text: Wide or Narrow? The Phenomenology of 750 GeV Diphotons Matthew R. Buckley1 1Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA I perform a combined analysis of the ATLAS and CMS diphoton data, using both Run-I and Run- II results, including those released at the 2016 Moriond conference. I ﬁnd combining the ATLAS and CMS results from Run-II increases the statistical signiﬁcance of the reported 750 GeV anomaly, assuming a spin-0 mediator coupling to gluons or heavy quarks with a width much smaller than the detector resolution. This signiﬁcance does not decrease when the 8 TeV data is included. A spin-2 mediator is disfavored compared to the spin-0 case. The cross section required to ﬁt the ATLAS anomaly is in tension with the aggregate data, all of which prefers a smaller value. The best ﬁt for all models I consider is a 4.0σ local signiﬁcance for a 750 GeV spin-0 mediator coupling to gluons with a cross section of 4 fb at 13 TeV (assuming narrow width) or 10 fb (assuming Γ = 45 GeV). I. INTRODUCTION ∼ The ATLAS announcement [1] of a 3.6σ local excess in diphotons with invariant masses near m 750 GeV in γγ the ﬁrst batch of LHC Run-II data, combined with the CMS Collaboration announcing [2] a 2.6σ local excess in the same channel and invariant mass, has sent the theoretical physics community into a frenzy of model-building, with 150 papers on the topic appearing on arXiv within a month. The “γγ Resonance that Stole Christmas” [3] is of course tremendously exciting. It is the LHC’s most statistically signiﬁcant deviation from the Standard Model − of particle physics made public since the discovery of the 125 GeV Higgs boson. The addition of 0.6 fb 1 of CMS data gathered without the presence of the solenoid ﬁeld [4], as well a reanalysis of the 13 and 8 TeV data by the ATLAS Collaboration [5] has increased the statistical preference for new physics, with CMS now reporting a 3.4σ local signiﬁcance in their combined data. At this stage, the information on which theorists can build models is somewhat minimal, assuming that these results are not just a somewhat unlikely statistical ﬂuctuation and there is actually new physics to be begin with. A 3.6σ signal is not very large, and additional slicing of the data in order to ask more detailed questions will typically yield only small statistical preference for any possible result. The goal of this paper is to investigate what information can be extracted from the combination of ATLAS and CMS data in the diphoton channel, including the most recent analyses released at the 2016 Moriond conference [4, 5]. In particular, I am interested in the statistical signiﬁcance of the combined 13 TeV Run-II data, and in the signiﬁcance of combinations with the Run-I 8 TeV data. In addition, the ATLAS Collaboration claims a better ﬁt (3.9σ local) to a resonance at 750 GeV with a width of 45 GeV, as opposed to a narrow resonance with width much less than the experimental resolution (as might be naively expected). Fitting this wide resonance into theoretical models has been the focus of many of the 150 papers on the topic. In this paper, I consider the combined statistical preference for this width. This paper should be viewed as a follow-up to the work of Refs. [6–8], which are some of the earliest theoretical papers on the 750 GeV anomaly, and have largely set the parameters which later papers have adopted. Where our results overlap, my results broadly agree with those found in Refs. [6–8]; in particular in terms of the best-ﬁt cross sections for the anomaly. This paper goes into somewhat more detail in ﬁtting the signal in the ATLAS and CMS data, by ﬂoating the ﬁt to the background functional forms as signal is added. I also perform ﬁts to spin-2 resonances, and investigate the preference of the combined data for a wide or narrow resonance. This paper does not attempt to address the implications of this anomaly for other search channels at the LHC, new physics models beyond the resonance itself, or derive any limits on fundamental couplings. Such questions have been well addressed in the literature. I restrict this paper solely to the input parameters for theoretical studies: the cross sections, widths, and statistical signiﬁcance of the anomaly itself. The next section describes my combination ﬁts to the ATLAS and CMS diphoton data, using both 13 and 8 TeV data. For simplicity (and due to the computational limitations of the event generation), I consider only local sig- niﬁcance in my combination ﬁts, rather than including the look-elsewhere-eﬀect. The statistical question at hand here is the signiﬁcance of the anomaly at 750 GeV speciﬁcally, thus somewhat justifying this choice. I parameterize the possible couplings of the mediator to gluons and light quarks separately, and consider both spin-0 and spin-2 resonances. I present additional results for second-generation and b-quark couplings in Appendix A. In Section III, I then turn to the question of the statistical preference for a wide or narrow resonance; using the best-ﬁt parameters from Section II. I will discuss my conclusions in Section IV, but I summarize some of the more salient points here:

text: Wide

label: title

text: or

text: Narrow?

text: The

text: Phenomenology

text: of

text: GeV

text: Diphotons

text: Matthew

label: author

text: R.

text: of

text: Physics

text: and

text: Astronomy,

text: Rutgers

text: University,

text: Piscataway,

text: NJ

text: USA

text: I

label: abstract

text: perform

text: a

text: combined

text: analysis

text: of

text: the

text: ATLAS

text: and

text: CMS

text: diphoton

text: data,

text: using

text: both

text: Run-I

text: and

text: Run-

text: II

text: results,

text: including

text: those

text: released

text: at

text: the

text: Moriond

text: conference.

text: I

text: ﬁnd

text: combining

text: the

text: ATLAS

text: and

text: CMS

text: results

text: from

text: Run-II

text: increases

text: the

text: statistical

text: signiﬁcance

text: of

text: the

text: reported

text: GeV

text: anomaly,

text: assuming

text: a

text: mediator

text: coupling

text: to

text: gluons

text: or

text: heavy

text: quarks

text: with

text: a

text: width

text: much

text: smaller

text: than

text: the

text: detector

text: resolution.

text: This

text: signiﬁcance

text: does

text: not

text: decrease

text: when

text: the

text: TeV

text: data

text: is

text: included.

text: A

text: mediator

text: is

text: disfavored

text: compared

text: to

text: the

text: case.

text: The

text: cross

text: section

text: required

text: to

text: ﬁt

text: the

text: ATLAS

text: anomaly

text: is

text: in

text: tension

text: with

text: the

text: aggregate

text: data,

text: all

text: of

text: which

text: prefers

text: a

text: smaller

text: value.

text: The

text: best

text: ﬁt

text: for

text: all

text: models

text: I

text: consider

text: is

text: a

text: local

text: signiﬁcance

text: for

text: a

text: GeV

text: mediator

text: coupling

text: to

text: gluons

text: with

text: a

text: cross

text: section

text: of

text: fb

text: at

text: TeV

text: (assuming

text: narrow

text: width)

text: or

text: fb

text: (assuming

text: Γ

text: GeV).

text: I.

text: INTRODUCTION

text: The

text: ATLAS

text: announcement

text: of

text: a

text: local

text: excess

text: in

text: diphotons

text: with

text: invariant

text: masses

text: near

text: m

text: GeV

text: in

text: γγ

text: the

text: ﬁrst

text: batch

text: of

text: LHC

text: Run-II

text: data,

text: combined

text: with

text: the

text: CMS

text: Collaboration

text: announcing

text: a

text: local

text: excess

text: in

text: the

text: same

text: channel

text: and

text: invariant

text: mass,

text: has

text: sent

text: the

text: theoretical

text: physics

text: community

text: into

text: a

text: frenzy

text: of

text: model-building,

text: with

text: papers

text: on

text: the

text: topic

text: appearing

text: on

text: arXiv

text: within

text: a

text: month.

text: The

text: “γγ

text: Resonance

text: that

text: Stole

text: Christmas”

text: is

text: of

text: course

text: tremendously

text: exciting.

text: It

text: is

text: the

text: LHC’s

text: most

text: statistically

text: signiﬁcant

text: deviation

text: from

text: the

text: Standard

text: Model

text: of

text: particle

text: physics

text: made

text: public

text: since

text: the

text: discovery

text: of

text: the

text: GeV

text: Higgs

text: boson.

text: The

text: addition

text: of

text: fb

text: of

text: CMS

text: data

text: gathered

text: without

text: the

text: presence

text: of

text: the

text: solenoid

text: ﬁeld

text: as

text: well

text: a

text: reanalysis

text: of

text: the

text: and

text: TeV

text: data

text: by

text: the

text: ATLAS

text: Collaboration

text: has

text: increased

text: the

text: statistical

text: preference

text: for

text: new

text: physics,

text: with

text: CMS

text: now

text: reporting

text: a

text: local

text: signiﬁcance

text: in

text: their

text: combined

text: data.

text: At

text: this

text: stage,

text: the

text: information

text: on

text: which

text: theorists

text: can

text: build

text: models

text: is

text: somewhat

text: minimal,

text: assuming

text: that

text: these

text: results

text: are

text: not

text: just

text: a

text: somewhat

text: unlikely

text: statistical

text: ﬂuctuation

text: and

text: there

text: is

text: actually

text: new

text: physics

text: to

text: be

text: begin

text: with.

text: A

text: signal

text: is

text: not

text: very

text: large,

text: and

text: additional

text: slicing

text: of

text: the

text: data

text: in

text: order

text: to

text: ask

text: more

text: detailed

text: questions

text: will

text: typically

text: yield

text: only

text: small

text: statistical

text: preference

text: for

text: any

text: possible

text: result.

