The Transit Photometry Method

This is by far the most effective method to find new exoplanets, and the majority of newly discovered exoplanets (particularly by space programs like Kepler) have been discovered this way.

Rather than look for the reflected light from an exoplanet, it relies on the fact that as an exoplanet orbits their host star, at some point, and if they are in the direct line-of-sight, the exoplanet obscures a small part of the host star as it transits. This causes a measurable drop in the light output, or flux, of the star, which cannot be explained by other means. In effect, as the exoplanet crosses in front of the host star, a “transit dip” occurs that continues for the life of the transit. At the end of the transit period, the host star’s flux rapidly increases back to its normal value.

After discounting other causes (stars can vary in flux for a host of other reasons) the shape of the flux-curve indicates whether an exoplanet has crossed in front of its star. A typical transit curve looks like that in fig.1. It is characterized by a very fast small drop in flux as the planet crosses in front of the star, that extends for the period of time that the exoplanet partially occludes the host star. As the exoplanet exits occlusion, the flux returns quickly to its prior steady-state value.

The shape of the transit dip is the biggest indicator that an exoplanet has crossed in front of a star. If the shape is “messy”, it may indicate that more than one exoplanet is crossing the host star at roughly the same time, or it could be the dip in flux is being caused by other means. Other explanations include binary or variable stars.

The second element that is an exoplanet give-away, is the period at which these dips occur. For example, if we see this distinctive dip on a regular basis, it indicates that the probable reason why a star’s flux is being reduced, is because an exoplanet is regularly orbiting in front of it. If the period between dips is not regular, it becomes less clear that an exoplanet is responsible for that dip, since it becomes difficult to imagine a solar orbit that would cause an irregular dip in flux (although it could happen if an exoplanet had an erratic orbit).

For these reasons, most obvious exoplanets are identified by a clear, regular dip in flux at predictable intervals. This matches to the regular orbit of a large exoplanet that is orbiting very close to its host star; i.e. closer than the orbit of Mercury to the Sun. Invariably, such large exoplanets are called “super Jupiters” (likely having similar characteristics as Jupiter, but larger).

These exoplanets, collectively known as “hot Jupiters”, are by far the most common exoplanet found so far. Unfortunately, many exoplanets are neither conveniently close to their host star or are very large in size, making it difficult to detect them (the dip in flux is extremely small and infrequent). For these exoplanets, the technology you use and the quality of the observations are critical for robust detection.

This means that as amateurs, although it is possible to identify exoplanets smaller than hot Jupiters, it becomes increasingly difficult to do so, since the relative drop in flux is much smaller and more difficult to measure precisely.

The Wobble Method

The second method to observe exoplanets, commonly known as the “Wobble Method”, requires such detailed observations that at this time, it is mostly the domain of a space telescope system. It requires precise spectrographic data to measure minute changes in the velocity of the star, caused by the gravitational pull of its orbiting exoplanet.

Differential Photometry

Before setting out to find hot Jupiters, one needs to understand how to go about it in some detail. It may seem easy enough but there are sever al pitfalls that catch out the unprepared. To accurately measure the amount of flux generated by a host star, we use a technique called differential photometry. If you have never done photometry before, do not worry, it is fairly straightforward; being realistic though, if you are not a proficient imager it is likely you will struggle to achieve reliable results.

This process also places high demands on equipment and before starting, one should check that you have the right calibre to make effective measurements. By and large, in order to observe and/or find new exoplanets, you ideally need the following, over and above regular imaging equipment:

• a high quality robotic mount, camera rotator and an auto-focus system

• a large aperture, quality optic (e.g. 14-inch+ Ritchey-Chrétien or Dall-Kirkham, or 6-inch refractor)

• quiet, small pixel CCD, (e.g. based on Sony HAD ICX834, ICX814 sensors)

• photometric filters (e.g. Johnson-Cousins V, B and Sloan g’, r’ i; Astrodon filters are recommended, since they are used by many NASA programs.)

