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#### Spatial and Temporal Averages for Elliptical Orbits

posted Dec 16, 2013, 5:45 AM by Abel Mendez Torres   [ updated ]

Mean orbital values for distance, stellar flux, and equilibrium temperature can be computed with respect to spatial or temporal coordinates. Spatial averages are usually calculated with respect to the mean anomaly (E) or true anomaly (f). Temporal averages are calculated with respect to time (t) or the eccentric anomaly (M). It is often assumed that the mean equilibrium temperature of a planet in an elliptic orbit can be simply calculated from its mean distance or flux, but this is not correct.

We calculated analytic solutions for both the spatial and temporal averages of distance, flux, and equilibrium temperature (Méndez et al., 2014, in preparation). Here a is the semi-major axis, e is the eccentricity, L is the stellar luminosity, A is the planet's bond albedo, and To = 278.5 K (i.e. the equilibrium temperature of Earth for zero albedo). A factor f is related to the effectiveness of atmospheric circulation and how the energy absorbed is transferred from the planet’s day to night sides (e.g. f = 1 for fast rotators and f = 2 for tidally locked planets without atmospheres)For convenience, the formulas derived here use 'exoplanet units' where distances are in AU, flux in solar units, and temperature in kelvins.

#### Mean Spatial Distance, Flux, and Equilibrium Temperature

 $\large&space;\bar{r}=a\sqrt{1-e^2}$ (1)

 $\large&space;\overline{F}=\frac{L(2+e^2)}{2a^2\(1-e^2)^2}$ (2)

 $\large&space;\overline{T}_{eq}=T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4} \frac{2}{\pi \sqrt{1-e}} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (3)

Here r̅ , F̅, and T̅eq are the mean spatial values (i.e. with respect to the true anomaly) for planet distance, stellar flux, and planet equilibrium temperature. E is the complete elliptic integral of the second kind of the argument within parenthesis.

#### Mean Temporal Distance, Flux, and Equilibrium Temperature

 $\large&space;\left \langle r \right \rangle=a\left ( 1+\frac{e^2}{2} \right )$ (4)

 $\large&space;\left \langle F \right \rangle=\frac{L}{a^2\sqrt{1-e^2}}$ (5)

 $\large&space;\left \langle T_{eq} \right \rangle=T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4}\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (6)

Here <r>, <F>, and <Teq> are the mean temporal values (i.e. with respect to time) for planet distance, stellar flux, and planet equilibrium temperature. E is the complete elliptic integral of the second kind of the argument within parenthesis.

All the previous equations become the well known expressions for circular orbits for e = 0, where there is no difference between spatial or temporal averages. For most applications the temporal averages are the ones necessary. Eq. 1, 4, and 5 are well known expressions (William and Pollard, 2002Perryman, 2011). Eq. 2, 3, and 6 are new derivations, but only Eq. 6 has practical implications. Eq. 4, 5, and 6 were also verified with a numerical simulation.

#### Using the Mean Temporal Equilibrium Temperature

The time-average stellar flux (Eq. 5) can't be used to calculate the time-average equilibrium temperature (Eq. 6) as if often assumed. The stellar flux increases to infinity as the eccentricity of the planet approaches one. However, the equilibrium temperature does not change accordingly, actually it decreases with eccentricity approaching a minimum of about 90% the value for a circular orbit. This seems contradictory but the stellar flux increases with 1/r2 while the equilibrium temperature with 1/√r. Therefore, the following expressions should not be used to calculate the average equilibrium temperature from the average distance or stellar flux since they produces large errors (>>10%) for highly eccentric orbits:

 $\large&space;\left \langle T_{eq} \right \rangle \neq T_o\left (\frac{fL(1-A)}{\left \langle r \right \rangle ^2}\right )^ \frac{1}{4} \neq T_o\left (\left \langle F \right \rangle f(1-A)\right )^ \frac{1}{4}$ (7)

where <r> is given from Eq. 4 and <F> is given from Eq. 5. These expression suggests that the equilibrium temperature increases with an increase in stellar flux due to eccentricity but it is quite the opposite.

