The Pluto System

Detection and Variability of Nix and Hydra

Generals Paper I

Alessondra Springmann

August 1, 2008

1

1 Introduction

Once thought to be a remote oddball of our outer Solar System, Pluto is now known to be

one of many objects in the Kuiper Belt. Its three moons—previously thought to be unique

for such a small, remote body—make Pluto one of the numerous binary or multiple-body

Kuiper Belt objects (KBOs) that have been discovered in the last 16 years. Up to 20% of

KBOs in various dynamical classes are multiple systems (Noll et al., 2008), implying that it

is not uncommon for a body residing in this region of the Solar System to be a binary or

multiple system.

Pluto’s largest moon Charon was discovered in 1978 (Christy & Harrington, 1978), mak-

ing Pluto a double planet until the 2005 discovery of Pluto’s smallest moons, Nix and Hydra

(Weaver et al., 2005). Recent dynamical simulations involving these two newly-discovered

moons present new constraints on the formation of the Pluto system. The satellites’ circular

orbits, coplanar with the orbit of Charon, suggest that these moons all formed from material

ejected into orbit around Pluto by a giant impact event (Canup, 2005; Stern et al., 2006b).

1.1 Motivation

Nix and Hydra’s recent discovery and their importance in our understanding the Pluto

system make them an important and timely mission goal for New Horizons, set for a 2015

encounter with Pluto. Providing limits on the rotation periods of these moons allows for more

efficient imaging of their surfaces by the New Horizons cameras, in addition to confirming the

period constraints determined by Buie et al. (2007) and allowing for more precise simulations

to be carried out of the Pluto system’s formation and evolution. In this paper, we discuss

attempts to detect these moons and constrain their rotation periods from data obtained with

the 6.5-meter Baade telescope at Las Campanas Observatory in July 2007.

2

2 Pluto System Overview

It is widely accepted that the Pluto system was formed in a giant impact collision (Canup,

2005; Stern et al., 2006b) with little or no exchange of heat or heating of the bodies’ materials

occuring, resulting in the formation of not only Pluto and Charon, but also Hydra and Nix.

This assumption is valid because the angular momentum of the Pluto system is quite high

(see references 9-12 in Stern et al., 2006b), thus Charon must have formed from a collision

with Pluto and was therefore likely not captured into orbit.

Smaller than Eris and red, Pluto has a surfaced varied in both color and albedo (Brown,

2008). The largest KBOs, including Pluto, have methane ice on their surfaces, in addi-

tion to N2, CO, ethane, and perhaps water ice (McKinnon et al., 2008). The density of

the Pluto-Charon-Nix-Hydra quaternary system (2.01 g cm−3, Tholen et al., 2008) implies

that the impactors that formed this system were initially accreted (McKinnon et al., 2008).

These impactors were differentiated by accretion and radiogenic heating, which resulted in

a differentiated Pluto. Taking the argument for Pluto’s being differentiated even further,

Hussmann et al. (2006) suggest the presence of a sub-surface ocean below Pluto’s low-density

ice. Although formation simulations that result in differentiation are accepted, scenarios that

result in subsurface oceans or cryovolcanism are not as widely considered to be as valid.

2.1 Charon

2.1.1 Formation

Charon formed from mantle material leftover from the collision that formed the entire

Pluto system. As this collision resulted in little or no melting of the impactors’ material,

there was little heat exchange, and thus a large portion of the impactors’ mantles remained

water ice and became the major constituents of Charon (Stern et al., 2006b).

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2.1.2 Mutual Events

The discovery of Charon allowed for more precise mass determination for Pluto. Soon

after the discovery of Charon, its orbit plane was oriented such that it was edge-on as seen

from Earth, resulting in mutual events of Pluto and Charon being visible for approximately

five years (Binzel & Hubbard, 1997), allowing for more precise measurements of color and

albedo of both bodies (Figure 2 shows a color map of Pluto’s surface). While we will not

experience another “season” of Pluto-Charon mutual events again in our lifetimes, improve-

ments in telescope technology and size have allowed us to study the bodies individually and

also through occultations (Noll et al., 2008). The discovery of additional binary or multiple

KBOs showcases the diversity of the Kuiper Belt, and allows us to apply what has been

learned from Pluto to other systems.

2.1.3 Physical Properties

Charon’s surface is dominated by water ice, in addition to a bit of ammonia (see McKinnon

et al., 2008, and references therein); quite a different surface from Pluto. Both Pluto and

Charon are tidally locked, the same faces of these two bodies always facing the companion.