• access to an exoplanet data reduction analysis tool: (e.g. one recommended by either the AAVSO or the BAA/RAS)

These will allow you to observe most exoplanets currently discovered (seeing-conditions permitting). If you do not have this level of equipment, however, one can still successfully observe known exoplanet transits, but only for the very brightest host stars.

fig.1 This shows a “large” exoplanet transiting across its host star. As it passes in front of the star, the total light output (flux) dips sharply, plateaus and then increases sharply again to its prior level, as the exoplanet passes out the line of sight. The flux reduction is tiny compared to the measurement variation between individual samples (shown with x).

Theory

This process consists of taking many, relatively short exposures using a low-noise CCD, ideally with very small pixels, before, during and ideally after the exoplanet transit. Although the general use of differential photometry does not necessarily require these CCD attributes, it makes life much easier if you do have them. At the same time, you need to be capable of taking exposures with minimal/zero drift, accurate tracking and be well-versed with plate solving, calibration and spreadsheets. As long as a single exposure does not over-expose a star, we can define a circle around each star, called an “aperture”, that we use to measure the flux that it generated for each exposure.

In practice, each star is surrounded by a common-sized aperture, that defines the area of the star (or in fact the precise pixels within the star) that are used to calculate the star’s flux measurement. Each star, at a near infinite imaging distance, should theoretically be a point of light. Optical effects (convolution) cause each star profile to be represented as a Gaussian blur (also called the point spreading function or PSF). The peak of the distribution represents the center of the star at its brightest point and as we move away from the center, the intensity rapidly drops. This continues until the star’s flux is indistinguishable from the background sky flux and the star merges into the background, as shown in fig.2.

This aperture defines the common area around each star that we define “is part of the star”. That is, we define a threshold and deem that everything above is part of the star, and everything below is not, and is part of the background sky. This may not be 100% accurate from a Physics standpoint, but it is accurate enough to enable us to perform the correct math.

To create a transit curve, similar to the one in fig.1, we measure the flux of the host star accurately in each image, allowing for any atmospheric differences that may occur at the time. The easiest way to do this is not by measuring an absolute flux of a host star (target star) but the relative change, or flux delta, from one image to another.

To do this, we measure the total flux of all pixels deemed to be part of a star, and compare it to the flux generated in a different comparison star, which we know to be stable and invariant. For example:

• If most stars dim by 0.01% from one image to another, the dimming is probably caused by atmospheric changes.

• If only the host star dims by 0.01% from one image to another, then it is more likely that the dimming is caused by an exoplanet transit.

The comparison star’s flux is used as the baseline measurement from one image to the next. If we always adjust the flux of the host star to make it relative to the comparison star, then it automatically compensates for changes in flux caused by other environmental factors. In this way, it can produce a numeric value of the flux generated by the host star from image to image, that excludes all factors, other than the host star dimming due to an exoplanet passing in front of it. This logic holds true as long as the comparison star is close to the host star, of similar surface temperature and of the same spectral type. In plain English, this only works if we compare apples with apples. So for example, if you use a comparison star that has a different color temperature, you will introduce an unknown variable into the comparison process, rendering the results unreliable and invalid for submission.

Acquiring Images

Let us assume you have decided to try this exciting pursuit by testing your observing skills on a known exoplanet. Unlike long-exposure imaging, or even regular variablestar photometric observations, exoplanet photometry is more challenging because the flux variation from the bottom to the top of a transit curve is usually 0.2% or less. In fact, since it requires multiple images across the transit period, it is potentially measuring milli-magnitude values; tiny changes in flux from one image to another, which are difficult to differentiate from the background noise generated by the camera and sky noise.

Having said that, if one follows the following process, you can achieve clear and value-adding exoplanet transit curves. It is essential to test your process on known exoplanets before trying to attempting to find new ones. Once you can replicate the accuracy of the transit curves produced by professionals, you can then start searching on your own. The remainder of this chapter assumes you initially observe a known exoplanet.