Eq. 6 is the analytic solution to the equilibrium temperatures for eccentric orbits. Errors from assuming a circular orbits can be up to 10% for highly eccentric orbits. Still, this is not a large error compared with larger uncertainties associated with f and A. Summarizing, given the equilibrium temperature for circular orbits, the equilibrium temperature for elliptical orbits is:

 $\large&space;\left \langle T_{eq} \right \rangle=T_{eqc}\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (8)

where the equilibrium temperature for circular orbits Teqc is given by:

 $\large&space;T_{eqc} = T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4} = T_\star \sqrt{ \frac{R_\star }{2a}}\left [ f\left ( 1-A \right ) \right ]^{1/4}$ (9)

and T* and R* are the effective temperature and radius of the star.

#### Notes

• There are many math libraries which include calculations for complete elliptic integral functions (e.g. Mathematica and GSL). The following expression could be used to simplify the calculation of Eq. 6 or 8. Errors from this approximation are less than 1% (less than 0.5% for e < 0.5).
 $\large&space;\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )\approx \sqrt{1+\left (\frac{8}{\pi^2} -1 \right )e^{5/2}}$ (10)

#### SER: First Look at Pluto

posted Nov 13, 2013, 9:34 AM by Abel Mendez Torres   [ updated Nov 13, 2013, 10:02 AM ]

 The Scientific Exoplanets Renderer (SER) is our core software to generate photorealistic-looking images of planets such as the ones for the Visible Paleo-Earth and the Habitable Exoplanets Catalog. SER was developed to simulate complex stellar transit events and interpret planetary light curves. It takes as little information from a planet as available to automatically create representations. The more input given the more the images become more scientific than artistic representations. Here with used SER to create a basic representation of Pluto (Figure 1 and 2) given its albedo maps obtained by Buie et al. (2010A, 2010B) from the Hubble Space Telescope (Figure 3). We will keep generating better and better representations as we get more data from Pluto, specially from New Horizons. We also plan to produce more creative versions by adding more surface features. It will be fun to compare our progress, starting from our first image, until the final close-up pictures of Pluto on July 2015. Figure 1. This representation of Pluto combines real and synthetic data to create a more photorealistic look. The image shows a possible scenario of Pluto that preserves the original basic albedo features. It uses a false color palette (similar to Triton) for only one of the two channels of the HST filters (F435W). Credit: PHL @ UPR Arecibo, NASA HST. Figure 2. Equirectangular projection of the Pluto map shown in Figure 1. Data for the south pole was extrapolated since it was not available in the original HST data. Credit: PHL @ UPR Arecibo, NASA HST. Figure 3. This is the most detailed view to date of the entire surface of the dwarf planet Pluto, as constructed from multiple NASA Hubble Space Telescope photographs taken from 2002 to 2003. Credit: NASA, ESA, and M. Buie (Southwest Research Institute).

#### Occurrence of Earth-like planets around GKM Stars

posted Nov 5, 2013, 8:12 AM by Abel Mendez Torres   [ updated Nov 5, 2013, 8:47 AM ]

Here we combine and interpret results from Kopparapu (2013) and Petigura et al. (2013) on the occurrence of Earth-like planets around red-dwarf (type M) and Sun-like (type G and K) stars, respectively. Both studies used NASA Kepler results. Based on their occurrence estimates we calculated the mean separation and 95% confidence distance for the solar neighborhood (<10pc). These numbers suggest that there is over a 95% probability of finding an Earth-like planet around a Sun-like star in the solar neighborhood. The probability is higher for those around M-dwarf stars.

Table 1. Occurrence (ηE), separation (δ), and 95% confidence distance (d95%) of Earth-like planets (0.5 — 2.0 RE in the HZ) around M-dwarfs and GK stars. The study of Petigura et al. (2013) only considered 1.0 — 2.0 RE planets. Calculations assume that there are 198 stellar systems with M-dwarf stars and 50 with GK-stars in the solar neighborhood (<10pc).