text: The

text: goal

text: of

text: this

text: paper

text: is

text: to

text: investigate

text: what

text: information

text: can

text: be

text: extracted

text: from

text: the

text: combination

text: of

text: ATLAS

text: and

text: CMS

text: data

text: in

text: the

text: diphoton

text: channel,

text: including

text: the

text: most

text: recent

text: analyses

text: released

text: at

text: the

text: Moriond

text: conference

text: In

text: particular,

text: I

text: am

text: interested

text: in

text: the

text: statistical

text: signiﬁcance

text: of

text: the

text: combined

text: TeV

text: Run-II

text: data,

text: and

text: in

text: the

text: signiﬁcance

text: of

text: combinations

text: with

text: the

text: Run-I

text: TeV

text: data.

text: In

text: addition,

text: the

text: ATLAS

text: Collaboration

text: claims

text: a

text: better

text: ﬁt

text: local)

text: to

text: a

text: resonance

text: at

text: GeV

text: with

text: a

text: width

text: of

text: GeV,

text: as

text: opposed

text: to

text: a

text: narrow

text: resonance

text: with

text: width

text: much

text: less

text: than

text: the

text: experimental

text: resolution

text: (as

text: might

text: be

text: naively

text: expected).

text: Fitting

text: this

text: wide

text: resonance

text: into

text: theoretical

text: models

text: has

text: been

text: the

text: focus

text: of

text: many

text: of

text: the

text: papers

text: on

text: the

text: topic.

text: In

text: this

text: paper,

text: I

text: consider

text: the

text: combined

text: statistical

text: preference

text: for

text: this

text: width.

text: This

text: paper

text: should

text: be

text: viewed

text: as

text: a

text: follow-up

text: to

text: the

text: work

text: of

text: Refs.

text: which

text: are

text: some

text: of

text: the

text: earliest

text: theoretical

text: papers

text: on

text: the

text: GeV

text: anomaly,

text: and

text: have

text: largely

text: set

text: the

text: parameters

text: which

text: later

text: papers

text: have

text: adopted.

text: Where

text: our

text: results

text: overlap,

text: my

text: results

text: broadly

text: agree

text: with

text: those

text: found

text: in

text: Refs.

text: in

text: particular

text: in

text: terms

text: of

text: the

text: best-ﬁt

text: cross

text: sections

text: for

text: the

text: anomaly.

text: This

text: paper

text: goes

text: into

text: somewhat

text: more

text: detail

text: in

text: ﬁtting

text: the

text: signal

text: in

text: the

text: ATLAS

text: and

text: CMS

text: data,

text: by

text: ﬂoating

text: the

text: ﬁt

text: to

text: the

text: background

text: functional

text: forms

text: as

text: signal

text: is

text: added.

text: I

text: also

text: perform

text: ﬁts

text: to

text: resonances,

text: and

text: investigate

text: the

text: preference

text: of

text: the

text: combined

text: data

text: for

text: a

text: wide

text: or

text: narrow

text: resonance.

text: This

text: paper

text: does

text: not

text: attempt

text: to

text: address

text: the

text: implications

text: of

text: this

text: anomaly

text: for

text: other

text: search

text: channels

text: at

text: the

text: LHC,

text: new

text: physics

text: models

text: beyond

text: the

text: resonance

text: itself,

text: or

text: derive

text: any

text: limits

text: on

text: fundamental

text: couplings.

text: Such

text: questions

text: have

text: been

text: well

text: addressed

text: in

text: the

text: literature.

text: I

text: restrict

text: this

text: paper

text: solely

text: to

text: the

text: input

text: parameters

text: for

text: theoretical

text: studies:

text: the

text: cross

text: sections,

text: widths,

text: and

text: statistical

text: signiﬁcance

text: of

text: the

text: anomaly

text: itself.

text: The

text: next

text: section

text: describes

text: my

text: combination

text: ﬁts

text: to

text: the

text: ATLAS

text: and

text: CMS

text: diphoton

text: data,

text: using

text: both

text: and

text: TeV

text: data.

text: For

text: simplicity

text: (and

text: due

text: to

text: the

text: computational

text: limitations

text: of

text: the

text: event

text: generation),

text: I

text: consider

text: only

text: local

text: sig-

text: niﬁcance

text: in

text: my

text: combination

text: ﬁts,

text: rather

text: than

text: including

text: the

text: look-elsewhere-eﬀect.

text: The

text: statistical

text: question

text: at

text: hand

text: here

text: is

text: the

text: signiﬁcance

text: of

text: the

text: anomaly

text: at

text: GeV

text: speciﬁcally,

text: thus

text: somewhat

text: justifying

text: this

text: choice.

text: I

text: parameterize

text: the

text: possible

text: couplings

text: of

text: the

text: mediator

text: to

text: gluons

text: and

text: light

text: quarks

text: separately,

text: and

text: consider

text: both

text: and

text: resonances.

text: I

text: present

text: additional

text: results

text: for

text: second-generation

text: and

text: b-quark

text: couplings

text: in

text: Appendix

text: A.

text: In

text: Section

text: III,

text: I

text: then

text: turn

text: to

text: the

text: question

text: of

text: the

text: statistical

text: preference

text: for

text: a

text: wide

text: or

text: narrow

text: resonance;

text: using

text: the

text: best-ﬁt

text: parameters

text: from

text: Section

text: II.

text: I

text: will

text: discuss

text: my

text: conclusions

text: in

text: Section

text: IV,

text: but

text: I

text: summarize

text: some

text: of

text: the

text: more

text: salient

text: points

text: here:


text: (a) (b) (c) Figure 4: Priors and posteriors for the patient with oscillating data (used to generate ﬁgure 3). In green are the priors of θ (a), θ (b), and θ (c). In blue 0 1 2 are histograms of the corresponding posterior samples. Figure 5: A diabetic patient with normal fasting OGTT and an Insulin resistance proﬁle. 8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

label: paragraph

text: (b)

text: (c)

text: Figure

text: Priors

label: caption

text: and

text: posteriors

text: for

text: the

text: patient

text: with

text: oscillating

text: data

text: (used

text: to

text: generate

text: ﬁgure

text: In

text: green

text: are

text: the

text: priors

text: of

text: θ

text: (a),

text: θ

text: (b),

text: and

text: θ

text: (c).

text: In

text: blue

text: are

text: histograms

text: of

text: the

text: corresponding

text: posterior

text: samples.

text: Figure

text: A

text: diabetic

text: patient

text: with

text: normal

text: fasting

text: OGTT

text: and

text: an

text: Insulin

text: resistance

text: proﬁle.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: (a) (b) (c) Figure 6: Priors and posteriors for the insulin resistant patient (used to generate ﬁgure 5). In green are the priors of θ (a), θ (b), and θ (c). In blue 0 1 2 are histograms of the corresponding posterior samples. glucose levels. For healthy patients and also for diabetics, it is uncommon for such data to become available for the duration of the test. θ is an important element of inference, and often data provide informa- 2 tion for it however, as yet, there is no clear relationship between the value of θ and a patient diagnosis. 2 We intend to provide a tool for proper diagnosis using OGTT. For this very important public health issue, there is a need for a more sophisticated tool than the direct recording of values read during the test. Covariates (weight, age, etc.) will be used to help in the analysis, creating a hierarchical model embedded into the ODE model, in more population based studies, using ideally a large sample of patients with a range of health conditions to tune the model parameter value ranges to establish a more comprehensive diagnosis tool than what is currently available. A more extensive validation of the model, and model parameters, is needed in order to have an eﬀective diagnosis tool. This will necessarily involve the OGTT tests, and their further analysis using our model, in a wide range of healthy individuals and individuals with a various prediabetic and diabetic conditions to establish a more comprehensive diagnosis tool than what is currently available. We are in the course of such research and our results will be published elsewhere. 9 ##LTFigure## ##LTFigure## ##LTFigure##

text: (a)

text: (b)

text: (c)

text: Figure

text: Priors

text: and

text: posteriors

text: for

text: the

text: insulin

text: resistant

text: patient

text: (used

text: to

text: generate

text: ﬁgure

text: In

text: green

text: are

text: the

text: priors

text: of

text: θ

text: (a),

text: θ

text: (b),

text: and

text: θ

text: (c).

text: In

text: blue

text: are

text: histograms

text: of

text: the

text: corresponding

text: posterior

text: samples.

text: glucose

text: levels.

text: For

text: healthy

text: patients

text: and

text: also

text: for

text: diabetics,

text: it

text: is

text: uncommon

text: for

text: such

text: data

text: to

text: become

text: available

text: for

text: the

text: duration

text: of

text: the

text: test.

text: θ

text: is

text: an

text: important

text: element

text: of

text: inference,

text: and

text: often

text: data

text: provide

text: informa-

text: tion

text: for

text: it

text: however,

text: as

text: yet,

text: there

text: is

text: no

text: clear

text: relationship

text: between

text: the

text: value

text: of

text: θ

text: and

text: a

text: patient

text: diagnosis.

text: We

text: intend

text: to

text: provide

text: a

text: tool

text: for

text: proper

text: diagnosis

text: using

text: OGTT.

text: For

text: this

text: very

text: important

text: public

text: health

text: issue,

text: there

text: is

text: a

text: need

text: for

text: a

text: more

text: sophisticated

text: tool

text: than

text: the

text: direct

text: recording

text: of

text: values

text: read

text: during

text: the

text: test.

text: Covariates

text: (weight,

text: age,

text: etc.)

text: will

text: be

text: used

text: to

text: help

text: in

text: the

text: analysis,

text: creating

text: a

text: hierarchical

text: model

text: embedded

text: into

text: the

text: ODE

text: model,

text: in

text: more

text: population

text: based

text: studies,

text: using

text: ideally

text: a

text: large

text: sample

text: of

text: patients

text: with

text: a

text: range

text: of

text: health

text: conditions

text: to

text: tune

text: the

text: model

text: parameter

text: value

text: ranges

text: to

text: establish

text: a

text: more

text: comprehensive

text: diagnosis

text: tool

text: than

text: what

text: is

text: currently

text: available.