Identify Which Exoplanet to Observe

It may seem trivial but you would be surprised at the number of amateurs who select an inappropriate exoplanet and then struggle. Most amateurs use the following websites to select (known) exoplanets: NASA, the Exoplanet Transit Database (ETD) and other more specific ones e.g. the AAVSO exoplanet database. Each database usually predicts when a transit will occur for a given latitude and longitude, the depth of the transit and its duration. Use this information to plan the observing session, allowing for any visual obstructions as you track the star and in particular, planning for meridian flips.

To correctly observe a transit, one not only needs to choose a host star with a known transit during your observing session, but also conform to the following conditions:

• The observing session captures the transit itself and a minimum of one hour on each side of the transit (two hours are better). This monitors the star when the flux-curve is flat and reveals when the flux-curve drops, making the transit curve more obvious.

• Select an exoplanet which has the largest delta in flux, i.e. has the deepest transit available to your location. This makes the measurement process easier, and since you are attempting to discover how the process of observing exoplanets operates, this significantly simplifies matters. Unless you have a 14-inch telescope (or larger) avoid selecting smaller “rocky Earths” sized exoplanets, where the corresponding dip in flux is closer to 0.06% or less.

• Select an exoplanet where the host star’s magnitude is within the visible range of your optics. It is difficult to observe a host star of magnitude 8 or 9, with a 14-inch telescope, as the star is easily over-exposed. Conversely, the intensity of a magnitude 13 host star with a 6-inch refractor is too weak and requires a long exposure to achieve an acceptable SNR (8/10 or better). This reduces the number of image events during the transit. Determine the brightest star magnitude that does not over-expose the CCD after an exposure of 60 seconds or so.

• Set the exposure within sensible limits, ensuring it does not saturate the CCD. If a transit’s elapsed time is 2 hours, 10-minute exposures are too long. Equally, short exposures of less than 5 seconds are not recommended either, due to exposure inconsistencies. 340

• Most CCDs become non-linear before they reach their maximum ADU. If a target or comparison star flux measurement becomes non-linear, a comparison is no longer valid. Fine-tune the exposure to keep all within the linear region; a rule of thumb assumes Sony HAD CCDs are linear up to about 85% of their ADU range and Kodak CCDs up to about 40–50%.

Setup Equipment for Photometry

Unlike long-exposure imaging, photometry requires one to be very careful about the quality of the captured data. In essence, whereas a galaxy or nebula is unlikely to change from night-to-night (unless there is a super-nova of course), an exoplanet transit will be changing every minute, so the tricks of the trade you might use to image deep sky objects, are not valid for imaging exoplanets.

Optimize Signal to Noise Ratio (SNR)

An image of a galaxy or nebula can be enhanced by taking many hours of images, to increase the SNR to 8/10 or better. Although it is possible to observe exoplanet transits with an SNR of 7/10, it is not recommended, since it is very close to the noise level of current CCD technology.

The process that calibrates, registers and integrates multiple images to produce a single image with a high SNR is not appropriate for exoplanet observing, as the host star’s flux will be varying second by second. The act of integrating multiple images averages the flux and introduces an approximation error that is virtually impossible to deconstruct later.

The aim is, however, to achieve a SNR of 8/10 or better, without integrating multiple images. This can be achieved partially by the use of low-noise CCDs, but also, through the use of calibration files taken in-situ.

Create Flats Before Session

It is good practice to take flats in advance of the session, with the rotator and camera in their observing session? positions (angle and focus extension). Ideally, these flats can be used, together with a second set taken post? meridian-flip, to calibrate the images from either side of the meridian.

Create Bias and Dark Frames Mid/Late Session

For normal imaging it is common practice to re-use bias and dark frames between sessions, for up to two or three months at a time or longer. It assumes the CCD does not vary significantly during this period. For exoplanet observing it is imperative to capture bias and dark frames, with conditions as close as possible to those when the data images were taken. This typically requires capturing calibration frames just before or after a transit observing session, so that the camera’s electronic characteristics are identical to those during the data capture process.