 M-dwarfs (Kopparapu, 2013) GK stars (Petigura et al., 2013) Habitable Zone Definition a (AU)* ηE δE (ly) dE 95% (ly) ηE δE (ly) dE 95% (ly) Conservative (1) 0.99 — 1.69 0.51 +0.10/-0.20 6.25 10.1 0.086 ± 0.03 17.9 28.9 Empirical (1) 0.75 — 1.84 0.61 +0.07/-0.15 5.89 9.51 0.14 ± 0.05 15.2 24.6 Simple Flux (2) 0.5 — 2 — — — 0.22 ± 0.08 13.1 21.1 Desert-H2 Planets (3) 0.38 — 10 — — — 0.52 ± 0.19 9.8 15.9

HZ References: (1)

Table Notes:
a = semi-major axis (values shown for G stars, check here for other types of stars).
ηE (eta Earth) = fraction of Earth-like planets per star.
δ(delta Earth) = mean distance between stellar systems with Earth-like planets.
d95% = 95% confidence distance for an Earth-like planet.

#### New Kepler Potential Planet Transit Signals

posted Nov 4, 2013, 8:00 AM by Abel Mendez Torres   [ updated Nov 4, 2013, 8:03 AM ]

 The NASA Kepler team released a revised global list of 16,285 potential planet transit signals (formally known as Threshold-Crossing Events or TCE) from quarters one to sixteen, the four years of operation of Kepler (Table 1). This list already includes most of the 3,602 Kepler Objects of interest (KOI) with 170 Kepler Confirmed Planets.Additional analysis is necessary to confirm the planetary nature of the over 12,000 remaining TCE objects. Many of these transits signals are expected to be false alarms, specially for long periods objects such as those orbiting within the habitable zone (HZ).Nearly 1,500 TCE objects match our standard criteria for potentially habitable planets (planet size 0.4 — 2.6 Earth radii orbiting within the empirical HZ), but only about 20% of them might turn out to be real planets, still a large number. Nevertheless, this analysis provides some upper limits to interesting objects in the Kepler TCE sample (Table 2).Table 1. Summary statistics of some planetary and stellar characteristics of the new 16,285 TCE list.---------------------------------------------------------------------------                      Property         MIN         AVG         MAX    Count---------------------------------------------------------------------------           Planet Radius (EU):        0.00       15.53     5312.00    16285         Planet Period (days):        0.50      197.33      707.28    16285  Planet Semi-Major Axis (AU):      0.0101      0.5349      2.3200    16285        Planet Temperature(K):       63.10      875.85    14200.00    16285               Star Mass (SU):        0.09        1.06        3.71    16285             Star Radius (SU):        0.12        4.18      219.50    16285         Star Temperature (K):     2661.00     5715.10    15896.00    16285         Star Luminosity (SU):      0.0006     48.2446  14036.8495    16285---------------------------------------------------------------------------EU = earth units, SU = solar unitsTable 2. Basic analysis of the TCE objects. ----------------------------------------------------------------------------Habitable Exoplanets Catalog (HEC) Statistics  Data Source: NASA Kepler Threshold Crossing Events (TCE)  [Nov  4, 2013] Planetary Habitability Laboratory (phl.upr.edu) ----------------------------------------------------------------------------                         Stellar Systems  9743                              Exoplanets 16285              Exoplanets in the Hot Zone 10982 (67.4%) Exoplanets in the Warm 'Habitable' Zone  3646 (22.4%)             Exoplanets in the Cold Zone  1657 (10.2%)                            Unclassified     0 ( 0.0%)          Potential Habitable Exoplanets  1486 ( 9.1%)   Expected Potential Habitable Exomoons   185 ( 1.1%)                               Eta Earth  15.3 %------------------------------------------------------------------------------------------------------------------------------------------------Number of Multiple Systems--------------------------------------------------------------------     1      2      3      4      5      6     7      8      9     10  6520   1832    537    292    228    204    93     30      3      4  66.9   18.8    5.5    3.0    2.3    2.1   1.0    0.3    <0.1  <0.1-------------------------------------------------------------------------------------------------------------------------------------------Planetary Class-----------------------------------------------------------------------Mercurian    Subterran     Terran    Superterran    Neptunian    Jovian42           327           1382      5766           5063         33550.3          2.0           8.5       35.4           31.1         20.6---------------------------------------------------------------------------------------------------------------Hot Zone            Category     Count   Percent----------------------------------------          Mercurians        19       0.1          Subterrans       274       1.7             Terrans      1123       6.9        Superterrans      3664      22.5          Neptunians      2695      16.5             Jovians      2858      17.5--------------------------------------------------------------------------------Warm 'Habitable' Zone            Category     Count   Percent----------------------------------------          Mercurians         3       0.0          Subterrans        29       0.2             Terrans       117       0.7        Superterrans      1340       8.2          Neptunians      1760      10.8             Jovians       396       2.4--------------------------------------------------------------------------------Cold Zone            Category     Count   Percent----------------------------------------          Mercurians        20       0.1          Subterrans        24       0.1             Terrans       142       0.9        Superterrans       762       4.7          Neptunians       608       3.7             Jovians       101       0.6----------------------------------------ReferencesDetection of Potential Transit Signals in Sixteen Quarters of Kepler Mission Data

