Properties of light and error analysis

• astronomy - learn about objects outside of our atmosphere. need information from them to do so: light!

• light, by quantum mechanics, is photons, has characteristics of both waves and particles. Wavelength/frequency corresponds to energy:
  E = h$$\nu$$ = $${h c \over \lambda}$$

• electromagnetic spectrum: gamma rays - X rays - UV - optical - IR - mm - radio. Review numerical wavelengths of different parts of spectrum: optical, near-UV, far-UV, near-IR, far-IR.

• We can describe the amount of light an object emits or that we receive by three fundamental quantities: luminosity, L, intensity or surface brightness, I, and flux F.

• For most sources, amount of light emitted is a function of wavelength, so we actually are often interested in estimates of the monochromatic flux/intensity/luminosity, sometimes known as flux/intensity/luminosity density = flux / unit wavelength (or unit frequency), also sometimes known as specific flux/intensity/luminosity.

  F_$\nu$ : flux per unit frequency. F_$\lambda$ : flux per unit wavelength.

  $$\int$$F_$\nu$d$$\nu$$ = - $$\int$$F_$\lambda$d$$\lambda$$
  F_$\nu$ = - F_$\lambda$d$$\lambda$$/d$$\nu$$
  = F_$\lambda$$${c \over \nu^2}$$ = F_$\lambda$$${\lambda^2\over c}$$

  Similarly, intensity and luminosity can be given per unit wavelength (or frequency). Note that a constant F_$\lambda$ implies a non-constant F_$\nu$ and vice versa!

• The surface brightness is defined as the amount of energy transmitted through a unit surface element per unit time per unit frequency (or wavelength) into a unit solid angle in the direction ( $\theta$,$\phi$ ), where $\theta$ is the angle away from the normal to the surface element, and $\phi$ the azimuthal angle. The flux is the amount of energy passing through a unit surface element in all directions, defined by
  F_$\nu$ = $$\int$$I_$\nu$cos($$\theta$$)d$$\Omega$$

  where d$\Omega$ is the solid angle element, and the integration is over the entire solid angle.

  The luminosity is the intrinsic energy emitted by the source per second. Thus we can also consider it as the power. For an isotropically emitting source,

  L = 4$$\pi$$d²F

  where d = distance to source.

• What do we measure for sources? For unresolved sources, we measure the flux, usually over some bandpass or wavelength interval; for resolved sources we can directly measure their surface brightness (intensity) distribution on the sky. We can only calculate the luminosity to a source if we know the distance.

• Questions: how do the three quantities, L, I, and F depend on distance to the source? What are the units of these three quantities? To what quantity is apparent magnitude of a star related? To what quantity the absolute magnitude?

• Units: astronomers often (not always) work in CGS units, although, as discussed below, they most often work in a dimensionless units... magnitudes.
  • You may also run into the Jansky, a flux unit corresponding to 10^-26W/m²/Hz

• Note, there are variations in terminology, especially between disciplines (e.g. astronomy and engineering) and you should make sure you understand what is adopted by someone presenting data. While astronomers are reasonably consistent in the use of these terms, you may run into things like radiant flux, irradiance, radiance, spectral intensity, and others, which are all related to intensity or flux.

• Terminology of measurements:
  • photometry (broad-band flux measurement),
  • spectroscopy (relative measurement of fluxes at different wavelengths),
  • spectrophotometry (absolute measurement of fluxes at different wavelengths).
  • astrometry/morphology: concerned with positional dependence of observed flux; often, absolute measurements are unimportant

• Generally, we measure flux with photometry, and flux density with spectroscopy (down to the resolution of the spectrograph). Also, with most detectors, we measure photon flux (photon counting devices), rather than energy flux (bolometers). The photon flux is given by
  photonflux = flux/energyperphoton
  = $$\int$$F_$\lambda$$${\lambda \over h c}$$d$$\lambda$$

Magnitudes and photometric systems

In astronomy, however, magnitude units are often used instead of flux units, where the two are related by:

m = - 2.5 log$${F\over F_0}$$

or

m = - 2.5 log F + 2.5 log F₀

where the coefficient of proportionality, F₀ , depends on the definition of photometric system; often, the quantity -2.5 log F₀ is known as the zeropoint. Inverting, one gets:

F = F₀10^-0.4m

Note that it is possible that F₀ is a function of wavelength!