It is critical that bias and dark frames are taken with the same duration and CCD temperature setting as the data images. As a consequence, this occurs after the exact exposure duration (that balances over-exposure and a decent SNR) has been established.

Select Photometric Filters

Although one can perform exoplanet observing with regular RGB filters (particularly R), in practice, it is much more valuable if you use photometric filters designed for scientific use. Many filter options exist, but it is useful to have what we refer to as the APASS photometric filter set. APASS is the most accurate photometric star survey carried out to date (by the AAVSO), with the aim to identify all stars that can be used as comparison stars across the entire sky.

APASS used a specific set of Astrodon filters, specified to be Johnson-Cousins V and B filters, and Sloan g’, r’ and i’ filters. For ease of comparison, it makes sense to use the same filter set. The UCAC4 database contains highly accurate magnitude values through those filters, for the 56 million APASS stars, making the task of finding a reliable comparison star much easier. In practice, if one uses any filter other than blue, it will maximize the result of observing an exoplanet. (Blue light is absorbed by the atmosphere, so the flux data captured for a star can be compromised by atmospheric absorption.) This is why some amateurs use Johnson-Cousins R, or Sloan r’ (red) photometric filters, or the Astrodon non-blue exoplanet filter.

Select the Correct Field of View

The selection of the correct comparison star is critical to the generation of accurate and reliable transit-curves. To this end, it is important that the CCD’s Field of View (FOV) is positioned so that it allows one to simultaneously capture target star data as well as identifiable and appropriate comparison stars. In some cases, depending on the size of the FOV, it may require an off-center target host star, to include nearby comparison stars of a similar surface temperature, of the same spectral type and which are not variable stars. The transit period should also occur within the observing period.

Select the Correct Exposure Time

For the same reason as selecting the appropriate field of view (i.e. the inclusion of comparison stars), the exposure time is carefully optimized so that it maximizes the number of photons captured during each image, without exceeding the linearity limit of the CCD. Assuming that your CCD’s linearity-loss point (the ADU value, above which the sensor response is non-linear) has been correctly calculated, the final exposure must not have any pixels, within a target or comparison star, that exceed this ADU value. Since these stars may significantly vary in magnitude, select an exposure time that creates the correct balance between not over-exposing any star, and maximizing the number of photons captured per pixel for the target’s host star.

For example, let us assume the target’s host star magnitude is 13.2, there are several nearby comparison stars that range in magnitude from 12.0 to 13.0 and the full depth of a transit curve, results in a 0.2% magnitude change. The target star’s pixel ADU, will similarly vary by 0.2%. Assuming the CCD’s linearity will only allow you to measure ADU values up to 55,000, then in order to not over-expose any comparison star, the maximum target star ADU is:

and the maximum transit depth ADU is:

To continue the example, assuming an exposure duration of 30 seconds and a transit egress of 30 minutes would acquire 60 exposures over the transition period. The ADU variance is as low as 36/60 = 0.6 ADU between exposures. This value is below a point where it can be reliably measured since ADUs are integer numbers produced by measuring photons hitting the pixel surface.

In this example it is better to discard the brightest comparison stars, and keep those that are closer to the target’s host star magnitude, to achieve a typical target star ADU count closer to 45,000. Applying the same equation gives us a measurable variance of 90/60 = 1.5 ADU between exposures. As a general guideline select an exposure time that produces a target host star ADU count per pixel in the range 40,000–50,000 (assuming CCD linearity continues to 55,000) and at the same time ensures the comparison star(s) are below 55,000 ADU.

Accurately Locate and Center on Target

Simply put, slew to the host star, plate-solve, re-position as required to capture the correct comparison stars, rotate the camera as necessary to bring the comparison stars and guide star into the FOV and autofocus. It is assumed that the guider is already calibrated and the telescope is auto-focused correctly to produce the sharpest possible exposures, although it is not strictly necessary. (Some observers deliberately de-focus bright stars to avoid over-saturation of pixels, but this can introduce a unique set of variables that cause difficulties later on.)