text: A

text: more

text: extensive

text: validation

text: of

text: the

text: model,

text: and

text: model

text: parameters,

text: is

text: needed

text: in

text: order

text: to

text: have

text: an

text: eﬀective

text: diagnosis

text: tool.

text: This

text: will

text: necessarily

text: involve

text: the

text: OGTT

text: tests,

text: and

text: their

text: further

text: analysis

text: using

text: our

text: model,

text: in

text: a

text: wide

text: range

text: of

text: healthy

text: individuals

text: and

text: individuals

text: with

text: a

text: various

text: prediabetic

text: and

text: diabetic

text: conditions

text: to

text: establish

text: a

text: more

text: comprehensive

text: diagnosis

text: tool

text: than

text: what

text: is

text: currently

text: available.

text: We

text: are

text: in

text: the

text: course

text: of

text: such

text: research

text: and

text: our

text: results

text: will

text: be

text: published

text: elsewhere.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: ∩ CH(R V ), except for uv. We note that γ∗ is a convex polygonal chain that shares its endpoints with γ i γ, and denote the region enclosed between γ and γ∗ as R . γ Let E denote the collection of all straight edges in the polygo- i T nal boundaries of the pseudo-trapezoids in ∗ and in the polygonal γ γ i U D 1 2 ch≤ains≤γ∗ corresponding to new exposed subarcs γ of ∂ ( j−1), γ1∗ γ2∗ 1 j i, which were created and polygonalized when adding the corresponding pseudo-disk D . See ﬁgure on the right. j When the pseudo-disks cover the plane. When the polygonalization process terminates, there are no more regions R , for boundary arcs γ of the union (because there is no boundary), so we are left γ with a straight-edge planar map M with E as its set of edges. (Invariant (I1) in Lemma 2.2 below m asserts that the edges in E do not cross each other.) By Euler’s formula, the complexity of M is O(m). m We then triangulate each face of M, and, as the analysis in the next subsection will show, obtain the desired triangulation. The general case. In general, the construction decomposes the union into (pairwise openly disjoint) triangles and crescent regions. To complete the construction, we decom- ≥ pose each crescent region into triangles and caps. A crescent region with t 2 vertices − − on its concave boundary can be decomposed into t 2 triangles and at most t 1 caps. ≥ The case t = 2 is vacuous, as the crescent is then a cap, so assume that t 3. To get such a decompo- sition, take an extreme edge of the concave polygonal chain, and extend it till it intersects the convex boundary of the crescent, at some point w, thereby chopping oﬀ a cap from the crescent. We then create the triangles that w spans with all the concave edges that it sees, and then recurse on the remaining − − crescent; see ﬁgure on the right. It is easily seen that this results in t 2 triangles and at most t 1 caps, as claimed. After this ﬁx-up, we get a decomposition of the union into triangles and caps. Here too, by Euler’s formula, the complexity of M is O(m). 2.3. Analysis The correctness of the construction is established in the following lemma. T Lemma 2.2. The pseudo-trapezoids in ∗ and the edges of E satisfy the following invariants: i i (I1) The segments in E do not cross one another. i U D ⊆ (I2) Each subarc γ of ∂ ( ) with endpoints u and v has an associated convex polygonal arc γ∗ E i i between u and v. The chains γ∗ are pairwise openly disjoint, and their union forms the boundary U ⊆ U D of a polygonal region ∗ ( ). i T i (cid:83) (I3) The pseudo-trapezoids in ∗ are pairwise openly disjoint, and each of them is fully contained in D i some pseudo-disk of . i U D \ (I4) ( ) τ∗ consists of a collection of pairwise openly disjoint holes. Each hole is a region i τ∗∈T ∗ i between two x-monotone convex chains or between two x-monotone concave chains, with common endpoints, where either both chains are polygonal, or one is polygonal and the other is a portion U D of the boundary of a single pseudo-disk that lies on ∂ ( ). (Each of the latter holes is a i crescent-like region of the form R , R , for some trapezoid τ, or R , for some exposed arc γ, τb τt γ 12 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CH(R

text: V

text: except

text: for

text: uv.

text: We

text: note

text: that

text: γ∗

text: is

text: a

text: convex

text: polygonal

text: chain

text: that

text: shares

text: its

text: endpoints

text: with

text: γ

text: i

text: γ,

text: and

text: denote

text: the

text: region

text: enclosed

text: between

text: γ

text: and

text: γ∗

text: as

text: R

text: γ

text: Let

text: E

text: denote

text: the

text: collection

text: of

text: all

text: straight

text: edges

text: in

text: the

text: polygo-

text: i

text: T

text: nal

text: boundaries

text: of

text: the

text: pseudo-trapezoids

text: in

text: and

text: in

text: the

text: polygonal

text: γ

text: γ

text: i

text: U

text: D

text: ch≤ains≤γ∗

text: corresponding

text: to

text: new

text: exposed

text: subarcs

text: γ

text: of

text: γ1∗

text: γ2∗

text: j

text: i,

text: which

text: were

text: created

text: and

text: polygonalized

text: when

text: adding

text: the

text: corresponding

text: pseudo-disk

text: D

text: See

text: ﬁgure

text: on

text: the

text: right.

text: j

text: When

text: the

text: pseudo-disks

text: cover

text: the

text: plane.

text: When

text: the

text: polygonalization

text: process

text: terminates,

text: there

text: are

text: no

text: more

text: regions

text: R

text: for

text: boundary

text: arcs

text: γ

text: of

text: the

text: union

text: (because

text: there

text: is

text: no

text: boundary),

text: so

text: we

text: are

text: left

text: γ

text: with

text: a

text: straight-edge

text: planar

text: map

text: M

text: with

text: E

text: as

text: its

text: set

text: of

text: edges.

text: (Invariant

text: in

text: Lemma

text: below

text: m

text: asserts

text: that

text: the

text: edges

text: in

text: E

text: do

text: not

text: cross

text: each

text: other.)

text: By

text: Euler’s

text: formula,

text: the

text: complexity

text: of

text: M

text: is

text: O(m).

text: m

text: We

text: then

text: triangulate

text: each

text: face

text: of

text: M,

text: and,

text: as

text: the

text: analysis

text: in

text: the

text: next

text: subsection

text: will

text: show,

text: obtain

text: the

text: desired

text: triangulation.

text: The

text: general

text: case.

text: In

text: general,

text: the

text: construction

text: decomposes

text: the

text: union

text: into

text: (pairwise

text: openly

text: disjoint)

text: triangles

text: and

text: crescent

text: regions.

text: To

text: complete

text: the

text: construction,

text: we

text: decom-

text: pose

text: each

text: crescent

text: region

text: into

text: triangles

text: and

text: caps.

text: A

text: crescent

text: region

text: with

text: t

text: vertices

text: on

text: its

text: concave

text: boundary

text: can

text: be

text: decomposed

text: into

text: t

text: triangles

text: and

text: at

text: most

text: t

text: caps.

text: The

text: case

text: t

text: is

text: vacuous,

text: as

text: the

text: crescent

text: is

text: then

text: a

text: cap,

text: so

text: assume

text: that

text: t

text: To

text: get

text: such

text: a

text: decompo-

text: sition,

text: take

text: an

text: extreme

text: edge

text: of

text: the

text: concave

text: polygonal

text: chain,

text: and

text: extend

text: it

text: till

text: it

text: intersects

text: the

text: convex

text: boundary

text: of

text: the

text: crescent,

text: at

text: some

text: point

text: w,

text: thereby

text: chopping

text: oﬀ

text: a

text: cap

text: from

text: the

text: crescent.

text: We

text: then

text: create

text: the

text: triangles

text: that

text: w

text: spans

text: with

text: all

text: the

text: concave

text: edges

text: that

text: it

text: sees,

text: and

text: then

text: recurse

text: on

text: the

text: remaining

text: crescent;

text: see

text: ﬁgure

text: on

text: the

text: right.

text: It

text: is

text: easily

text: seen

text: that

text: this

text: results

text: in

text: t

text: triangles

text: and

text: at

text: most

text: t

text: caps,

text: as

text: claimed.

text: After

text: this

text: ﬁx-up,

text: we

text: get

text: a

text: decomposition

text: of

text: the

text: union

text: into

text: triangles

text: and

text: caps.

text: Here

text: too,

text: by

text: Euler’s

text: formula,

text: the

text: complexity

text: of

text: M

text: is

text: O(m).

text: Analysis

text: The

text: correctness

text: of

text: the

text: construction

text: is

text: established

text: in

text: the

text: following

text: lemma.

text: T

text: Lemma

text: The

text: pseudo-trapezoids

text: in

text: and

text: the

text: edges

text: of

text: E

text: satisfy

text: the

text: following

text: invariants:

text: i

text: i

text: The

text: segments

text: in

text: E

text: do

text: not

text: cross

text: one

text: another.

text: i

text: U

text: D

text: Each

text: subarc

text: γ

text: of

text: with

text: endpoints

text: u

text: and

text: v

text: has

text: an

text: associated

text: convex

text: polygonal

text: arc

text: γ∗

text: E

text: i

text: i

text: between

text: u

text: and

text: v.

text: The

text: chains

text: γ∗

text: are

text: pairwise

text: openly

text: disjoint,

text: and

text: their

text: union

text: forms

text: the

text: boundary

text: U

text: U

text: D

text: of

text: a

text: polygonal

text: region

text: i

text: T

text: i

text: The

text: pseudo-trapezoids

text: in

text: are

text: pairwise

text: openly

text: disjoint,

text: and

text: each

text: of

text: them

text: is

text: fully

text: contained

text: in

text: D

text: i

text: some

text: pseudo-disk

text: of

text: i

text: U

text: D

text: τ∗

text: consists

text: of

text: a

text: collection

text: of

text: pairwise

text: openly

text: disjoint

text: holes.