Because of the relative sizes of Pluto and Charon (d\ = 2390 km; dC = 1207 km), these two

bodies are occasionally called a “double planet”.

2.2 Nix & Hydra

Nix and Hydra likely formed in the same collision that produced Charon, probably origi-

nating as debris from this giant impact between two large, differentiated bodies (Stern et al.,

2006b). Their coplanar orbits with Charon also suggest that they formed in the same impact

from which formed Charon. Combining their coplanar orbits with their neutral colors, it

can be assumed that Nix and Hydra are similar in composition to Charon, i.e. that their

4

constituents are mostly water ice.

Another argument for Nix and Hydra forming at the same time as Charon is that they are

larger than the critical size boundary for catastrophic breakup over the age of the solar system

(Durda & Stern, 2000; Stern et al., 2006b). Forming along with Pluto and Charon, Nix and

Hydra are likely the same age as the former two bodies and have withstood disruption

forces that would have torn apart smaller bodies. It is unlikely that these small moons

were captured, because of their coplanar, circular orbits with Charon, and if they had been

captured, these two small moons would be in neither coplanar nor circular orbits, and would

not be orbiting in a common direction (Ward & Canup, 2006).

Formation of these moons alongside Pluto and Charon was likely, as they are presently

in highly circular orbits—e = 0.01504 for Nix and e = 0.00870 for Hydra (Tholen et al.,

2008)—implying that they were not dynamically captured, as the circularization timescales

are greater than the age of the Solar System (Stern et al., 2006b).

Knowing the rotation rates of these moons will help verify whether they are in any sort

of spin-orbit resonances with Pluto, and perhaps place important constraints on the rotation

rates of moons of other KBOs, as mentioned in Section 2.3. Numerous simulations by Canup

(2005); Lissauer (2006); Stern et al. (2006b); Ward & Canup (2006); Lithwick & Wu (2008);

Tholen et al. (2008) have guessed at the specifics of origin of the Pluto system, but without

rotational constraint on Nix and Hydra, knowing the particulars of how they formed and

evolved to their current orbits will elude dynamicists.

Even if Nix and Hydra have resisted disruption by impacts or tidal forces, they may

have lost up to 10% of their masses due to impact erosion (Stern, 2008a,c). Material ejected

from their surfaces by collisions with other KBO fragments could help feed the hypothetical,

tenuous rings or ring arcs in the Pluto system1.

1Binzel (2006) and Stern et al. (2006b) propose the possibility of rings (or ring arcs) in the Pluto system,of time-varying optical depth of approximately τ = 5× 10−6, of the same order of magnitude as the opticaldepth of the tenuous rings around Jupiter (Showalter et al., 1987; Throop et al., 2004). As Pluto is far away

5

The proximity of Nix and Hydra to corotation—more literally, co-orbital—resonances

also implies a common impact origin for these smaller moons, meaning that Nix and Hydra

were driven outward by Charon’s migration (Ward & Canup, 2006). A corotation resonance

occurs where the object’s semi-major axis oscillates, rather than its eccentricity. At some

point, the outward migration of Nix and Hydra ceased, most likely due to the end of Charon’s

outward migration. Nix likely escaped from its corotation resonance with Charon first, before

the end of Charon’s outward semi-major axis migration due to tidal expansion. Ward &

Canup (2006) also show that these particular corotation resonances (4:1 and 6:1) are the

most conducive to moon formation around Pluto, as other corotation resonances are close

to the Hill sphere and other dynamically unstable regions. Nix and Hydra’s probable initial

locations in the 4:1 and 6:1 corotation resonances could have lead to the aggregation of

material at these points and thus helped contribute to the long-term survival of these moons

over the age of the Solar System.

Though the orbits of these two small moons are relatively well known (Tholen et al., 2008),

not much is known about either the exact dimensions of Nix and Hydra or their rotation

rates. Stern et al. (2006b) assume minimal masses for Nix and Hydra and calculate that

these moons are likely not in synchronous rotation with Pluto, unless they formed much

closer to Pluto and migrated outward (Ward & Canup, 2006). Thus, understanding the

rotation rates of Nix and Hydra helps constrain formation models of multiple-body systems,

including other KBOs.

from the perturbing forces of the Sun, the possibility of there existing rings of collisional ejecta from Nixand Hydra in the Pluto system is high, though the arrival of New Horizons will ultimately confirm or denythe prescence of rings. Stern (2008a,b) also suggests the potential resurfacing of Nix and Hydra by impactejecta, up to 70 meters on each small moon if the process is efficient.

