameters: temperatures, masses, distances, often with model assumptions.

Metadata matters

In well-formed catalogs, the unit and description are not only in the column header but also in metadata. FITS tables, for example, can store units in header keywords. When you load a table, check whether units are attached or whether you must assign them manually.

Common pitfalls with tables

• Silent unit mismatch: a column labeled “RA” might be in degrees in one table and in hours in another.
• Sentinel values: missing values may be encoded as -999, 99.99, NaN, or blank strings. Treat them consistently before computing statistics.
• Logarithmic columns: some columns store log10 of a quantity (e.g., logg, logL). Do not average logs unless that is what you intend.
• Flags ignored: quality flags often indicate that a measurement is unreliable. Filtering by flags can change results dramatically.
• Duplicate sources: cross-matched catalogs can contain multiple matches per object; decide how to choose or combine them.

Units: Conversions, Consistency, and Dimensional Thinking

Units are the grammar of physical meaning. In astronomy you frequently switch between angular units (degrees, arcminutes, arcseconds), time units (days, seconds), and physical units (meters, parsecs, km/s). A good workflow keeps everything consistent internally and only converts for presentation at the end.

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Common astronomical units and relationships

• Angles: 1 degree = 60 arcmin = 3600 arcsec.
• Time: 1 day = 86400 s.
• Distance: 1 parsec (pc) ≈ 3.0857×10¹⁶ m; 1 kpc = 1000 pc; 1 Mpc = 10⁶ pc.
• Speed: km/s is common for radial velocities.
• Flux and magnitude: flux is linear; magnitude is logarithmic (a difference of 5 mag corresponds to a factor of 100 in flux).

Dimensional checks as error detectors

Before trusting a derived column, do a dimensional check: if you compute a speed from distance/time, the result should have units of length/time. If you compute a luminosity from flux and distance, the result should scale as distance squared. These checks catch mistakes like using degrees where radians are required in trigonometric functions.

Practical step-by-step: unit hygiene in a table workflow

• Step 1: Identify units for every numeric column. If the file does not specify units, consult documentation and annotate them yourself.
• Step 2: Convert to a consistent internal unit system. For example: angles in radians for trig, distances in parsecs or meters, times in days or seconds.
• Step 3: Store the converted values as new columns rather than overwriting the originals. This preserves traceability.
• Step 4: When combining tables, convert both to the same units before merging or comparing.
• Step 5: Only at the plotting/reporting stage convert to “human-friendly” units (arcsec, km/s, etc.).

Plots: Seeing Structure Without Misleading Yourself

Plots are not decoration; they are diagnostic tools. In astronomy, a plot often reveals systematics, selection effects, or outliers that would be invisible in summary statistics. The most common plot types are scatter plots, histograms, time series, and residual plots (data minus model). The key is to choose axes, scales, and error display that match the physics and the measurement process.

Scatter plots with units and uncertainties

A scatter plot should communicate: what is measured, in what units, and how uncertain it is. If uncertainties are significant compared to the spread, add error bars. If there are many points, consider transparency (alpha) or density plots to avoid overplotting.

Log scales and why they are common

Many astronomical quantities span orders of magnitude (flux, luminosity, mass). Log scales compress dynamic range and often linearize power-law relationships. However, log scales require positive values; if your data includes zeros or negatives (common after background subtraction), you must handle them explicitly (mask, offset, or use a different visualization).

Histograms and selection effects

Histograms show distributions, but the binning choice can hide or invent features. Use multiple bin widths or use kernel density estimates when appropriate. Also remember that catalogs are often incomplete below a detection threshold, so the low end of a histogram may reflect selection rather than astrophysics.

Time series: cadence, gaps, and aliasing

Time series plots (brightness vs time, radial velocity vs time) should show the time unit and reference system (e.g., MJD). Gaps in data can create misleading apparent trends. If you later compute periods, irregular sampling can introduce aliases; a simple time plot is your first check for sampling issues.