Expose Images

Take many (500+) short star images (typically less than 60s) and manage the meridian flip. It is important to ensure the host star is in exactly the same position during the observing session (for each side of the meridian) and for that reason, disable any dither functions between exposures. Plate-solvers are common-place but they are not all created equal. Use the most accurate plate-solver available, and one that can use the UCAC4 catalog (it is the most accurate to date and it has the added benefit of including the 56 million-star APASS catalog in full).

Manage the Meridian Flip

If your acquisition program does not automatically flip, plate-solve, self-center and rotate the camera to pixel precision, trap the flip before it happens and manage the meridian flip manually. This includes optionally rotating the camera, plate solving and slewing the mount to an identical observing position and continuing the imaging sequence. If the observing session requires a meridian flip, remember to take additional flats with the camera at that position (e.g. dawn flats at the end of the observing session). Note that some practitioners rotate the camera back to the same orientation, post meridian flip, to remove the effect of any potential CCD variation across the sensor surface. It is something worth testing with an experiment.

Take Bias and Dark Frames

Allow 20 minutes or so to take a minimum of 16 bias and 16 dark frames (at the same time and temperature as the observing session). Clearly, it is not advisable to capture this data during the ingress or egress of the transiting exoplanet, but they can be taken while the exoplanet is fully occluding the host star, or just before/after the transit itself. Flat frames are not particularly temperature dependent (on account of the short exposure) and most analysis tools will produce dark-flats for your flat frames.

This is one reason why Sony HAD CCDs are preferred for observing exoplanets. They have a read noise of under 1.5 electrons (compared to a typical 8 or 9 electrons of a Kodak 8300 CCD) and a very low dark current. In practice, a slight camera temperature variation during the observing session should not be a concern. Typically, with Sony HAD cameras, once the camera is below -15°C, any temperature fluctuation produces negligible dark-noise variation.

Remember to take flat frames before and after the observing session, on each side of the meridian flip and with the camera at the precise orientation that it was at each stage in the observing session. Some practitioners have attempted to use artificial LED light sources in the middle of an observing session, with limited success. Its light output does not match the broadband spectrum of pure white light.

Identify the Stars Against a Known Star Catalog

This first stage is necessary so that we identify the target stars as well as the comparison stars that we use as a baseline. As previously discussed, differential photometry is the process in which the flux of a target star is compared against that of a suitable comparison star. By measuring the difference in target star flux output between one image and another, we can produce a flux curve for that particular target star. Unless we are lucky enough to be using an orbiting telescope, however, the sky quality will vary slightly during the observing session. For example, if a small cloud passes across the FOV during an exposure, some or all of the stars in the image will appear less bright (and possibly with a brighter background too) than in the exposures where there was no obscuration.

To make matters worse, sky quality can additionally vary if there is turbulence or high levels of humidity in the upper atmosphere. These events are usually invisible to the naked eye, but are sufficient to partially obscure the stars in the image, creating a noticeable dimming effect. Since the target flux variances are very small (~0.2%) we must compensate for atmospheric and seeing changes. This is the purpose of differential photometry and requires the observer to determine the published magnitude of a comparison star, measured through specified photometric filter(s).

Most older star catalogs contain information on the brightness (or magnitude) of the star, measured using Johnson-Cousins filters (usually B and V). More recent star catalogs are comprised of sky surveys that used the more accurate modern photometric filters; usually the Sloan u’, g’, r’, i’ and z’ filter set.

In practice, it is recommended that you utilize the UCAC4 star catalog, which contains sufficient detail to identify most stars in your field of view for telescopes up to 400–500 mm aperture, for photometric filters B, V, g’, r’ and i’.

As mentioned earlier, always use the most accurate star catalog and plate-solver. Although most plate-solvers work well in a dense starfield, choose one that also performs well with a handful of bright stars in an image. It is not uncommon for substantial errors to creep in, which result in the incorrect identification of stars that are in close proximity to one another.