6

2.3 Other Kuiper Belt Binaries

Because up to a fifth of Kuiper Belt objects are binary or have multiple satellites

(Stephens & Noll, 2006), studying the multiple system of Pluto allows us to better under-

stand the evolution and current states of not only the Pluto system but also other multiple

Kuiper Belt objects. Although there are plenty of binary KBOs, it is significantly harder to

observe smaller satellites of 100-km class TNOs (Noll et al., 2008), especially for narrowly-

separated objects. Comparing the frequencies of nearly equal-sized binaries with objects

with smaller companions is difficult, due to observational limits from seeing, telescope size,

differences in detectors, etc. Pluto, being one of the brightest KBOs, allows us to most

readily study its multiple satellites, making it an ideal target for New Horizons (Section 3).

3 New Horizons

A collaboration between the John Hopkins Applied Physics Laboratory and the South-

west Research Institute, New Horizons was launched in January 2006 and is expected to

fly by Pluto on July 14 in 2015. New Horizons’ science goals include imaging the surfaces

and thus mapping the compositions of the four main bodies in the Pluto system, studying

Pluto’s atmosphere, searching for an atmosphere around Charon, along with imaging the en-

tire Pluto Hill sphere, searching for presently undiscovered moons and probable rings (Stern

et al., 2006b; Young et al., 2007; Stern, 2008a).

The recent discovery of Nix and Hydra is very timely in allowing target planning and

new science goals for the New Horizons mission. If the rotation rates of these moons are

verified, New Horizons will be able to more optimally image their surfaces.

After visiting Pluto New Horizons will swing by at least one yet-to-be-determined KBO

to further study the other icy bodies of the outer Solar System, linking them with other

dynamical populations such as Centaurs and comets. Hopefully in the process, New Horizons

7

will begin to answer our questions about the origins and compositions of the cold leftovers

of our Solar System’s formation and evolution.

Nix and Hydra’s recent discovery and their importance in understanding of the Pluto

system make them an important and timely mission goal for New Horizons. Until New Hori-

zons arrives at Pluto, a good deal will remain uncertain about the Pluto system, including

the absolute radius of Pluto, the nature of its atmosphere, the rotation rates of Nix and

Hydra, and the masses and composition of these moons.

4 Observations

To determine or constrain potential short period rotation of Nix and Hydra, images of

the Pluto system were obtained with the 6.5-meter Walter Baade Magellan telescope at the

Las Campanas Observatory, Chile. Repeated imaging sequences covering a total of 10 hours

on UT 2007 July 11 and 12 were executed using the Inamori Magellan Areal Camera and

Spectrograph (IMACS) in the Baade f/4 direct imaging mode. A total of 815 images with

integration times of 6 seconds (cadence of 30 seconds) in a Sloan i′ filter were collected under

0.4-0.7 arcsecond seeing. Only Chip 2 out of the eight chips that compose IMACS was used

and was further subrastered to reduce image readout time. A sample image sequence is in

Figure 1, moving 0.44 arcseconds per hour.

5 Data Reduction and Analysis

The data were calibrated using bias and flat frame images. Routines from the IRAF/DAOPHOT

package were used to perform stellar photometry on the Pluto fields.

The point spread functions (PSFs) of Pluto and Charon overwhelm any signal from Nix

and Hydra. Therefore, in order to determine the rotation rates of these small moons, much

8

Figure 1: Six-second integrations of the Pluto system taken 1 hour apart on UT 2007 July11. The Pluto-Charon system is the bright object near the center of the frames that changesposition between exposures and is indicated by the blue circle.

less detect their signal in the science frames, the light from Pluto and Charon must be

modeled and subtracted from the images to reveal the signatures of Nix and Hydra.

This section is separated into three subsections pertaining to the physics of rotational

lightcurves, the astronomy of limiting magnitude calculations, and the mathematics of point

spread function fitting:

1. Rotational lightcurves and detection (Section 5.1)

2. Astrometry/limiting magnitude calculation (Section 5.2)

3. Point spread function modeling and subtraction (Section 5.3)

5.1 Rotational lightcurves

Rotation rates of small Solar System bodies are determined by measuring their changes

in brightness via aperture photometry. As a non-uniform body rotates, it reflect varying

amounts of light due to changes in albedo, surface area, and other surface features, producing

changes in brightness that can be measured by a detector. Plotting the change in brightness

or the magnitude of the object as a function of time produces a lightcurve. Analyzing

9

the lightcurve yields information regarding the rotation period of the object and also more

detailed information regarding surface features and body shape with sufficient data.