Note that since magnitudes are logarithmic, the difference between magnitudes corresponds to a ratio of fluxes; ratios of magnitudes are generally unphysical! If one is just doing relative measurements of brightness, this can be done without knowledge of F₀ (or the zeropoint). Often people talk about instrumental magnitudes, -2.5 log F ; this is ``instrumental" because it clearly depends on what instrument (and conditions) you are observing under (e.g. size of telescope, efficiency of telescope/detector, etc.).

The flux density for astronomical objects may be specified by a monochromatic magnitude:

F_$\lambda$ = F₀($$\lambda$$)10^{-0.4m($\lambda$)}

We start by describing two simple photometric systems, the STMAG and the ABNU mag system, which are conceptually easy; however, they are not currently the most widely used system in astronomy!

In the STMAG system, F₀ = 3.60E - 9ergs/cm²/s/Å , which is the flux of Vega at 5500Å ; hence a star of Vega's brightness at 5500Å is defined to have m=0. Alternatively, we can write

m = - 2.5 log F_$\lambda$ - 21.1

(for F_$\lambda$ in cgs units).

In the ABNU system, things are defined for F_$\nu$ instead of F_$\lambda$ , and we have

F_$\nu$ = 3.63 x 10^-20erg/cm²/s/Hz10^-0.4m_$\nu$

or

m = - 2.5 log F_$\nu$ - 48.6

(for F_$\nu$ in cgs units). Again, the constant comes from the flux of Vega.

Related are integrated (over a spectral bandpass) magnitude systems; the STMAG and ABMAG integrated systems are defined relative to sources of constant F_$\lambda$ and F_$\nu$ systems, respectively.

m_ST = - 2.5 log$${\int F_\lambda \lambda d\lambda\over \int 3.6\times 10^{-9} \lambda d\lambda}$$

(the factor of $\lambda$ comes in for photon counting detectors).

m_AB = - 2.5 log$${\int (F_\nu/\nu) d\nu\over \int (3.6\times 10^{-20}/\nu) d\nu}$$

Note that these systems differ by more than a constant, because one is defined by units of F_$\lambda$ and the other by F_$\nu$ , so the difference between the systems is a function of wavelength.

The standard UBVRI broadband photometric system, as well as several other magnitude systems, however, are not defined for a constant F_$\lambda$ or F_$\nu$ spectrum; rather, they are defined relative to the spectrum of an A0V star. Most systems are defined (or at least were originally) to have the magnitude of Vega be zero in all bandpasses (VEGAMAGS); if you ever get into this in detail, note that this is not exactly true for the UBVRI system.

For the broadband UBVRI system, we have

m_UBVRI $$\approx$$ -2.5 log$${\int_{UBVRI} F_\lambda(object) \lambda d\lambda\over \int_{UBVRI} F_\lambda(Vega) \lambda d\lambda}$$

(as above, the factor of $\lambda$ comes in for photon counting detectors).

Here is a plot to demonstrate the difference between the different systems.

Why do the different systems exist? While it seems that STMAG and ABNU systems are more straightforward, in practice it is difficult to measure absolute fluxes, and much easier to measure relative fluxes between objects. Hence, historically observations were tied to observations of Vega (or to stars which themselves were tied to Vega), so VEGAMAGS made sense, and the issue of determining physical fluxes boiled down to measuring the physical flux of Vega. Today, in some cases, it may be more accurate to measure the absolute throughput of an instrumental system, and using STMAG or ABNU makes more sense.

Working in magnitudes, the difference in magnitudes between different bandpasses (called the color index) is related to the flux ratio between the bandpasses, i.e., the color. In the UBVRI system, the difference between magnitudes gives the ratio of the fluxes in different bandpasses relative to the ratio of the fluxes of an A0V star in the different bandpasses (for VEGAMAG). Note the typical colors of astronomical objects - which are different for the different photometric systems!