Practical step-by-step: building an honest plot

• Step 1: Label axes with quantity and unit (e.g., “Radial velocity (km/s)”).
• Step 2: Decide linear vs log scale based on expected relationships and dynamic range.
• Step 3: Visualize uncertainties with error bars or shaded bands when they matter.
• Step 4: Inspect outliers and check whether they correlate with flags, low signal-to-noise, or edge-of-field positions.
• Step 5: Add a residual plot when comparing to a model; residuals reveal systematics better than the main plot.

Uncertainty: What It Is and How to Use It

Uncertainty quantifies how much a measurement could vary due to noise and limitations of the measurement process. In data tables, uncertainty often appears as a separate column (e.g., flux_err, mag_err, rv_err). Treat these columns as essential: they determine how strongly each point should influence fits and averages.

Random vs systematic uncertainty

• Random uncertainty varies from measurement to measurement (e.g., photon noise). It often decreases when you average many independent measurements.
• Systematic uncertainty shifts measurements in a correlated way (e.g., calibration offset). It does not necessarily decrease with averaging.

Catalogs often provide per-point random uncertainties but may not fully capture systematics. When you compare datasets from different instruments or surveys, systematics can dominate.

Standard deviation, standard error, and “sigma” language

In many contexts, uncertainties are reported as 1σ standard deviations assuming an approximately normal (Gaussian) error distribution. The standard deviation describes the spread of repeated measurements; the standard error describes the uncertainty on a mean (standard deviation divided by √N for independent points). Confusing these leads to under- or over-confident results.

Signal-to-noise ratio (SNR)

SNR is a compact way to assess measurement quality: SNR = value / uncertainty (for quantities where that makes sense, such as flux). Low SNR points can dominate scatter and bias fits if not handled with appropriate weighting or filtering.

Propagation of Uncertainty: From Measured Columns to Derived Quantities

When you compute a new quantity from measured ones, its uncertainty depends on the uncertainties of the inputs. A practical approach is to use standard propagation formulas for simple functions and to use Monte Carlo sampling for complex transformations.

Core propagation rules (independent uncertainties)

Let x and y be measured quantities with uncertainties σ_x and σ_y, assumed independent.

• Addition/subtraction: z = x ± y ⇒ σ_z = √(σ_x² + σ_y²)
• Multiplication/division: z = x·y or z = x/y ⇒ (σ_z/|z|) = √((σ_x/x)² + (σ_y/y)²)
• Power: z = xⁿ ⇒ (σ_z/|z|) = |n|·(σ_x/|x|)

These are approximations that work well when uncertainties are small compared to the values and the functions are smooth near the measured values.

Example: color index and its uncertainty

Suppose a star has magnitudes g and r with uncertainties σ_g and σ_r. The color is c = g − r. The uncertainty is σ_c = √(σ_g² + σ_r²). This is a common derived quantity in catalogs, and it illustrates why you should keep uncertainty columns: without them, you cannot tell whether a color difference is significant.

Example: converting parallax to distance (and why it is tricky)

Parallax p is often given in milliarcseconds (mas). A naive distance estimate is d(pc) = 1 / p(arcsec). If p is in mas, then d(pc) = 1000 / p(mas). For small fractional uncertainties, you can propagate: σ_d ≈ (σ_p/p)·d. However, when p is small or has large relative uncertainty, the inverse transformation becomes highly non-linear and can produce biased distances. In that regime, a Monte Carlo approach (sampling p from its uncertainty distribution and transforming each sample) is safer for representing asymmetric distance uncertainties.

Practical step-by-step: Monte Carlo uncertainty propagation

• Step 1: Choose the input distributions. Often you start with Gaussian distributions centered on the measured values with widths equal to the reported 1σ uncertainties.
• Step 2: Draw many samples (e.g., 10,000) for each input quantity.
• Step 3: Compute the derived quantity for each sample using the same formula you use for the nominal value.
• Step 4: Summarize the result using the median and percentiles (e.g., 16th and 84th percentiles for a “1σ-like” interval).
• Step 5: Plot the distribution if it is skewed; report asymmetric uncertainties when appropriate.