Some applications attempt to simplify matters by allowing one to measure stars without identifying them first. More specifically, having identified the stars in the first image, its assumes that each star will remain at exactly the same position from one image to the next. Be wary of this, as it requires perfect mount tracking (no drift or PE) over the entire observing session. In practice, it is easier to allow the mount to drift slightly during the observing session, and then use an application that uses accurate plate-solving to correctly identify the stars in each image.

Identify the Target Stars

This activity is straightforward, providing one has executed an accurate plate-solve. First, look up the catalog number of the star in question and locate it on the plate-solved image. (You can alternatively compare a star map to the image in front of you, and determine which star is which.)

Simple photometric analysis tools can produce a flux curve, after manually selecting the stars you are interested in (by visual comparison to a star map), but they are not terribly accurate, and can be easily out-witted. In many cases, if you meridian flipped halfway through the observing session, the application will not realize that the stars are in completely different positions, and will produce some very peculiar-looking transit flux curves. When measuring exoplanet flux curves, it pays to be cautious, pedantic and thorough.

Identify the Comparison Star(s)

As discussed earlier, the way in which differential photometry negates any seeing or atmospheric effects, is by measuring the target star’s difference in flux when compared to a specified comparison star, from one exposure to another. This assumes that if the seeing or atmospherics have changed from one exposure to another, all stars in the field of view, or at least stars close together, will be affected in the same way. In fact, the equations for differential photometry can be proven mathematically to be highly robust, providing they follow these guidelines, repeated here for clarity:

• the comparison star is the same stellar type as the target star

• the comparison star has the same stellar surface temperature as the target star

• the comparison star is close to the target star

In practice, rather than comparing surface temperatures of each star, we measure and compare particular color intensities. Specifically, we measure the difference in star magnitude for the same star through two known photometric filters. A common metric used is the difference between the blue and green color measurements for a given star; which using Johnson-Cousins filters, equate to:

If the target star has a very different B – V value to the comparison star, it indicates they have very different surface temperatures. In this situation, the normal differential flux comparison equation will fail, as it assumes that the target and comparison stars have similar surface temperatures, hence the recommendation to select comparison and target stars with a similar B – V value.

Similarly, measurement accuracy improves by selecting comparison stars that are close to the target stars. Close stars are more likely to vary similarly with atmospheric obstructions and transmission. An important element of the differential photometry equation is the altitude of each star, which is compromised if the comparison star has a substantially different altitude to the target star. Starlight is absorbed and scattered by the Earth’s atmosphere and passes obliquely through more atmosphere at lower altitudes. At the same time select frequencies of light are progressively attenuated (this is why the Sun appears red near the horizon). It is important to avoid the situation where the flux from the target and comparison stars are dissimilarly affected by atmospheric affects.

In practice, it is a good idea to select more than one comparison star to use against a single target star. Using the flux data from multiple valid comparison stars improves the SNR measurement of the baseline flux, which in turn improves the target star’s flux difference measurement robustness. In other words, the more the merrier!

Finally, do not forget to ensure that each comparison star has a valid star catalog magnitude value and is consistent with the photometric filters used during the observing session. If the tool has not correctly extracted the magnitude value from the catalog, find the value using an offline resource and enter the value manually. For our purposes, the precise magnitude of the exoplanet flux curve is less important than evidencing the characteristic transit dip in the magnitude measurement.

Identify the Check Stars

Check stars are treated in the same way as target stars, with one exception; you use them to check that the comparison stars are good enough to enable the production of a reliable flux curve. In theory, if the data capture process has been executed correctly, then applying the differential photometry algorithm and processes to the check stars should produce flux curves that match the expected flux curve and magnitude. In this way, you can confirm that all aspects of the data analysis process are working as they should.

Having analyzed all the exposures and confirmed the stated magnitude for each comparison star is accurately reflecting the catalog value, implies that your analysis and process is sound. Moving on, any exoplanet flux curves produced from the data reduction process are more likely to be accurate and reliable. As with comparison stars, select check stars that are similar to the target/comparison stars in terms of stellar type and color temperature and are not known variable stars. Choose those with a transit within the observing period for this analysis.