Given UBVRI magnitudes of an object in the desired band, filter profiles (e.g. Bessell 1990, PASP 102,1181), and absolute spectrophotometry of Vega (e.g., Bohlin & Gilliland 2004, AJ 127, 3508, one can determine the flux. Alternatively, if one is given just the V magnitude of an object, a spectral energy distribution (SED), and the filter profiles, one can compute the integral of the SED over the V bandpass, determine the scaling by comparing with the integral of the Vega spectrum over the same bandpass, then use the normalized SED to compute the flux in any desired bandpass. Some possibly useful references for SEDs are: Bruzual, Persson, Gunn, & Stryker; Hunter, Christian, & Jacoby; Kurucz).

Things are certainly a bit simpler in the ABNU or STMAG system, and there has been some movement in this direction: the STScI gives STMAG calibrations for HST instruments, and the SDSS photometric system is close to an ABNU system.

Note, however, that even when the systems are conceptually well defined, determining the absolute calibration of any photometric system is very difficult in reality, and determining absolute fluxes to the 1% level is very challenging.

Note that some people, in particular, the SDSS, have adopted a modified type of magnitudes, called asinh magnitudes, which behave like normal (also known as Pogson) magnitude for brighter objects, but have different behavior for faint objects; see Lupton, Gunn, & Szalay 1999 AJ 118, 1406 for details.

Observed fluxes and the count equation

The intrinsic photon flux from the source is not trivial to determine from the number of photons that you count. To get the number of photons that you count in an observation, you need to take into account the area of your photon collector (telescope), photon losses and gains from the Earth's atmosphere (which changes with conditions), and the efficiency of your collection/detection apparatus (which can change with time). Generally, the astronomical signal (which might be a flux or a surface brightness, depending on whether the object is resolved) can be written

S = Tt$$\int$$$${F_\lambda\over{hc\over\lambda}}$$q_$\lambda$a_$\lambda$d$$\lambda$$

where T is the telescope collecting area, t is the integration time, a_$\lambda$ is the atmospheric transmission (more later) and q_$\lambda$ is the system efficiency. I refer to this as the count equation.

Usually, however, one doesn't use this information to go backward from S to F_$\lambda$ because it is very difficult to measure all of the terms precisely, and some of them (e.g. a and q are time-variable). Instead, most observations are performed differentially to a set of standard stars, which themselves have been (usually painfully) calibrated. We'll discuss this process in more depth later. Note that even this requires correcting for different atmospheric transmission between object(s) and standard(s). Sometimes just relative brightness between an object and another object in the field is measured (differential photometry), which is sufficient for variability studies, or even for absolute studies if the reference object is relatively bright and can have its absolute brightness determined at some other time (photometric weather) relatively quickly.

The count equation is used, however, very commonly, for computing the approximate number of photons you will receive from a given source in a given amount of time for a given observational setup. This number is critical to know in order to estimate your expected errors and exposure times in observing proposals, observing runs, etc. Understanding errors in absolutely critical in all sciences, and maybe even more so in astronomy, where objects are faint, photons are scarce, and errors are not at all insignificant. The count equation provides the basis for exposure time calculator (ETC) programs.

Errors in photon rates

For a given rate of emitted photons, there's a probability function which gives the number of photons we detect, even assuming 100% detection efficiency, because of statistical uncertainties. In addition, there may also be instrumental uncertainties. Consequently, we now turn to the concepts of probability distributions, with particular interest in the distribution which applies to the detection of photons.