text: Each

text: hole

text: is

text: a

text: region

text: i

text: τ∗∈T

text: i

text: between

text: two

text: x-monotone

text: convex

text: chains

text: or

text: between

text: two

text: x-monotone

text: concave

text: chains,

text: with

text: common

text: endpoints,

text: where

text: either

text: both

text: chains

text: are

text: polygonal,

text: or

text: one

text: is

text: polygonal

text: and

text: the

text: other

text: is

text: a

text: portion

text: U

text: D

text: of

text: the

text: boundary

text: of

text: a

text: single

text: pseudo-disk

text: that

text: lies

text: on

text: (Each

text: of

text: the

text: latter

text: holes

text: is

text: a

text: i

text: crescent-like

text: region

text: of

text: the

text: form

text: R

text: R

text: for

text: some

text: trapezoid

text: τ,

text: or

text: R

text: for

text: some

text: exposed

text: arc

text: γ,

text: τb

text: τt

text: γ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3.4. Crossing properties of the planar subdivision − Recall that our construction computes a t2-division of the xy-projection L(cid:48) of L where t := (cn/r (cid:28) k k 43.5r)/9βr (recall that k = n/r, r n, and β is a constant). Our goal in the rest of this section is to ↑ show that the lifting (∆) of any triangle ∆ contained in the convex hull C of a subgraph (“piece”) P k of this decomposition intersects at most ck planes of H. We prove this for bounded triangles, the proof for unbounded triangles is similar. ↑ Recall that for a point p in the xy-plane, we denote by (p) the (unique) point that lies on L and k ∈ R k is co-vertical with p. The crossing distance cr(p, q) between any pair of points p, q 3, with respect to H, is the number of planes of H that intersect the closed segment pq. The crossing distance is a ⊆ R quasi-metric, in that it is symmetric and satisﬁes the triangle inequality. For a connected set X 3, the crossing number cr(X) of X is the number of planes of H intersecting X (thus cr(p, q) is the crossing number of the closed segment pq). ∈ ↑ Lemma 3.6. Let p, q, r be three collinear points in the xy-plane, such that q pr, and let p(cid:48) = (p), ↑ ↑ k q(cid:48) = (q), and r(cid:48) = (r); these points, that lie on the k-level, are in general not collinear. Let q(cid:48)(cid:48) be the k k ≤ intersection of the vertical line through q with the segment p(cid:48)r(cid:48). Then we have cr(q(cid:48)(cid:48), q(cid:48)) 1cr(p(cid:48), r(cid:48))+8.5. 2 Proof: For a point q we denote by level(q) the number of planes lying vertically strictly below q. Put k(cid:48)(cid:48) = level(q(cid:48)(cid:48)). The point p(cid:48) lies at level k, which is the closure of all points of level k. So the number of planes lying vertically strictly below q is k if p(cid:48) is in the relative interior of a face of level k, at least − − k 1 if p(cid:48) is in the relative interior of an edge of level k, and at least k 2 if p(cid:48) is a vertex in level k. In ≥ − either case we have level(p(cid:48)) k 2, and similarly for r(cid:48), and thus ≥ | − | ≥ | − | − cr(p(cid:48), q(cid:48)(cid:48)) level(p(cid:48)) level(q(cid:48)(cid:48)) k k(cid:48)(cid:48) 2 , and ≥ | − | ≥ | − | − cr(q(cid:48)(cid:48), r(cid:48)) level(p(cid:48)) level(q(cid:48)(cid:48)) k k(cid:48)(cid:48) 2 . On the other hand we have p0 q00 L ≤ | − | ≤ | − | k cr(q(cid:48), q(cid:48)(cid:48)) k(cid:48)(cid:48) level(q(cid:48)) + 3 k k(cid:48)(cid:48) + 5 . r0 | − | ≤ | − − | ≤ | − | q0 (Indeed, if q(cid:48)(cid:48) lies above q(cid:48) then k(cid:48)(cid:48) level(q(cid:48)) k(cid:48)(cid:48) (k 2) k(cid:48)(cid:48) k +2, | − | ≤ | − | p q r and if q(cid:48) lies above q(cid:48)(cid:48) then k(cid:48)(cid:48) level(q(cid:48)) k(cid:48)(cid:48) k . In addition, the diﬀerence in the levels of q(cid:48) and q(cid:48)(cid:48) does not count the at most three planes that intersect q(cid:48)q(cid:48)(cid:48) at q(cid:48)(cid:48) if q(cid:48)(cid:48) is above q(cid:48) and at q(cid:48) otherwise). Hence, ≤ 1 cr(q(cid:48), q(cid:48)(cid:48)) (cr(p(cid:48), q(cid:48)(cid:48)) + cr(q(cid:48)(cid:48), r(cid:48)) + 4) + 5 2 ≤ 1 1 (cr(p(cid:48), r(cid:48)) + 3) + 7 = cr(p(cid:48), r(cid:48)) + 8.5. 2 2 In what follows, we consider polygonal regions contained in L , where each such region R is a k connected union of some of the faces of L . The xy-projection of R is a connected polygon in the k xy-plane, and, for simplicity, we refer to R itself also as a polygon. π Lemma 3.7. Let H be a set of n non-vertical planes in R3 in general 2 q0 u0 position. Let P (cid:48) be a bounded connected polygon with t edges that lies on A the k-level L of (H), such that all the boundary edges of P (cid:48) are edges π1 q00 k of Lk. Let p(cid:48) be a vertex of the external boundary of P (cid:48), and let q be p0 p v0 any point in the convex hull C of the xy-projection P of P (cid:48). Then the ↑ u crossing distance between p(cid:48) and q(cid:48) = (q) is at most 3t + 14.5. q k v 23 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Crossing

text: properties

text: of

text: the

text: planar

text: subdivision

text: Recall

text: that

text: our

text: construction

text: computes

text: a

text: of

text: the

text: xy-projection

text: of

text: L

text: where

text: t

text: (cn/r

text: k

text: k

text: (recall

text: that

text: k

text: n/r,

text: r

text: n,

text: and

text: β

text: is

text: a

text: constant).

text: Our

text: goal

text: in

text: the

text: rest

text: of

text: this

text: section

text: is

text: to

text: show

text: that

text: the

text: lifting

text: of

text: any

text: triangle

text: contained

text: in

text: the

text: convex

text: hull

text: C

text: of

text: a

text: subgraph

text: (“piece”)

text: P

text: k

text: of

text: this

text: decomposition

text: intersects

text: at

text: most

text: ck

text: planes

text: of

text: H.

text: We

text: prove

text: this

text: for

text: bounded

text: triangles,

text: the

text: proof

text: for

text: unbounded

text: triangles

text: is

text: similar.

text: Recall

text: that

text: for

text: a

text: point

text: p

text: in

text: the

text: xy-plane,

text: we

text: denote

text: by

text: (p)

text: the

text: (unique)

text: point

text: that

text: lies

text: on

text: L

text: and

text: k

text: R

text: k

text: is

text: co-vertical

text: with

text: p.

text: The

text: crossing

text: distance

text: cr(p,

text: q)

text: between

text: any

text: pair

text: of

text: points

text: p,

text: q

text: with

text: respect

text: to

text: H,

text: is

text: the

text: number

text: of

text: planes

text: of

text: H

text: that

text: intersect

text: the

text: closed

text: segment

text: pq.

text: The

text: crossing

text: distance

text: is

text: a

text: R

text: quasi-metric,

text: in

text: that

text: it

text: is

text: symmetric

text: and

text: satisﬁes

text: the

text: triangle

text: inequality.

text: For

text: a

text: connected

text: set

text: X

text: the

text: crossing

text: number

text: cr(X)

text: of

text: X

text: is

text: the

text: number

text: of

text: planes

text: of

text: H

text: intersecting

text: X

text: (thus

text: cr(p,

text: q)

text: is

text: the

text: crossing

text: number

text: of

text: the

text: closed

text: segment

text: pq).

text: Lemma

text: Let

text: p,

text: q,

text: r

text: be

text: three

text: collinear

text: points

text: in

text: the

text: xy-plane,

text: such

text: that

text: q

text: pr,

text: and

text: let

text: (p),

text: k

text: (q),

text: and

text: (r);

text: these

text: points,

text: that

text: lie

text: on

text: the

text: k-level,

text: are

text: in

text: general

text: not

text: collinear.

text: Let

text: be

text: the

text: k

text: k

text: intersection

text: of

text: the

text: vertical

text: line

text: through

text: q

text: with

text: the

text: segment

text: Then

text: we

text: have

text: Proof:

text: For

text: a

text: point

text: q

text: we

text: denote

text: by

text: level(q)

text: the

text: number

text: of

text: planes

text: lying

text: vertically

text: strictly

text: below

text: q.

text: Put

text: The

text: point

text: lies

text: at

text: level

text: k,

text: which

text: is

text: the

text: closure

text: of

text: all

text: points

text: of

text: level

text: k.

text: So

text: the

text: number

text: of

text: planes

text: lying

text: vertically

text: strictly

text: below

text: q

text: is

text: k

text: if

text: is

text: in

text: the

text: relative

text: interior

text: of

text: a

text: face

text: of

text: level

text: k,

text: at

text: least

text: k

text: if

text: is

text: in

text: the

text: relative

text: interior

text: of

text: an

text: edge

text: of

text: level

text: k,

text: and

text: at

text: least

text: k

text: if

text: is

text: a

text: vertex

text: in

text: level

text: k.