Determine the Star Flux Measurement Aperture

The final element is to determine how much of a star’s area should be included in the calculation of star flux. We know the image of a star represents the visual description of a point light source (i.e. the stars are at optical infinity relative to Earth). It appears as a Gaussian blur; brightest at the center point, with surrounding pixels dropping off in brightness with their distance from the center. When the ADU value of each pixel drops to the background sky level, we visualize this as the boundary of the star. Bright stars appear “fatter” than faint ones, and occupy a larger number of pixels on the image.

For this reason, we utilize a mathematical calculation of how big a star is, to represent at what point do we distinguish between pixels being part of a star, and pixels being part of the background. Commonly, the Gaussian blur is assumed symmetrical and a slice through the middle produces an intensity profile resembling a Gaussian curve. (If you are looking at a stretched 16-bit image, the apparent boundary of the star will vary, depending on the scale of the image stretch.) In practice, we use two common metrics to define the boundary of a star, those being:

• Full Width Half Max: the point at which the pixel ADU value is less than 50% of the maximum ADU value of any pixel in the star (i.e. the star center)

• Half Flux Diameter: the point at which the ADU sum of all the pixels inside the star, equal the ADU sum of all the pixels outside the star. It follows that this only works when you define an arbitrary outer boundary for the outside of the star.

We commonly use the FWHM value to indicate if a star is in focus or not (an out-of-focus star generates a larger FWHM value). Once a star is in focus, the FWHM value reduces to a minimum, set by the constraints of optics and the seeing quality of that location. Pin sharp stars will never be exactly one pixel in size and we will always see a fall-off from the center of the star in light intensity. Understanding what constitutes the boundary of a star is a critical factor in the analysis; without knowing where the star’s boundary is, we do not know which pixels to include in the flux measurement.

As described earlier, most data reduction/photometric applications will ask you to define a circular boundary or “aperture” of the star; in which the ADU values of each pixel are measured. In order to compare robustly, we must select an aperture value that is applied identically to all the stars being measured. (Recall the target and comparison stars will be of similar magnitudes.) Many academics have analyzed at length what the optimum boundary point might be, for amateurs, a good starting point is:

A critical element in the calculation, is the extent to which those pixels on the boundary are fully measured (partial pixels). This requirement is not supported by all analysis tools but is required to match the translation of a perfect circle into the orthogonal pixel grid on the CCD surface. (For example, a simple 9-pixel group is square, not a circle). Several photometric data reduction tools, used for detecting exoplanets, were originally designed for the analysis of variable stars. The common approximations made by these tools do not significantly affect the measurement of variable stars but introduce large errors into the more demanding analysis of exoplanet flux curves. It is a good idea to check that the data reduction tool you use properly accounts for partial pixels.

Determine the Background Sky Flux

The temperature of interplanetary vacuum is never at absolute zero and any pixels that represent the background sky (i.e. the space between stars), will by definition have a non-zero flux value (even in zero light pollution). For the differential photometry equation to be effective, it must calculate and allow for the element of a star’s flux that is actually due to the background sky flux.

Differential photometry uses a circular area outside the star’s aperture to measure the sky flux. This area is separated from the aperture by a thin annulus, a no-man’s land in which the pixels are not used for any measurement. The outer circular boundary forms another annulus, the pixels of which are selected to measure the sky flux (fig.2). This is done by measuring the ADU values within the outer annulus and calculating the mean ADU value per pixel. This value is defined as the sky background radiation and is deducted from the calculated background ADU value from each pixel that makes up a star (i.e. the pixels within the aperture) before summing up the ADU values and comparing them to the adjusted ADU values in the comparison star. In practice, the tool does not actually compare star ADU values; they are all converted by a mathematical equation into star magnitude values. It is these magnitude values that are compared between the target and comparison stars. Having multiple comparison stars requires us to repeat the process for each comparison star, and calculate a mean magnitude to compare against the calculated magnitude of each target star.