Distributions and characteristics thereof

• concept of a distribution : define p(x)dx as probability of event occuring in x + dx :
  $$\int$$p(x)dx = 1

Some definitions relating to values which characterize a distribution:

mean $$\equiv$$ $$\mu$$ = $$\int$$xp(x)dx

variance $$\equiv$$ $$\sigma^{2}_{}$$ = $$\int$$(x - $$\mu$$)²p(x)dx

standarddeviation $$\equiv$$ $$\sigma$$ = $$\sqrt{{variance}}$$

median : mid-point value.

$${\int_{-\infty}^{x_{median}} p(x) dx \over \int_{-\infty}^{\infty} p(x) dx}$$ = $${1\over 2}$$

mode : most probable value

Note that the geometric interpretation of above quantities depends on the nature of the distribution; although we all carry around the picture of the mean and the variance for a Gaussian distribution, these pictures are not applicable to other distributions, but the quantities are still well-defined.

Also, note that there is a difference between the sample mean, variance, etc. and the population quantities. The latter apply to the true distribution, while the former are estimates of the latter from some finite sample (N measurements) of the population. The sample quantities are derived from:

sample mean : $$\bar{x}$$ $$\equiv$$ $${\sum x_i \over N}$$

sample variance $$\equiv$$ $${\sum(x_i - \bar x)^2 \over N-1}$$ = $${\sum x_i^2 - (\sum x_i)^2/N \over N-1}$$

The sample mean and variance approach the true mean and variance as N approaches infinity. But note for small samples, your estimate of the mean and variance may differ from their true (population) values (more below)!

Now we consider what distribution is appropriate for the detection of photons. The photon distribution can be derived from the binomial distribution, which gives the probability of observing the number, x , of some possible event, given a total number of events n , and the probability of observing the particular event (among all other possibilities) in any single event, p :

P(x, n, p) = $${n! p^x (1-p)^{n-x} \over x! (n-x)!}$$

For the binomial distribution, one can derive:

mean $$\equiv$$ $$\int$$xp(x)dx = np

variance $$\equiv$$ $$\sigma^{2}_{}$$ $$\equiv$$ $$\int$$(x - $$\mu$$)²p(x)dx = np(1 - p)

The Poisson distribution

In the case of detecting photons, n is the total number of photons emitted, and p is the probability of detecting a photon during our observation out of the total emitted. We don't know either of these numbers! However, we do know that p < < 1 and we know, or at least we can estimate, the mean number detected:

$$\mu$$ = np

.

In this limit, the binomial distribution asymtotically approaches the Poisson distribution:

P(x,$$\mu$$) = $${\mu^x \exp^{-\mu} \over x!}$$

From the expressions for the binomial distribution in this limit, the mean of the distribution is $\mu$ , and the variance is

variance = $$\sum_{x}^{}$$[(x - $$\mu$$)²p(x,$$\mu$$)]

variance = np = $$\mu$$

This is an important result.

Note that the Poisson distribution is generally the appropriate distribution not only for counting photons, but for any sort of counting experiment where a series of events occurs with a known average rate, and are independent of time since the last event.

What does the Poisson distribution look like? Plots for $\mu$ = 2,$\mu$ = 10,$\mu$ = 10000 .

The normal, or Gaussian, distribution

Note, for large $\mu$ , the Poisson distribution is well-approximated around the peak by a Gaussian, or normal distribution:

P(x,$$\mu$$,$$\sigma$$) = $${1 \over \sqrt{2\pi} \sigma}$$e^{${-{(x-\mu)^2\over 2\sigma^2}}$}

This is important because it allows us to use ``simple'' least squares techniques to fit observational data, because these generally assume normally distributed data. However, beware that in the tails of the distribution, and at low mean rates, the Poisson distribution can differ significantly from a Gaussian distribution. In these cases, least-squares may not be appropriate to model observational data; instead, one might need to consider maximum likelihood techniques instead.

The normal distribution is also important because many physical variables seem to be distributed accordingly. This may not be an accident because of the central limit theorem: if a quantity depends on a number of independent random variables with ANY distribution, the quantity itself will be distributed normally (see statistics texts). In observations, we encounter the normal distribution because one important source of instrumental noise, readout noise, is distributed normally.