text: In

text: either

text: case

text: we

text: have

text: k

text: and

text: similarly

text: for

text: and

text: thus

text: k

text: and

text: k

text: On

text: the

text: other

text: hand

text: we

text: have

text: L

text: k

text: k

text: (Indeed,

text: if

text: lies

text: above

text: then

text: (k

text: k

text: p

text: q

text: r

text: and

text: if

text: lies

text: above

text: then

text: k

text: In

text: addition,

text: the

text: diﬀerence

text: in

text: the

text: levels

text: of

text: and

text: does

text: not

text: count

text: the

text: at

text: most

text: three

text: planes

text: that

text: intersect

text: at

text: if

text: is

text: above

text: and

text: at

text: otherwise).

text: Hence,

text: In

text: what

text: follows,

text: we

text: consider

text: polygonal

text: regions

text: contained

text: in

text: L

text: where

text: each

text: such

text: region

text: R

text: is

text: a

text: k

text: connected

text: union

text: of

text: some

text: of

text: the

text: faces

text: of

text: L

text: The

text: xy-projection

text: of

text: R

text: is

text: a

text: connected

text: polygon

text: in

text: the

text: k

text: xy-plane,

text: and,

text: for

text: simplicity,

text: we

text: refer

text: to

text: R

text: itself

text: also

text: as

text: a

text: polygon.

text: π

text: Lemma

text: Let

text: H

text: be

text: a

text: set

text: of

text: n

text: non-vertical

text: planes

text: in

text: in

text: general

text: position.

text: Let

text: P

text: be

text: a

text: bounded

text: connected

text: polygon

text: with

text: t

text: edges

text: that

text: lies

text: on

text: A

text: the

text: k-level

text: L

text: of

text: (H),

text: such

text: that

text: all

text: the

text: boundary

text: edges

text: of

text: P

text: are

text: edges

text: π1

text: k

text: of

text: Lk.

text: Let

text: be

text: a

text: vertex

text: of

text: the

text: external

text: boundary

text: of

text: P

text: and

text: let

text: q

text: be

text: p

text: any

text: point

text: in

text: the

text: convex

text: hull

text: C

text: of

text: the

text: xy-projection

text: P

text: of

text: P

text: Then

text: the

text: u

text: crossing

text: distance

text: between

text: and

text: (q)

text: is

text: at

text: most

text: q

text: k

text: v

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) A · to 4r/n times the complexity of (H), namely,4 O((r/n) n3) = O(n2r). | | To estimate the expected value of X , we note that each pair (e, e(cid:48)) that is counted in this i i sum belongs to the set, call it X , of pairs that are accounted for in the bound in Lemma 4.2, with 0 q = 7n/(4r), but our pairs constitute∈only a s(cid:0)mall subset of X0(cid:1). Speciﬁcally, by our choice of random levels, the probability of a pair (e, e(cid:48)) X to appear in one of the sets X is at most proportional to 0 i · | | · · (4r/n)2 X = O (r/n)2 n3 (n/r) = O(n2r). 0 (cid:32) (cid:33) (cid:18) (cid:19) Substituting the separate bounds obtain(cid:88)ed so far in Eq. (4.1), we get that the expected size of Ξ satisﬁes | | 1 r | | r2 · Ξ = O M∗ = O n2r = O(r3). t2 i n2 i=1 By applying this construction with r(cid:48) = r/c instead of r, for some suﬃciently large constant c, we can guarantee that each prism does not intersect more than n/r planes, at the cost of increasing the number of prisms by a constant factor. We therefore obtain the following result. R Theorem 4.4. For a set H of n non-vertical planes in 3 in general position, and a parameter r < n, A one can construct a layered (1/r)-cutting of (H) of size O(r3). Speciﬁcally, we cover space by a set Ξ of O(r3) vertical triangular prisms, such that each point is covered at most twice, and each prism is crossed ≤ by n/r planes of H. The top triangles of the prisms form r polyhedral terrains, each approximating A a suitable level of (H), and the bottom triangles of the prisms form r other polyhedral terrains of a (cid:0) (cid:1) similar nature. One can construct such a cutting eﬃciently by using a relative 1, 1 -approximation of size O(r log r) A r 2 as in Section . We construct the arrangement (S) of such a sample S, apply the construction as A described above to obtain a (1/r(cid:48))-cutting of (S) for r(cid:48) = r/c where c is a suitable ﬁxed constant, and − A claim that this is, with probability 1 1/rO(1), also (1/r)-cutting of (H). It takes linear time to sample | | A S and then O( S 3) = O(r3 log3 r) time to construct (S) and apply to it the algorithm described above. One can construct the conﬂict lists using a standard range reporting data structure. We preprocess ≥ · H into a data structure of size s n, in time O(s polylog(n)), so that, for each query segment e, we can report all the k planes of H that e crosses in time O( n polylog(n)) + k ); See [Mat92b, Cha12] e (cid:80) (cid:80)s1/3 e for details. We query the structure with the O(r3) edges of the prisms of the cutting, and assemble from the outputs, in a straightforward manner, the conﬂict lists of the prisms. The overall running time is O(( r3n + s)polylog(n) + k ). We have that k = O(r3(n/r)) = O(nr2). We choose { s1/3 } e e e e s = max r9/4n3/4, n ; this makes the running time O(nr2polylog(n)), as is easily checked. 4.2. Approximate halfspace range counting R In its dual setting, the problem is: Let H be a set of n nonvertical planes in 3 in general position, and let ε > 0 be an error parameter. We wish to preprocess H into a data structure that supports queries of the form: For a query point q, count the number of planes lying below q, up to a multiplicative factor ± − of 1 ε. That is, if q lies at level k, the answer should be between (1 ε)k and (1 + ε)k. A A 4Each vertex of (H) appears in three consecutive levels, and each edge appears in two, so features of (H) may be drawn more than once, but at most three times. 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: to

text: times

text: the

text: complexity

text: of

text: (H),

text: O((r/n)

text: To

text: estimate

text: the

text: expected

text: value

text: of

text: X

text: we

text: note

text: that

text: each

text: pair

text: (e,

text: that

text: is

text: counted

text: in

text: this

text: i

text: i

text: sum

text: belongs

text: to

text: the

text: set,

text: call

text: it

text: X

text: of

text: pairs

text: that

text: are

text: accounted

text: for

text: in

text: the

text: bound

text: in

text: Lemma

text: with

text: q

text: but

text: our

text: pairs

text: constitute∈only

text: a

text: subset

text: of

text: Speciﬁcally,

text: by

text: our

text: choice

text: of

text: random

text: levels,

text: the

text: probability

text: of

text: a

text: pair

text: (e,

text: X

text: to

text: appear

text: in

text: one

text: of

text: the

text: sets

text: X

text: is

text: at

text: most

text: proportional

text: to

text: i

text: X

text: O

text: (n/r)

text: Substituting

text: the

text: separate

text: bounds

text: so

text: far

text: in

text: Eq.

text: we

text: get

text: that

text: the

text: expected

text: size

text: of

text: Ξ

text: satisﬁes

text: r

text: Ξ

text: O

text: M∗

text: O

text: i

text: By

text: applying

text: this

text: construction

text: with

text: r/c

text: instead

text: of

text: r,

text: for

text: some

text: suﬃciently

text: large

text: constant

text: c,

text: we

text: can

text: guarantee

text: that

text: each

text: prism

text: does

text: not

text: intersect

text: more

text: than

text: n/r

text: planes,

text: at

text: the

text: cost

text: of

text: increasing

text: the

text: number

text: of

text: prisms

text: by

text: a

text: constant

text: factor.

text: We

text: therefore

text: obtain

text: the

text: following

text: result.

text: R

text: Theorem

text: For

text: a

text: set

text: H

text: of

text: n

text: non-vertical

text: planes

text: in

text: in

text: general

text: position,

text: and

text: a

text: parameter

text: r

text: n,

text: A

text: one

text: can

text: construct

text: a

text: layered

text: of

text: (H)

text: of

text: size

text: Speciﬁcally,

text: we

text: cover

text: space

text: by

text: a

text: set

text: Ξ

text: of

text: vertical

text: triangular

text: prisms,

text: such

text: that

text: each

text: point

text: is

text: covered

text: at

text: most

text: twice,

text: and

text: each

text: prism

text: is

text: crossed

text: by

text: n/r

text: planes

text: of

text: H.

text: The

text: top

text: triangles

text: of

text: the

text: prisms

text: form

text: r

text: polyhedral

text: terrains,

text: each

text: approximating

text: A

text: a

text: suitable

text: level

text: of

text: (H),

text: and

text: the

text: bottom

text: triangles

text: of

text: the

text: prisms

text: form

text: r

text: other

text: polyhedral

text: terrains

text: of

text: a

text: similar

text: nature.

text: One

text: can

text: construct

text: such

text: a

text: cutting

text: eﬃciently

text: by

text: using

text: a

text: relative

text: -approximation

text: of

text: size

text: O(r

text: log

text: r)

text: A

text: r

text: as

text: in

text: Section

text: We

text: construct

text: the

text: arrangement

text: (S)

text: of

text: such

text: a

text: sample

text: S,

text: apply

text: the

text: construction

text: as

text: A

text: described

text: above

text: to

text: obtain

text: a

text: of

text: (S)

text: for

text: r/c

text: where

text: c

text: is

text: a

text: suitable

text: ﬁxed

text: constant,

text: and

text: A

text: claim

text: that

text: this

text: is,

text: with

text: probability

text: also

text: of

text: (H).