Weighted Statistics: Means, Fits, and Why Errors Change the Answer

If measurements have different uncertainties, treating them equally is usually wrong. Weighted statistics give more influence to more precise points.

Weighted mean

Given measurements x_i with uncertainties σ_i, a common choice of weights is w_i = 1/σ_i². The weighted mean is:

x̄ = (Σ w_i x_i) / (Σ w_i)

The uncertainty on the weighted mean (under standard assumptions) is:

σ_x̄ = 1 / sqrt(Σ w_i)

This is widely used for combining repeated measurements of the same quantity (for example, multiple radial velocity measurements of a stable star).

When weighting can mislead

If uncertainties are underestimated, weights become too large and the weighted mean becomes overconfident. A practical diagnostic is to compute the scatter of residuals relative to uncertainties. If residuals are systematically larger than expected, you may need to add an extra “jitter” term (an additional variance component) or revisit quality cuts.

Quality Flags, Masks, and Data Cleaning Without Guesswork

Cleaning data is not about making plots look nice; it is about ensuring that the remaining points satisfy the assumptions of your analysis. Astronomical catalogs often include flags that encode known issues. Use them.

Practical step-by-step: a defensible cleaning workflow

• Step 1: Start with a copy of the raw table and never overwrite it.
• Step 2: Define missing values (NaN, sentinel values) and convert them to a consistent representation.
• Step 3: Apply flag-based masks to remove measurements marked as saturated, blended, or otherwise problematic.
• Step 4: Apply uncertainty or SNR cuts (e.g., keep points with SNR > 5) if your analysis requires reliable measurements.
• Step 5: Track how many rows you remove at each step; large losses can indicate overly strict cuts or a misunderstanding of flags.
• Step 6: Re-plot after each major cut to confirm that you removed known artifacts rather than real structure.

Joining and Cross-Matching Tables: Keeping Meaning Intact

Many projects require combining information from multiple catalogs: photometry from one survey, astrometry from another, spectroscopy from a third. This introduces two main challenges: matching the same object across tables and reconciling differences in units, epochs, and definitions.

Key ideas when merging catalogs

• Join keys: sometimes you have a shared ID; often you do not.
• Coordinate-based matching: match objects by sky position within a radius (e.g., 1 arcsec). This requires consistent coordinate systems and epochs.
• Many-to-one matches: crowded fields can produce multiple candidates within the match radius; you need a rule (closest match, best quality, probabilistic match).
• Column name collisions: two tables may both have a column named mag but in different bands; rename columns before joining.

Practical step-by-step: safe table joining checklist

• Step 1: Standardize coordinate units (usually degrees) and confirm the reference frame if provided.
• Step 2: Decide a match radius based on positional uncertainties and source density.
• Step 3: Perform the match and record the separation for each match as a diagnostic column.
• Step 4: Inspect the separation distribution; a clean match often shows a strong peak near zero with a tail of chance alignments.
• Step 5: Resolve duplicates using a consistent rule and document it.
• Step 6: Harmonize units and definitions before computing combined quantities.

Reproducible Analysis: Documenting Transformations and Assumptions

Astronomical data analysis often involves many small transformations: unit conversions, filtering, derived columns, and plotting choices. If you cannot reproduce your own steps later, it is hard to trust the result. Reproducibility does not require complex tooling; it requires discipline in recording what you did.

Practical habits for reproducibility

• Keep a data dictionary: a short table describing each column, its unit, and its meaning after transformations.
• Record selection criteria: flags used, SNR thresholds, match radius, and any manual exclusions.
• Version derived tables: save intermediate outputs with clear names (e.g., catalog_clean_v1).
• Prefer explicit columns: store distance_pc and distance_err_pc rather than a single ambiguous d.
• Plot diagnostics: separation histograms for cross-matches, residual plots for fits, and uncertainty vs value plots to reveal heteroscedasticity.