Importance of error distribution analysis

You need to understand the expected errors in your observations in order to:

• predict the amount of observing time you'll need to get errors as small as you need them to do your science,

• answer the question: is observed data consistent with expected errors? If the answer is no, they you know you've either learned some astrophysics or you don't understand something about your observations. This is especially important in astronomy where objects are faint and many projects are pushing down into the noise as far as possible. Really we can usually only answer this probabilistically. Generally, tests compute the probability that the observations are consistent with the expected distribution (the null hypothesis). You can then look to see if this probability is low, and if so, you can reject the null hypothesis.

Confidence levels

For example, say we want to know whether some single point is consistent with expectations, e.g., we see a bright point in multiple measurements of a star, and want to know whether the star flared. Say we have a time sequence with known mean and variance, and we obtain a new point, and want to know whether it is consistent with known distribution?

If the form of the probability distribution is known, then you can calculate the probability of getting a measurement more than some observed distance from the mean. In the case where the observed distribution is Gaussian (or approximately so), this is done using the error function (sometimes called erf(x)), which is the integral of a gaussian from some starting value.

Some simple guidelines to keep in mind follow (the actual situation often requires more sophisticated statistical tests). First, for Gaussian distributions, you can calculate that 68% of the points should fall within plus or minus one sigma from the mean, and 95.3% between plus or minus two sigma from the mean. Thus, if you have a time line of photon fluxes for a star, with N observed points, and a photon noise $\sigma$ on each measurement, you can test whether the number of points deviating more than 2$\sigma$ from the mean is much larger than expected. To decide whether any single point is really significantly different, you might want to use more stringent criterion, e.g., 5$\sigma$ rather than a 2$\sigma$ criterion; a 5$\sigma$ has much higher level of significance. On the other hand, if you have far more points in the range 2 - -4$\sigma$ brighter or fainter than you would expect, you may also have a significant detection of intensity variations (provided you really understand your uncertainties on the measurements!).

Also, note that your observed distribution should be consistent with your error estimates given the above guidelines. If you have a whole set of points, that all fall within 1$\sigma$ of each other, something is wrong with your error estimates (or perhaps your measurements are correlated with each other)!

Noise equation: how do we predict expected errors?

Signal noise

Astronomers often describe errors in terms of the fractional error, e.g. the amplitude of the error divided by the amplitude of the quantity being measured; often, the inverse of this, referred to as the signal-to-noise ratio is used. Given an estimate the number of photons expected from an object in an observation, we can calulate the signal-to-noise ratio:

$${S\over N}$$ = $${S \over \sqrt{\sigma^2}}$$

which is the inverse of the predicted fractional error (N/S ).

Consider an object with photon flux (per unit area and time), S^$\prime$ , leading to a signal, S = S^$\prime$Tt where T is the telescope area and t is the exposure time. In the simplest case, the only noise source is Poisson statistics from the source, in which case:

$$\sigma^{2}_{}$$ = S = S^$\prime$Tt

$${S\over N}$$ = $$\sqrt{{S}}$$ = $$\sqrt{{S^\prime T t}}$$

In other words, the S/N increases as the square root of the object brightness, telescope area, efficiency, or exposure time. Note that S is directly observable, so one can calculate the S/N for an observation without knowing the telescope area or exposure time! We've just broken S down so that you can specifically see the dependence on telescope area and/or exposure time.

Background noise

A more realistic case includes the noise contributed from Poisson statistics of background light (more on the physical nature of this later), B^$\prime$ , which has units of flux per area on the sky (i.e. a surface brightness); note that this is also usually given in magnitudes.

B^$\prime$ = $$\int$$$${B_\lambda\over{hc\over\lambda}}$$q_$\lambda$d$$\lambda$$

The amount of background in our measurement depends on how we choose to make the measurement (how much sky area we observe). Say we just use an aperture with area, A , so the total observed background counts is

B = B^$\prime$ATt

Again, B^$\prime$Tt is the directly observable quantity, but we split it into the quantities on which it depends to understand what factors are important in determining S/N .