To first approximation, a rotating moon such as Nix or Hydra can be approximated by a

triaxial ellipsoid (Figure 2). As the moon rotates, it presents different faces to the Sun and

thus reflects varying amounts of light at the observer, generating a lightcurve, which can be

approximated by a Fourier series of sinusoidal curves. Two cases that would generate flat,

or featureless, lightcurves include a uniformly spherical rotating object, or a rotating object

viewed from its pole. Should either of these cases apply to Nix or Hydra, no discernible

change in lightcurve will be detected from that moon.

Using the standard magnitude equation,

∆m = m2 −m1 = 2.5 log10

(b1b2

)(1)

where m indicates magnitude, b is the luminosity, and t is the exposure time, and then

substituting for b = F/t while assuming that t1 = t2, we obtain:

∆m = 2.5 log10

(F1

F2

). (2)

Assuming that flux is proportional to the projected area of the target, the maximum

points of the lightcurve will correspond to when the largest face of the target is reflecting

light at the observer. Likewise, the minimum points of the lightcurve correspond to when

the object is presenting the smallest amount of reflected light to the observer and thus yield

the smallest amount of flux. Taking the area of an ellipse is π · a · b, the projected area of

the largest face of the ellipsoid will be Al = πac,while the smallest face will have a projected

area of As = πbc. From this, we obtain:

∆m = 2.5 log10

(πac

πbc

)= 2.5 log10

(a

b

).

10

Thus, the change in lightcurve amplitude as the body rotates is proportional to the ratio

of the semi-major over the semi-minor axis. A target body with an axial ratio of 1.25 will

ideally show a change in lightcurve semiamplitude of 0.24 (Figure 3).

Figure 2: A triaxial ellipsoid with axes of different lengths (a 6= b 6= c).

During the observations conducted in July 2007, the phase angle of Pluto was 0◦.07 and

increasing, making Pluto quite close to opposition and thus at the hight of its brightness as

seen from Earth. This makes the Sun, Pluto, and Earth effectively in a line, since the angle

between the Sun, Pluto, and the Earth is less than one degree (Figure 4).

5.2 Limiting Magnitude Calculation

In order to determine if Nix and Hydra are detectable in our science images, we calculated

how many images would need to be coadded to obtain sufficient signal-to-noise on Nix. Nix

was chosen as the limiting case because it is the dimmest (apparent V magnitude of 23.7

(Stern et al., 2006a)) of these moons, to bring them to the same brightness as the dimmest

stars in our exposures.

Using interactive IDL astrom routines from Marc Buie2, we matched the locations of the

stars in our frames to US Naval Observatory stellar locations. Initally, the predicted catalog

2http://www.boulder.swri.edu/~buie/idl/

11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!0.5

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5

rotational phase

! m

agni

tude

Figure 3: A synthetic lightcurve of an object with an axial ratio of 1.25 over one rotationperiod. The two maxima and two minima correspond to the two large and two small facesof the object, respectively. The lightcurve is approximated to first order by a sine curve.

Figure 4: The viewing geometry of the Pluto system as seen from Earth on UT July 11 2007.(Note: orbits and body radii not to scale.)

12

positions do not match the positions of the stars on the chip very well. The positions of

the stars on the chip are manually aligned with the predicted catalog positions (Figure 5).

astrom then calculates the vectors between the catalog and chip locations of the stars, then

determines the standard magnitudes of the stars in the exposure.

Figure 5: Alignment between the USNO star catalog and star chip positions. Yellow circlesindicate a match between star catalog and chip positions; red circles indicate that the routinewas unable to find a match between predicted and chip positions.

To calculate how many images of Nix and Hydra need to be coadded, we used the

astrom calculations of the limiting magnitude star in each exposure to determine the dimmest

magnitude visible of a star in our images. The magnitudes we reference from the USNO

catalog magnitudeare in R. We realize this is suboptimal for comparing to photometry

performed in Sloan i′; however, it makes a reasonable estimate.