text: It

text: takes

text: linear

text: time

text: to

text: sample

text: A

text: S

text: and

text: then

text: O(

text: S

text: r)

text: time

text: to

text: construct

text: (S)

text: and

text: apply

text: to

text: it

text: the

text: algorithm

text: described

text: above.

text: One

text: can

text: construct

text: the

text: conﬂict

text: lists

text: using

text: a

text: standard

text: range

text: reporting

text: data

text: structure.

text: We

text: preprocess

text: H

text: into

text: a

text: data

text: structure

text: of

text: size

text: s

text: n,

text: in

text: time

text: O(s

text: polylog(n)),

text: so

text: that,

text: for

text: each

text: query

text: segment

text: e,

text: we

text: can

text: report

text: all

text: the

text: k

text: planes

text: of

text: H

text: that

text: e

text: crosses

text: in

text: time

text: O(

text: n

text: polylog(n))

text: k

text: See

text: e

text: e

text: for

text: details.

text: We

text: query

text: the

text: structure

text: with

text: the

text: edges

text: of

text: the

text: prisms

text: of

text: the

text: cutting,

text: and

text: assemble

text: from

text: the

text: outputs,

text: in

text: a

text: straightforward

text: manner,

text: the

text: conﬂict

text: lists

text: of

text: the

text: prisms.

text: The

text: overall

text: running

text: time

text: is

text: O((

text: s)polylog(n)

text: k

text: We

text: have

text: that

text: k

text: We

text: choose

text: e

text: e

text: e

text: e

text: s

text: max

text: n

text: this

text: makes

text: the

text: running

text: time

text: as

text: is

text: easily

text: checked.

text: Approximate

text: halfspace

text: range

text: counting

text: R

text: In

text: its

text: dual

text: setting,

text: the

text: problem

text: is:

text: Let

text: H

text: be

text: a

text: set

text: of

text: n

text: nonvertical

text: planes

text: in

text: in

text: general

text: position,

text: and

text: let

text: ε

text: be

text: an

text: error

text: parameter.

text: We

text: wish

text: to

text: preprocess

text: H

text: into

text: a

text: data

text: structure

text: that

text: supports

text: queries

text: of

text: the

text: form:

text: For

text: a

text: query

text: point

text: q,

text: count

text: the

text: number

text: of

text: planes

text: lying

text: below

text: q,

text: up

text: to

text: a

text: multiplicative

text: factor

text: of

text: ε.

text: That

text: is,

text: if

text: q

text: lies

text: at

text: level

text: k,

text: the

text: answer

text: should

text: be

text: between

text: ε)k

text: and

text: ε)k.

text: A

text: A

text: vertex

text: of

text: (H)

text: appears

text: in

text: three

text: consecutive

text: levels,

text: and

text: each

text: edge

text: appears

text: in

text: two,

text: so

text: features

text: of

text: (H)

text: may

text: be

text: drawn

text: more

text: than

text: once,

text: but

text: at

text: most

text: three

text: times.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [Aga91b] P. K. Agarwal. Intersection and Decomposition Algorithms for Planar Arrangements. Cam- bridge University Press, New York, 1991. [AM95] P. K. Agarwal and J. Matouˇsek. Dynamic half-space range reporting and its applications. Algorithmica, 13:325–345, 1995. [AMS94] B. Aronov, J. Matouˇsek, and M. Sharir. On the sum of squares of cell complexities in hyperplane arrangements. J. Combinat. Theory Ser. A, 65:311–321, 1994. [AT14] P. Afshani and K. Tsakalidis. Optimal deterministic shallow cuttings for 3d dominance ranges. In Proc. 25th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 1389–1398, 2014. [BR52] R. P. Bambah and C. A. Rogers. Covering the plane with convex sets. J. London Math. Soc., s1-27(3):304–314, 1952. [CEG+90] K. L. Clarkson, H. Edelsbrunner, L. J. Guibas, M. Sharir, and E. Welzl. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom., 5:99– 160, 1990. [CEGS91] B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir. A singly-exponential stratiﬁcation scheme for real semi-algebraic varieties and its applications. Theoret. Comput. Sci., 84:77– 105, 1991. Also in Proc. 16th Int. Colloq. on Automata, Languages and Programming, pages 179–193, 179–193. [CF90] B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229–249, 1990. [Cha93] B. Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9(2):145–158, 1993. ≤ [Cha00] T. M. Chan. Random sampling, halfspace range reporting, and construction of ( k)-levels in three dimensions. SIAM J. Comput., 30(2):561–575, 2000. [Cha04] B. Chazelle. Cuttings. In D. P. Mehta and S. Sahni, editors, Handbook of Data Structures and Applications, chapter 25. Chapman and Hall/CRC, Boca Raton, Florida, 2004. [Cha05] T. M. Chan. Low-dimensional linear programming with violations. SIAM J. Comput., 34(4):879–893, 2005. [Cha10] T. M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. Assoc. Comput. Mach., 57(3):1–15, 2010. Art. 16. [Cha12] T. M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661–690, 2012. [Cla87] K. L. Clarkson. New applications of random sampling in computational geometry. Discrete Comput. Geom., 2:195–222, 1987. [CS89] K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387–421, 1989. [CT15] T. M. Chan and K. Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. In Proc. 31st Int. Annu. Sympos. Comput. Geom. (SoCG), pages 719–732, 2015. 30

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text: E.

text: Ezra,

text: D.

text: Halperin,

text: and

text: M.

text: Sharir.

text: Speeding

text: up

text: the

text: incremental

text: construction

text: of

text: the

text: union

text: of

text: geometric

text: objects

text: in

text: practice.

text: Comput.

text: Geom.

text: Theory

text: Appl.,

text: G.

text: N.

text: Frederickson.

text: Fast

text: algorithms

text: for

text: shortest

text: paths

text: in

text: planar

text: graphs,

text: with

text: applications.

text: SIAM

text: J.

text: Comput.,

text: S.

text: Har-Peled.

text: Constructing

text: planar

text: cuttings

text: in

text: theory

text: and

text: practice.

text: SIAM

text: J.

text: Comput.,

text: S.

text: Har-Peled,

text: H.

text: Kaplan,

text: and

text: M.

text: Sharir.

text: Approximating

text: the

text: k-level

text: in

text: three-dimensional

text: plane

text: arrangements.

text: In

text: Proc.

text: ACM-SIAM

text: Sympos.

text: Discrete

text: Algs.

text: (SODA),

text: pages

text: S.

text: Har-Peled

text: and

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text: Sharir.

text: Relative

text: (p,

text: ε)-approximations

text: in

text: geometry.

text: Discrete

text: Comput.

text: Geom.,

text: K.

text: Kedem,

text: R.

text: Livne,

text: J.

text: Pach,

text: and

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text: Sharir.

text: On

text: the

text: union

text: of

text: Jordan

text: regions

text: and

text: collision-

text: free

text: translational

text: motion

text: amidst

text: polygonal

text: obstacles.

text: Discrete

text: Comput.

text: Geom.,

text: P.

text: N.

text: Klein,

text: S.

text: Mozes,

text: and

text: C.

text: Sommer.

text: Structured

text: recursive

text: separator

text: decompositions

text: for

text: planar

text: graphs

text: in

text: linear

text: time.

text: In

text: Proc.

text: Annu.

text: ACM

text: Sympos.

text: Theory

text: Comput.

text: (STOC),

text: pages

text: V.

text: Koltun.

text: Almost

text: tight

text: upper

text: bounds

text: for

text: vertical

text: decompositions

text: in

text: four

text: dimensions.

text: J.

text: Assoc.

text: Comput.

text: Mach.,

text: H.

text: Kaplan,

text: E.

text: Ramos,

text: and

text: M.

text: Sharir.

text: Range

text: minima

text: queries

text: with

text: respect

text: to

text: a

text: random

text: permutation,

text: and

text: approximate

text: range

text: counting.

text: Discrete

text: Comput.

text: Geom.,

text: V.

text: Koltun

text: and

text: M.

text: Sharir.

text: Curve-sensitive

text: cuttings.

text: SIAM

text: J.

text: Comput.,

text: R.

text: J.

text: Lipton

text: and

text: R.

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text: A

text: separator

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text: for

text: planar

text: graphs.

text: SIAM

text: J.

text: Appl.

text: Math.,

text: J.

text: Matouˇsek.

text: Construction

text: of

text: ε-nets.

text: Discrete

text: Comput.

text: Geom.,

text: J.

text: Matouˇsek.

text: Eﬃcient

text: partition

text: trees.

text: Discrete

text: Comput.

text: Geom.,

text: J.

text: Matouˇsek.

text: Range

text: searching

text: with

text: eﬃcient

text: hierarchical

text: cutting.

text: Discrete

text: Comput.

text: Geom.,

text: J.

text: Matouˇsek.

text: Reporting

text: points

text: in

text: halfspaces.

text: Comput.

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text: Theory

text: Appl.,

text: J.

text: Matouˇsek.

text: On

text: constants

text: for

text: cuttings

text: in

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text: plane.

text: Discrete

text: Comput.

text: Geom.,

text: G.

text: L.

text: Miller.