The total number of photons observed, O , is

O = S + B = (S^$\prime$ + B^$\prime$A)Tt

The variance of the total observed counts, from Poisson statistics, is:

$$\sigma_{O}^{2}$$ = O = S + B = (S^$\prime$Tt + B^$\prime$ATt)

To get the desired signal from the object only , we will need to measure separately the total signal and the background signal to estimate:

S $$\equiv$$ S^$\prime$Tt = O - B

where B is some estimate we have obtained of the background level multiplied by the area A ( B = B^$\prime$ATt ). The noise in the measurement is

$$\sigma_{S}^{2}$$ $$\approx$$ $$\sigma_{O}^{2}$$ = (S^$\prime$ + B^$\prime$A)Tt

where the approximation is accurate if the background is determined to high accuracy, which one can do if one measures the background over a large area, thus getting a large number of background counts (with correspondingly small fractional error in the measurement).

This leads to a common form of the noise equation:

$${S\over N}$$ = $${S \over \sqrt{S + B}}$$

Breaking out the dependence on exposure time and telescope area, this is:

$${S\over N}$$ = $${S^\prime \over \sqrt{S^\prime + B^\prime A}}$$$$\sqrt{{T}}$$$$\sqrt{{t}}$$

In the signal-limited case, S^$\prime$ > > B^$\prime$A , we get

$${S\over N}$$ = $$\sqrt{{S}}$$ = $$\sqrt{{S^\prime t T}}$$

In the background limited case, B^$\prime$A > > S^$\prime$ , and

$${S\over N}$$ = $${S\over \sqrt{B}}$$ = $${S^\prime\over \sqrt{B^\prime A}}$$$$\sqrt{{t T}}$$

As one goes to fainter objects, the S/N drop, and it drops faster when you're background limited. This illustrates the importance of dark-sky sites, and also the importance of good image quality.

Consider two telescopes of collecting area, T₁ and T₂ . If we observe for the same exposure time on each and want to know how much fainter we can see with the larger telescope at a given S/N, we find:

S₂ = $${T_1\over T_2}$$S₁

for the signal-limited case, but

S₂ = $$\sqrt{{T_1\over T_2}}$$S₁

for the background-limited case.

Instrumental noise

In addition to the errors from Poisson statistics (statistical noise), there may be additional terms from instrumental errors. A common example of this that is applicable for CCD detectors is readout noise, which is additive noise (with zero mean) that comes from the detector and is independent of signal level. For a detector whose readout noise is characterized by $\sigma_{{rn}}^{}$ ,

$${S\over N}$$ = $${S \over \sqrt{S + B A + \sigma_{rn}^2}}$$

if a measurement is made in a single pixel. If an object is spread over N_pix pixels, then

$${S\over N}$$ = $${S \over \sqrt{S + B A + N_{pix}\sigma_{rn}^2}}$$

For large $\sigma_{{rn}}^{}$ , the behavior is the same as the background limited case. This makes it clear that if you have readout noise, image quality (and/or proper optics to keep an object from covering too many pixels) is important for maximizing S/N. It is also clear that it is critical to have minimum read-noise for low background applications (e.g., spectroscopy).

There are other possible additional terms in the noise equation, arising from things like dark current, digitization noise, errors in sky determination, errors from photometric technique, etc. (we'll discuss some of these later on), but in most applications, the three sources discussed so far - signal noise, background noise, and readout noise - are the dominant noise sources.

Error propagation

So now we know how to estimate uncertainties of observed count rates. Let's say we want to make some calculations (e.g., calibration, unit conversion, averaging, conversion to magnitudes, calculation of colors, etc.) using these observations: we need to be able to estimate the uncertainties in the calculated quantities that depend on our measured quantities.

Consider what happens if you have several known quantities with known error distributions and you combine these into some new quantity: we wish to know what the error is in the new quantity.

x = f (u, v,....)

The question is what is $\sigma_{x}^{}$ if we know $\sigma_{u}^{}$ , $\sigma_{v}^{}$ , etc.?