For exposure 101 with exposure time t0 = 6 seconds) on UT July 11, the sky back-

ground, µ, is 53.19 counts, while the standard deviation, σ, is ±10 counts. The dimmest

star detectable by the photometry routines used would have S1 = µ + 10σ counts, in this

13

case, 153.19 counts above background for a signal to noise ratio of 10σ/σ = 10. For a As a

test to scale the limiting magnitude, we used a star located at RA 17:47:33.56, declination

-16:25:36.39 having a catalog R magnitude of 16.12. This image records a signal, S0, of

10910.3 counts with the sky subtracted. Using the standard magnitude equation (equation

1), where brightness, b, is related to flux, F , and integration time, t, by b = F/t, it is possible

to calculate the difference in magnitudes between the star with signal S0 and a star with

signal S1:

∆m = 2.5 log10

(S0

S1

)= 2.5 log10

(10910.3

100

)= 5.09

Thus, a star with 10910 counts corresponds to a star in the Sloan i′ filter with magnitude

16.12, while a star with 100 counts in the same filter corresponds to a magnitude of 16.12 +

5.09 = 21.21, which is the limiting magnitude for a six-second exposure. The results from

the astrom routines also indicate that the average limiting magnitude for all of our science

frames from UT July 11 2007 is roughly 21.

Using the V magnitudes from Stern et al. (2006a) we estimate the magnitude of Nix as

24.0. In order for Nix to be detected in our science frames, some amount more signal on

these moons will need to be obtained, which will correspond to an increase of n times in

exposure length to obtain a SNR of 10 on this moon.

Using the magnitude equation (Equation 1) again we find that:

∆m = 24.0− 21.21 = 2.79 = 2.5 log10

(m[N,H]

mlimiting

)= 2.5 log10

(mlimiting · nmlimiting

)

2.79

2.5= log10(n); n = 10

2.792.5

n = 13.02

Therefore, Nix needs a factor of 13 more flux for a 10 σ detection of this moon from

the background signal, corresponding to a 65-second exposure, approximately 11 6-second

14

exposures.

5.3 Point Spread Function Models

In order to subtract the signal from Pluto and Charon, models of the point spread

functions (PSFs) that these objects create on the chip are modeled. As the PSF changes

from exposure to exposure, a different PSF must be generated for each image.

To create a PSF for the image, a series of tasks from the DAOPHOT routines are used.

First, the daofind task creates an initial star list by locating all the star-like sources in an

image within a certain width and range of counts, then tvmark marks the stars from the

star list on the displayed FITS file. Using both phot and daofind, the background values

and initial magnitudes for the stars in the star list are calculated.

To generate a PSF from the stars in the image, between seven and nine stars brighter

than Pluto are manually selected with pstselect and an initial PSF is created with the psf

routine, where the type of PSF (Lorentz, Gauss, or Moffat) can be specified.

The nstar task then takes the PSF generated in the previous step and fits it to the rest

of the stars in the field. The PSFs of the fitted PSF stars and their neighbors are then

subtracted from the image using substar. This process is then repeated on the subtracted

images until a clean PSF is obtained, usually after three or four iterations.

5.4 PSF Modeling

Several different point spread function (PSF) types were used to accurately model the light

from Pluto and Charon and thus subtract these two bodies’ PSFs. Three different models

have been used for approximating the point spread functions of the data. Davis (1994) de-

scribe Gauss, Moffat, and Lorentz models for IRAF/DAOPHOT PSF modeling as discussed

in Appendix A. However, the Moffat model best describes the shape of the PSFs obtained

15

with IMACS and thus was used for creating PSFs in this paper.

After the stars have been modeled and subtracted in each frame, approximately 65 sec-

onds (integration time, not clock time) of images are coadded together to obtain sufficient

signal to noise with respect to Nix and Hydra, as described in Section 5.2.

Although the Moffat25 models were better than the Gauss and Lorentz PSF models,

they were not sufficient to completely remove the signal from Pluto, Charon, and other

stars neighboring the locations of Nix and Hydra in our science frames. Thus, coadding 11

exposures did not yield a detection of either Nix or Hydra, discussed further in the next

section.

6 Discussion and Future Work

Our observations yield continuous coverage with integration times of six seconds over a

span of seven hours on the first night of observing. The high sampling frequency should

allow or constrain rotation periods of Nix and Hydra under 24 hours, should these moons

be detected.

Improvements to the PSF fitting routines would most likely yield a positive detection of

these moons, such that photometry could be performed on them to provide limits to their

rotation rates. Pluto does not appear as a point source in the science images; it is resolved

as a disk. Thus, being able to convolve a point spread function into a “disk spread function”

would allow the signal from Pluto to be modeled better than by simply using a Moffat or

other Gaussian model.