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text: simple

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text: separators

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text: J.

text: Comput.

text: Syst.

text: Sci.,


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text: Draft version May 28, 2020 Preprint typeset using LATEX style emulateapj v. 08/22/09 HST HOT-JUPITER TRANSMISSION SPECTRAL SURVEY: CLEAR SKIES FOR COOL SATURN WASP-39b Patrick D. Fischer1, Heather A. Knutson1, David K. Sing2, Gregory W. Henry3, Michael W. Williamson3, Jonathan J. Fortney4, Adam S. Burrows5, Tiffany Kataria2, Nikolay Nikolov2, Adam P. Showman6, Gilda E. Ballester6, Jean-Michel De´sert7, Suzanne Aigrain8, Drake Deming9, Alain Lecavelier des Etangs10, Alfred Vidal-Madjar10 1Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA 2Astrophysics Group, School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK 3Center of Excellence in Information Systems, Tennessee State University, Nashville, TN 37209, USA 4Department of Astronomy and Astrophysics, University of California Santa Cruz, CA 95064, USA 5Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA 6Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA 7CASA, Department of Astrophysical and Planetary Sciences, University of Colorado, 389-UCB, Boulder, CO 80309, USA 8Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK 9Department of Astronomy, University of Maryland, College Park, MD 20742, USA and 10CNRS, Institut dAstrophysique de Paris, UMR 7095, 98bis boulevard Arago, F-75014 Paris, France Draft version May 28, 2020 ABSTRACT We present HST STIS optical transmission spectroscopy of the cool Saturn-mass exoplanet WASP- 39b from 0.29-1.025 µm, along with complementary transit observations from Spitzer IRAC at 3.6 and 4.5 µm. The low density and large atmospheric pressure scale height of WASP-39b make it particularly amenable to atmospheric characterization using this technique. We detect a Rayleigh scattering slope as well as sodium and potassium absorption features; this is the ﬁrst exoplanet in which both alkali features are clearly detected with the extended wings predicted by cloud-free atmosphere models. The full transmission spectrum is well matched by a clear, H -dominated atmosphere or one containing a 2 weak contribution from haze, in good agreement with the preliminary reduction of these data presented in Sing et al. (2016). WASP-39b is predicted to have a pressure-temperature proﬁle comparable to that of HD 189733b and WASP-6b, making it one of the coolest transiting gas giants observed in our HST STIS survey. Despite this similarity, WASP-39b appears to be largely cloud-free while the transmission spectra of HD 189733b and WASP-6b both indicate the presence of high altitude clouds or hazes. These observations further emphasize the surprising diversity of cloudy and cloud-free gas giant planets in short-period orbits and the corresponding challenges associated with developing predictive cloud models for these atmospheres. Subject headings: planetary systems — planets and satellites: atmospheres — stars: individual (WASP-39) — techniques: spectroscopic 1. INTRODUCTION reﬂective cloud layers (Demory et al. 2013; Shporer & Hu 2015). These observations indicate that there is a range Over the past twenty years, ongoing radial velocity and of hot Jupiter albedos, consisting of planets with rela- transit surveys have detected more than 200 short-period tively low (<0.1) to somewhat higher (0.3-0.4) albedos gas giant planets transiting bright, nearby stars. This in the optical Kepler bandpass. population of planets, which are often referred to as hot Although a high albedo can indicate the presence of Jupiters, has provided an unprecedented opportunity to clouds, some planets may have relatively tenuous and/or study the properties of hydrogen-dominated atmospheres low-albedo cloud layers. By measuring the wavelength- at temperatures more akin to cool stars than solar sys- dependent transit depth or transmission spectrum of tem gas giants (Seager & Deming 2010; Heng & Show- these planets as they pass in front of their host stars, we man 2015). Despite more than a decade of study, many can detect the presence of trace clouds and hazes located aspects of hot Jupiter atmospheres remain poorly under- near the planet’s day-night terminator. During the tran- stood, as demonstrated by the detection of clouds and sit light from the star travels along a slant optical path hazes in a subset of these atmospheres. through the upper part of the planet’s atmosphere, where We can determine the presence or absence of clouds even relatively small amounts of haze or cloud particles in hot Jupiter atmospheres via several complementary can result in a signiﬁcant scattering opacity (e.g. Fortney techniques. Observations of the secondary eclipse, when et al. 2003, 2005; Pont et al. 2008). For cloud-free hot the planet passes behind the star, can be used to con- Jupiters, sodium and potassium are predicted to produce strain the planet’s visible-light albedo and detect the strong absorption at optical wavelengths, while water is signatures of reﬂective cloud layers in the upper atmo- the strongest absorber in the near-infrared (e.g. Seager sphere (e.g. Heng & Demory 2013; Evans et al. 2013; & Sasselov 2000; Hubbard et al. 2001; Sudarsky et al. Sheets & Deming 2014; Angerhausen et al. 2014). By 2003). However in many of the systems observed, the observing changes in the planet’s albedo as a function of predicted absorption features from Na, K, and water are orbital phase, we can map the relative locations of these

text: Draft

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: HST

text: HOT-JUPITER

text: TRANSMISSION

text: SPECTRAL

text: SURVEY:

text: CLEAR

text: SKIES

text: FOR

text: COOL

text: SATURN

text: Patrick

text: D.

text: Heather

text: A.

text: David

text: K.

text: Gregory

text: W.

text: Michael

text: W.

text: Jonathan

text: J.

text: Adam

text: S.

text: Tiffany

text: Nikolay

text: Adam

text: P.

text: Gilda

text: E.

text: Jean-Michel

text: Suzanne

text: Drake

text: Alain

text: Lecavelier

text: des

text: Alfred

text: of

text: Geological

text: and

text: Planetary

text: Sciences,

text: California

text: Institute

text: of

text: Technology,

text: Pasadena,

text: CA

text: USA

text: Group,

text: School

text: of

text: Physics,

text: University

text: of

text: Exeter,

text: Stocker

text: Road,

text: Exeter,

text: UK

text: of

text: Excellence

text: in

text: Information

text: Systems,

text: Tennessee

text: State

text: University,

text: Nashville,

text: TN

text: USA

text: of

text: Astronomy

text: and

text: Astrophysics,

text: University

text: of

text: California

text: Santa

text: Cruz,

text: CA

text: USA

text: of

text: Astrophysical

text: Sciences,

text: Peyton

text: Hall,

text: Princeton

text: University,

text: Princeton,

text: NJ

text: USA

text: and

text: Planetary

text: Laboratory,

text: University

text: of

text: Arizona,

text: Tucson,

text: AZ

text: USA

text: Department

text: of

text: Astrophysical

text: and

text: Planetary

text: Sciences,

text: University

text: of

text: Colorado,

text: Boulder,

text: CO

text: USA

text: of

text: Physics,

text: University

text: of

text: Oxford,

text: Denys

text: Wilkinson

text: Building,

text: Keble

text: Road,

text: Oxford

text: UK

text: of

text: Astronomy,

text: University

text: of

text: Maryland,

text: College

text: Park,

text: MD

text: USA

text: and

text: Institut

text: dAstrophysique

text: de

text: Paris,

text: UMR

text: boulevard

text: Arago,

text: Paris,

text: France

text: Draft

text: version

text: May

text: ABSTRACT

text: We

text: present

text: HST

text: STIS

text: optical

text: transmission

text: spectroscopy

text: of

text: the

text: cool

text: Saturn-mass

text: exoplanet

text: WASP-

text: from

text: µm,

text: along

text: with

text: complementary

text: transit

text: observations

text: from

text: Spitzer

text: IRAC

text: at

text: and

text: µm.

text: The

text: low

text: density

text: and

text: large

text: atmospheric

text: pressure

text: scale

text: height

text: of

text: make

text: it

text: particularly

text: amenable

text: to

text: atmospheric

text: characterization

text: using

text: this

text: technique.

text: We

text: detect

text: a

text: Rayleigh

text: scattering

text: slope

text: as

text: well

text: as

text: sodium

text: and

text: potassium

text: absorption

text: features;

text: this

text: is

text: the

text: ﬁrst

text: exoplanet

text: in

text: which

text: both

text: alkali

text: features

text: are

text: clearly

text: detected

text: with

text: the

text: extended

text: wings

text: predicted

text: by

text: cloud-free

text: atmosphere

text: models.

text: The

text: full

text: transmission

text: spectrum

text: is

text: well

text: matched

text: by

text: a

text: clear,

text: H

text: -dominated

text: atmosphere

text: or

text: one

text: containing

text: a

text: weak

text: contribution

text: from

text: haze,

text: in

text: good

text: agreement

text: with

text: the

text: preliminary

text: reduction

text: of

text: these

text: data

text: presented

text: in

text: Sing

text: et

text: al.

text: is

text: predicted

text: to

text: have

text: a

text: pressure-temperature

text: proﬁle

text: comparable

text: to

text: that

text: of

text: HD

text: and

text: making

text: it

text: one

text: of

text: the

text: coolest

text: transiting

text: gas

text: giants

text: observed

text: in

text: our

text: HST

text: STIS

text: survey.

text: Despite

text: this

text: similarity,

text: appears

text: to

text: be

text: largely

text: cloud-free

text: while

text: the

text: transmission

text: spectra

text: of

text: HD

text: and

text: both

text: indicate

text: the

text: presence

text: of

text: high

text: altitude

text: clouds

text: or

text: hazes.

text: These

text: observations

text: further

text: emphasize

text: the

text: surprising

text: diversity

text: of

text: cloudy

text: and

text: cloud-free

text: gas

text: giant

text: planets

text: in

text: short-period

text: orbits

text: and

text: the

text: corresponding

text: challenges

text: associated

text: with

text: developing

text: predictive

text: cloud

text: models

text: for

text: these

text: atmospheres.

text: Subject

text: headings:

text: planetary

text: systems

text: planets

text: and

text: satellites:

text: atmospheres

text: stars:

text: individual

text: techniques:

text: spectroscopic

text: INTRODUCTION

text: reﬂective

text: cloud

text: layers

text: (Demory

text: et

text: al.