As long as errors are small:

x_i - < x > $$\sim$$ (u_i - < u > )$$\left(\vphantom{{{\partial{x}}\over{\partial {u}}}}\right.$$$${{\partial{x}}\over{\partial {u}}}$$$$\left.\vphantom{{{\partial{x}}\over{\partial {u}}}}\right)$$ + (v_i - < v > )$$\left(\vphantom{{{\partial{x}}\over{\partial {v}}}}\right.$$$${{\partial{x}}\over{\partial {v}}}$$$$\left.\vphantom{{{\partial{x}}\over{\partial {v}}}}\right)$$ + ...

$$\sigma_{x}^{2}$$ = lim(N - > $$\infty$$)$${1\over N}$$$$\sum$$(x_i - < x > )²

= $$\sigma_{u}^{2}$$$$\left(\vphantom{{{\partial{x}}\over{\partial {u}}}}\right.$$$${{\partial{x}}\over{\partial {u}}}$$$$\left.\vphantom{{{\partial{x}}\over{\partial {u}}}}\right)^{2}_{}$$ + $$\sigma_{v}^{2}$$$$\left(\vphantom{{{\partial{x}}\over{\partial {v}}}}\right.$$$${{\partial{x}}\over{\partial {v}}}$$$$\left.\vphantom{{{\partial{x}}\over{\partial {v}}}}\right)^{2}_{}$$ +2$$\sigma_{{uv}}^{2}$$$${{\partial{x}}\over{\partial {u}}}$$$${{\partial{x}}\over{\partial {v}}}$$ + ...

The last term is the covariance, which relates to whether errors are /textitcorrelated.

$$\sigma_{{uv}}^{2}$$ = lim(n - > $$\infty$$)$${1\over N}$$$$\sum$$(u_i - < u > )(v_i - < v > )

If errors are uncorrelated, then $\sigma_{{uv}}^{}$ = 0 because there's equal chance of getting opposite signs on v_i for any given u_i .

Examples:

• addition/subtraction:
  x = u + v
  $$\sigma_{x}^{2}$$ = $$\sigma_{u}^{2}$$ + $$\sigma_{v}^{2}$$ +2$$\sigma_{{uv}}^{2}$$

  In this case, errors are said to add in quadrature.

• multiplication/division:
  x = uv
  $$\sigma_{x}^{2}$$ = v²$$\sigma_{u}^{2}$$ + u²$$\sigma_{v}^{2}$$ +2uv$$\sigma_{{uv}}^{2}$$

• logs:
  x = ln u
  $$\sigma_{x}^{2}$$ = $$\sigma_{u}^{2}$$/u²

Distribution of resultant errors

When propagating errors, even though you can calculate the variances in the new variables, the distribution of errors in the new variables is not, in general, the same as the distribution of errors in the original variables, e.g. if errors in individual variables are normally distributed, errors in output variable is not necessarily.

When two variables are added, however, the output is normally distributed.

Determining sample parameters: averaging measurements

We've covered errors in single measurements. Next we turn to averaging measurements. Say we have multiple observations and want the best estimate of the mean and variance of the population, e.g. multiple measurements of stellar brightness. Here we'll define the best estimate of the mean as the value which maximizes the likelihood that our estimate equals the true parent population mean.

For equal errors, this estimate just gives our normal expression for the sample mean:

$$\mu$$ = $${\sum x_i \over N}$$

Using error propagation, the estimate of the error in the sample mean is given by:

$$\sigma_{{\mu}}^{2}$$ = $$\sum$$$${\sigma_i^2\over N^2}$$ = $${\sigma^2\over N}$$

But what if errors on each observation aren't equal, say for example we have observations made with several different exposure times? Then we determine the sample mean using a:

weightedmean = $${\sum {x_i\over\sigma_i^2} \over \sum {1\over\sigma_i^2}}$$

and the estimated error in this value is given by:

$$\sigma_{\mu}^{2}$$ = $$\sum$$$${{\sigma_i^2\over \sigma_i^4} \over ({\sum {1\over \sigma_i^2}})^2}$$ = $${1 \over \sum{1\over\sigma_i^2}}$$

where the $\sigma_{i}^{}$ 's are individual weights.