In practice, the important thing to remember is that:

• Select an aperture that fully encloses the largest star that is being measured.

• Define a thin spacing between the aperture and the start of the annulus.

• Select an annulus that does NOT include pixels that form part of other stars.

Remember, these aperture values apply to all stars being measured and must be chosen with care. Most applications allow you to visualize what each aperture and annulus will look like on each star being measured; check that they do not inadvertently include partial elements of other stars within the annulus (this inflates the background sky flux value and will artificially reduce the flux calculation of that star). Conversely, avoid making the aperture too small as it may exclude flux elements of the star being measured. This is very easy to do if the stars are of substantially different size or area and will generate “static” star flux values between exposures, as a significant percentage of a star’s pixels are not being measured.

A careless selection at this point, that includes the wrong bits of stars, generates major discrepancies in the analysis. In practice, most amateurs will repeatedly recalculate exoplanet flux curves, by changing the values of the aperture and annulus. The difference in results, between a set of aperture and annulus values to another, has to be seen to be believed. This is why some more modern applications dispense with the definition of an annulus altogether, and apply complex mathematical equations to correctly deduct the flux value of all stars from the image (pixel by pixel). This technique is extensively used by professionals but can largely be ignored for our purposes, as long as the data reduction tool you use is capable of performing this function for you.

Measuring Flux of the Target Stars etc

Having done all the above, the last stage is repetitive and is best left to the automatic data reduction tool. Each exposure is analyzed as above, using the values set for star aperture and annulus. The ADU value per target and check star is converted into a magnitude value, and is compared against the mean magnitude value of the comparison stars. To compare exposures of differing duration (which will produce brighter/fainter stars) the flux values are converted into a flux value per second of exposure. This ensures a robust comparison (again, as long as none of the stars are over-exposed).

Once a calculated magnitude value is generated for each target and check star, based on the relative value of flux in the comparison stars, it is plotted on a time axis. The flux curve is then examined to see if one can spot the characteristic exoplanet dip, corresponding to the start of an exoplanet transit; the period when it is fully in front of the star, and the period when it is exiting the star.

In practice, it is common to run this analysis stage several times using the same data but differing selections for comparison and check stars. The investment in analyzing the data in various ways, improves the certainty that the flux curve has the highest degree of accuracy.

Best-Fit modeling

The final stage in the process, applies complex “best-fit” modeling techniques to predict if what appears to be an exoplanet flux curve, is actually an exoplanet transit. This is also referred to as Procrustes modeling, named after the mythological Greek hotelier who used to chop off the extremities of any individual unlucky enough to overhang his deliberately small hotel bed. In other words, it “forces” the data to fit the constraints.

fig.2 This pictorial shows a typical star image and its intensity profile. Although technically the Airy pattern, the central profile is principally a Gaussian curve. Here the flux measurement aperture is set to 1.44x the FWHM value and an outer annulus used for the sky background flux measurement. The thin annulus between these is excluded from either measurement. As the circle becomes smaller, the measurement software needs to consider the orthogonal pixel approximation to a circle and compensate by using a proportion of the boundary pixel values, referred to as “partial pixels”.

Modeling is a powerful technique that statistically correlates data against an assumed outcome or model. It is not cheating, as the output additionally indicates the probability of the data fit to the model. Different analysis tools do this differently, and it is best to familiarize yourself with how your specific tool operates. If in doubt follow the advice of the AAVSO or BAA/RAS exoplanet groups.

And Finally

If you have got this far, you will have produced one or more exoplanet flux curves to rival the results produced by the professionals. The surprising aspect to this activity, is the ease in which amateurs can produce complex astronomical analyses, purely using the equipment we have had for producing amazing astronomical pictures. Frame the flux curve; you are now one of a small number of amateur astronomers worldwide who can correctly identify exoplanets. Examples of professional data curves can be found on the NASA Exoplanet Archive website:

http://exoplanetarchive.ipac.caltech.edu

If you have overwhelmed your PC with data and your brain hurts, why not try astrophotography, it will now appear easy by comparison ;)