text: Shporer

text: Hu

text: These

text: observations

text: indicate

text: that

text: there

text: is

text: a

text: range

text: Over

text: the

text: past

text: twenty

text: years,

text: ongoing

text: radial

text: velocity

text: and

text: of

text: hot

text: Jupiter

text: albedos,

text: consisting

text: of

text: planets

text: with

text: rela-

text: transit

text: surveys

text: have

text: detected

text: more

text: than

text: short-period

text: tively

text: low

text: to

text: somewhat

text: higher

text: albedos

text: gas

text: giant

text: planets

text: transiting

text: bright,

text: nearby

text: stars.

text: This

text: in

text: the

text: optical

text: Kepler

text: bandpass.

text: population

text: of

text: planets,

text: which

text: are

text: often

text: referred

text: to

text: as

text: hot

text: Although

text: a

text: high

text: albedo

text: can

text: indicate

text: the

text: presence

text: of

text: Jupiters,

text: has

text: provided

text: an

text: unprecedented

text: opportunity

text: to

text: clouds,

text: some

text: planets

text: may

text: have

text: relatively

text: tenuous

text: and/or

text: study

text: the

text: properties

text: of

text: hydrogen-dominated

text: atmospheres

text: low-albedo

text: cloud

text: layers.

text: By

text: measuring

text: the

text: wavelength-

text: at

text: temperatures

text: more

text: akin

text: to

text: cool

text: stars

text: than

text: solar

text: sys-

text: dependent

text: transit

text: depth

text: or

text: transmission

text: spectrum

text: of

text: tem

text: gas

text: giants

text: (Seager

text: Deming

text: Heng

text: Show-

text: these

text: planets

text: as

text: they

text: pass

text: in

text: front

text: of

text: their

text: host

text: stars,

text: we

text: man

text: Despite

text: more

text: than

text: a

text: decade

text: of

text: study,

text: many

text: can

text: detect

text: the

text: presence

text: of

text: trace

text: clouds

text: and

text: hazes

text: located

text: aspects

text: of

text: hot

text: Jupiter

text: atmospheres

text: remain

text: poorly

text: under-

text: near

text: the

text: planet’s

text: day-night

text: terminator.

text: During

text: the

text: tran-

text: stood,

text: as

text: demonstrated

text: by

text: the

text: detection

text: of

text: clouds

text: and

text: sit

text: light

text: from

text: the

text: star

text: travels

text: along

text: a

text: slant

text: optical

text: path

text: hazes

text: in

text: a

text: subset

text: of

text: these

text: atmospheres.

text: through

text: the

text: upper

text: part

text: of

text: the

text: planet’s

text: atmosphere,

text: where

text: We

text: can

text: determine

text: the

text: presence

text: or

text: absence

text: of

text: clouds

text: even

text: relatively

text: small

text: amounts

text: of

text: haze

text: or

text: cloud

text: particles

text: in

text: hot

text: Jupiter

text: atmospheres

text: via

text: several

text: complementary

text: can

text: result

text: in

text: a

text: signiﬁcant

text: scattering

text: opacity

text: (e.g.

text: Fortney

text: techniques.

text: Observations

text: of

text: the

text: secondary

text: eclipse,

text: when

text: et

text: al.

text: Pont

text: et

text: al.

text: For

text: cloud-free

text: hot

text: the

text: planet

text: passes

text: behind

text: the

text: star,

text: can

text: be

text: used

text: to

text: con-

text: Jupiters,

text: sodium

text: and

text: potassium

text: are

text: predicted

text: to

text: produce

text: strain

text: the

text: planet’s

text: visible-light

text: albedo

text: and

text: detect

text: the

text: strong

text: absorption

text: at

text: optical

text: wavelengths,

text: while

text: water

text: is

text: signatures

text: of

text: reﬂective

text: cloud

text: layers

text: in

text: the

text: upper

text: atmo-

text: the

text: strongest

text: absorber

text: in

text: the

text: near-infrared

text: (e.g.

text: Seager

text: sphere

text: (e.g.

text: Heng

text: Demory

text: Evans

text: et

text: al.

text: Sasselov

text: Hubbard

text: et

text: al.

text: Sudarsky

text: et

text: al.

text: Sheets

text: Deming

text: Angerhausen

text: et

text: al.

text: By

text: However

text: in

text: many

text: of

text: the

text: systems

text: observed,

text: the

text: observing

text: changes

text: in

text: the

text: planet’s

text: albedo

text: as

text: a

text: function

text: of

text: predicted

text: absorption

text: features

text: from

text: Na,

text: K,

text: and

text: water

text: are

text: orbital

text: phase,

text: we

text: can

text: map

text: the

text: relative

text: locations

text: of

text: these


text: 11 Table 2 Transit Parameters from Global Fit. Instrument Date Wavelength (˚A) Parameter Value ± STIS G430L 2013 Feb 8, 12 2896 - 5706 Planet radius (Rp/R∗) 0.14463 0.00069 STIS G430L 2013 Feb 8 2896 - 5706 Transit center BJDTDB (days) 2456332.45857+−00..0000003351 STIS G430L 2013 Feb 12 2896 - 5706 Transit center BJDTDB (days) 2456336.51323+−00..0000003334 ± STIS G750L 2013 Mar 17 5259 - 10251 Planet radius (Rp/R∗) 0.14452 0.00059 STIS G750L 2013 Mar 17 5259 - 10251 Transit center BJDTDB (days) 2456368.95549+−00..0000002246 ± IRAC 3.6-µm 2013 Apr 18 31618 - 39284 Planet radius (Rp/R∗) 0.14513 0.00148 IRAC 3.6-µm 2013 Apr 18 31618 - 39284 Transit center BJDTDB (days) 2456401.39733+−00..0000002326 ± IRAC 4.5-µm 2013 Oct 10 39735 - 50198 Planet radius (Rp/R∗) 0.14591 0.00096 IRAC 4.5-µm 2013 Oct 10 39735 - 50198 Transit center BJDTDB (days) 2456575.77465+−00..0000002241 ± Combined Semi-major axis (a/R◦ ∗) 11.55 ± 0.13 Combined Orbital inclination ( ) 87.93 0.14 Table 3 STIS Transmission Spectrum Results. Wavelength (˚A) Rp/R∗ STIS G430L ± 2900 - 3700 0.14429 ± 0.00230 3700 - 3950 0.14408 ± 0.00143 3950 - 4113 0.14467 ± 0.00089 4113 - 4250 0.14500 ± 0.00088 4250 - 4400 0.14653 ± 0.00087 4400 - 4500 0.14491 ± 0.00093 4500 - 4600 0.14396 ± 0.00085 4600 - 4700 0.14376 ± 0.00073 4700 - 4800 0.14447 ± 0.00086 4800 - 4900 0.14461 ± 0.00076 4900 - 5000 0.14381 ± 0.00081 5000 - 5100 0.14301 ± 0.00081 5100 - 5200 0.14361 ± 0.00089 5200 - 5300 0.14541 ± 0.00077 5300 - 5400 0.14330 ± 0.00081 5400 - 5500 0.14309 ± 0.00122 5500 - 5600 0.14437 ± 0.00098 5600 - 5700 0.14558 0.00083 STIS G750L ± 5500 - 5650 0.14381 ± 0.00115 5650 - 5880 0.14550 ± 0.00126 5880 - 5910 0.14890 ± 0.00282 5910 - 6060 0.14496 ± 0.00109 6060 - 6300 0.14531 ± 0.00111 6300 - 6450 0.14491 ± 0.00091 6450 - 6600 0.14520 ± 0.00121 6600 - 6800 0.14315 ± 0.00091 6800 - 7100 0.14301 ± 0.00092 7100 - 7650 0.14423 ± 0.00057 7650 - 7710 0.14532 ± 0.00283 7710 - 8100 0.14604 ± 0.00109 8100 - 8500 0.14451 ± 0.00200 8500 - 9000 0.14425 ± 0.00122 9000 - 10250 0.14179 0.00161 Draft version May 28, 2020 Preprint typeset using LATEX style emulateapj v. 08/22/09 HST HOT-JUPITER TRANSMISSION SPECTRAL SURVEY: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Transit

text: Parameters

text: from

text: Global

text: Fit.

text: Instrument

label: table

text: Date

text: Wavelength

text: (˚A)

text: Parameter

text: Value

text: STIS

text: Feb

text: Planet

text: radius

text: (Rp/R∗)

text: STIS

text: Feb

text: Transit

text: center

text: BJDTDB

text: (days)

text: STIS

text: Feb

text: Transit

text: center

text: BJDTDB

text: (days)

text: STIS

text: Mar

text: Planet

text: radius

text: (Rp/R∗)

text: STIS

text: Mar

text: Transit

text: center

text: BJDTDB

text: (days)

text: IRAC

text: Apr

text: Planet

text: radius

text: (Rp/R∗)

text: IRAC

text: Apr

text: Transit

text: center

text: BJDTDB

text: (days)

text: IRAC

text: Oct

text: Planet

text: radius

text: (Rp/R∗)

text: IRAC

text: Oct

text: Transit

text: center

text: BJDTDB

text: (days)

text: Combined

text: Semi-major

text: axis

text: (a/R◦

text: Combined

text: Orbital

text: inclination

text: Table

text: STIS

text: Transmission

text: Spectrum

text: Results.

text: Wavelength

text: (˚A)

text: Rp/R∗

text: STIS

text: STIS

text: Draft

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: HST

text: HOT-JUPITER

text: TRANSMISSION

text: SPECTRAL

text: SURVEY:

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