This is a standard result for determining sample means from a set of observations with different weights.

However, there can sometimes be a subtlety in applying this formula, which has to do with the question: how do we go about choosing the weights, $\sigma_{i}^{}$ ? We know we can estimate $\sigma$ using Poisson statistics for a given count rate, but remember that this is a sample variance (which may be based on a single observation!) not the true variance. This can lead to biases.

Consider observations of a star made on three nights, with measurements of 40, 50, and 60 photons. It's clear that the mean observation is 50 photons. However, beware of the being trapped by your error estimates. From each observation alone, you would estimate errors of $\sqrt{{40}}$ , $\sqrt{{50}}$ , and $\sqrt{{60}}$ . If you plug these error estimates into a computation of the weighted mean, you'll get a mean rate of 48.64!

Using the individual estimates of the variances, we'll bias values to lower rates, since these will have estimated higher S/N.

Note that it's pretty obvious from this example that you should just weight all observations equally. However, note that this certainly isn't always the right thing to do. For example, consider the situation in which you have three exposures of different exposure times. Here you clearly want to give the longer exposures higher weight. In this case, you again don't want to use the individual error estimates or you'll introduce a bias. There is a simple solution here also: just weight the observations by the exposure time. This works fine for Poisson errors (variances proportional to count rate), but not if there are instrumental errors as well which don't scale with exposure time. For example, the presence of readout noise can have this effect. With readout noise, the longer exposures should be weighted even higher than expected for the exposure time ratios. The only way to properly average measurements in this case is to estimate a sample mean, then use this value scaled to the appropriate exposure times as the input for the Poisson errors.

Can you split exposures?

Although from S/N considerations, one can determine the required number of counts you need (exposure time) to do your science, when observing, one must also consider the question of whether this time should be collected in single or in multiple exposures, i.e. how long individual exposures should be. There are several reasons why one might imagine that it is nicer to have a sequence of shorter exposures rather than one single longer exposure (e.g., tracking, monitoring of photometric conditions, cosmic ray rejection), so we need to consider whether doing this results in poorer S/N.

Consider the object with photon flux S^$\prime$ , background surface brightness B^$\prime$ , and detector with readout noise $\sigma_{{rn}}^{}$ . A single short exposure of time t has a variance:

$$\sigma_{S}^{2}$$ = S^$\prime$Tt + B^$\prime$ATt + N_pix$$\sigma_{{rn}}^{2}$$

If N exposures are summed, the resulting variance will be

$$\sigma_{N}^{2}$$ = N$$\sigma_{S}^{2}$$

If a single long exposure of length Nt is taken, we get

$$\sigma_{L}^{2}$$ = S^$\prime$TNt + B^$\prime$ATNt + N_pix$$\sigma_{{rn}}^{2}$$.

The ratio of the noises, or the inverse ratio of the S/N (since the total signal measured is the same in both cases), is

$${\sigma_N\over \sigma_L}$$ = $$\sqrt{{ S^\prime T Nt + B^\prime A T Nt + NN_{pix} \sigma_{rn}^2 \over S^\prime T Nt + B^\prime A T Nt + N_{pix} \sigma_{rn}^2 }}$$

In the signal- or background-limited regimes, exposures can be added with no loss of S/N. However, if readout noise is significant, then splitting exposures leads to reduced S/N.

Random errors vs systematic errors

So far, we've been discussing random errors. There is an additional, usually more troublesome, type of errors known as systematic errors. These don't occur randomly but rather are correlated with some, possibly unknown, variable relating to your observations.

EXAMPLE : flat fielding

EXAMPLE : WFPC2 CTE

Note also that in some cases, systematic errors can masquerade as random errors in your test observations (or be missing altogether if you don't take data in exactly the same way), but actually be systematic in your science observations.

EXAMPLE: flat fielding, subpixel QE variations.

Note that error analysis from expected random errors may be the only clue you get to discovering systematic errors. To discover systematic errors, plot residuals vs. everything!