The high cadence of observations complements the observing sequence of the Hubble

telescope by Buie et al. (2007), which should constrain the moons’ rotation rates on the

order of days or longer. The Buie team should be able to discern if Nix and Hydra are

variable on a time scale on the order of 12 hours due to the durations of time between their

16

exposures on the Hubble Space Telescope. Between our measurements and those of the Buie

team, the rotation rates of these moons should be discerned in the near future and made

available to the New Horizons team in order to maximize the imaging capabilities of this

spacecraft during its flyby of these moons.

The arrival of New Horizons at Pluto will begin the in situ exploration of the Kuiper

Belt and the icy regions beyond Neptune, helping us understand better the nature of not

only Plutos moons but also their dwarf planet companion and other Kuiper Belt objects.

By imaging the surfaces of these moons and other objects beyond Pluto, we will be able to

determine the compositions of these moons and also their orbital characteristics, giving us

further clues as to their dynamical evolutions as well as those of other Kuiper Belt objects.

Almost 30 years after the realization that Pluto was a double planet, the discovery of

1998 WW31 demonstrated that Pluto was no longer unique in its binary status (Veillet et al.,

2001). According to the frequency predictions of Noll et al. (2008), there could be up to tens

of thousands of Kuiper Belt objects with satellites or that are binaries (Stephens & Noll,

2006). Though we know of only three moons in orbit around Pluto, there could have been

more moons that were disrupted or destabilized throughout the dynamical history of the

Pluto system. In addition to moons that once were, more satellites could be lurking below

the detection thresholds of current telescopes, along with rings or ring arcs, not just around

the tightly-bound Pluto system but around other KBOs. Maybe there exist KBO systems

less tightly-bound than Pluto, such as ones with captured satellites in highly eccentric orbits.

The arrival of New Horizons in the Kuiper Belt will answer numerous questions about the

history of this region, but will likely also raise new ones as the spacecraft’s science instruments

begin to probe the chilly remnants of planet formation. The objects that compose Pluto’s

cohort beyond Neptune should continue to yield more information and also more mysteries

as to their nature, for as Hal Levison says, “. . . the Kuiper belt is very strange.”

17

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Strobel, D., Summers, M. E., & Tyler, G. L. 2007, Space Science Reviews, 709

Appendix

A Point Spread Function Models

Three different models were used to approximate the point spread functions of the data.

Davis (1994) describe Gaussian, Moffat, and Lorentz models for PSFs as discussed below.

A.1 Moffat25

The Moffat function, as described by Trujillo et al. (2001), is

PSF(r) =β − 1

πα2

[1 +

(r

α

)2]−β

, (3)

with the full width at half maximum, FWHM=2α√

21/β − 1, where PSF(FWHM/2) =

(1/2)PSF(0) and the total flux is normalized to 1. According to Davis (1994), the Mof-

fat family of functions are good for modeling under-sampled ground-based data. For the

DAOPHOT Moffat25 model, β is set as 2.5.

20

A.2 Gauss

The Gauss model for a PSF is a limiting case of the Moffat PSF (when β →∞) (Trujillo

et al., 2001):

PSF(r) = 4(21/β − 1

) β − 1

πF 2

[1 + 4(21/β − 1)

(r

F

)2]−β

. (4)

As β →∞, we can substitute 21/β − 1 with (ln 2)/β, so:

limβ→∞

PSF(r) = limβ→∞

β − 1

β

4 ln 2

πF 2

[1 +

4 ln 2

β

(r

F

)2]−β

. (5)

Using limm→∞(1 + z/m)m = ez we have

limβ→∞

PSF(r) =4 ln 2

πF 2e−

4 ln 2F2 r2 . (6)

Finally, writing F 2 = 8σ2 ln 2 we obtain:

limβ→∞

PSF(r) =1

2πσ2e−

12( rσ )

2

. (7)

The IRAF/DAOPHOT package describes the Gauss model as being the “best choice for

well-sampled, fwhmpsf ≥ 2.5pixels, ground-based images because the interpolation errors

are small and the evaluation is efficient as the function is separable in x and y.”

A.3 Lorentz

The Lorentz, or Cauchy, distribution for a PSF is an elliptical function with extended

wings that can be aligned along an arbitrary position angle. It is described mathematically

as follows:

PSF(r) =

[γ2

(x− x0)2 + γ2

](8)

21

where Γ describes the width of the function. As opposed to the Gauss distribution model,

the Lorentz model’s extended wings do not decrease to zero as rapidly. A large Gauss

distribution is exponentially related to the square of the deviation, whereas a large Lorentz

distribution is inversely related to the square of the deviation.

22