#### 50th Anniversary of the Arecibo Observatory

posted Oct 21, 2013, 2:15 PM by Abel Mendez Torres   [ updated Oct 27, 2013, 7:20 PM ]

#### A Binomial Nomenclature for Common Names of Exoplanets

posted Sep 27, 2013, 12:09 AM by Abel Mendez Torres   [ updated Sep 27, 2013, 12:20 AM ]

 When naming exoplanets create common names for both the stellar system and the exoplanetsThere are nearly one thousand exoplanets already confirmed and many more waiting for confirmation. The general astronomy community is interested on naming exoplanets with the help of the general public (1). Individual naming of such discoveries is a time consuming labor but both usable by the scientific community and general public.Simple procedures for the individual naming of thousands or even millions of objects are nothing new to science. For example the binomial nomenclature by genera and species in Biology has been used to name over 1.2 million species and has the potential to name more than the estimated 8.7 million species on Earth (2). These are used as scientific names, equivalent to the catalog names in astronomy, but in many cases are easier to remember and even used as common names (i.e. E. coli).We propose a similar binomial nomenclature for exoplanets to create simple hierarchical common names. The first part of the name is the stellar system and the second part is the individual planet name. If the stars have already a common name then it is used as the stellar system names (i.e. Fomalhaut). The planets or other stars of the system are named with the usual alphabetical letters until a proper name is given.This procedure reduces the complexity of the catalog star names until the planets are individually named. As an hypothetical example, the planet Alpha Centauri B b, could become Alcen-B Rakhat, where Alcen is the name of the stellar system, B denotes the second star of the system, and Rakhat the name of the planet. The actual selection process of the individual names for the system and planets are out of the scope of this proposal.Therefore, any exoplanet naming campaign should concentrate first efforts on naming the stellar systems of interest and then any individual planets, avoiding naming exoplanets before the stellar systems. The stellar names could also be used to guide the naming of its planets. Our suggestion describes a minimum effort plan for naming exoplanets and provides a logical and hierarchical system not much different from the current catalog names, but for common names too.In the end, it is expected that some exoplanets will be better known by their full name (using both system and planet names) and others just by their individual name. A new definition of planets that includes both solar and extrasolar planets would also be appropriate before any naming campaign (3).ReferencesA similar version of this letter was submitted to the IAU request for exoplanets names on September 15, 2013.

#### About 40 potentially habitable worlds by the end of 2015?

posted Sep 20, 2013, 3:12 AM by Abel Mendez Torres

We started with just two planets in our Habitable Exoplanets Catalog almost two years ago on December 2011. At that time having a catalog for just two planets was overkill and we were not expecting much change anyway until many years, but we were surprised. We ended the first years with seven and now we have twelve, and the year is not over yet. That certainly exceeded our expectations. The exoplanets field was moving much faster toward the detection of smaller Earth-size planets by both the radial velocity and transit methods. A catalog is now a necessary tool to track these discoveries.

The 2010's is the decade when we finally started to discover potentially habitable worlds (the exception is Gliese 581d discovered on 2007). We have enough data now to give some preliminary prediction on the expected number of habitable exoplanets to be discovered in the next years. If we follow the present trend (Figure 1) there could be a total of 21 by 2014 and 38 by 2015. That still sounds like too many for just two years but there are 36 waiting for confirmation from the NASA Kepler mission alone. Similar predictions can be derived for the total number of confirmed exoplanets (Figure 2). Lets wait and see what happens.
Figure 1. Cumulative number of potentially habitable exoplanets detected in the last four years (red dots). The data was fitted with an exponential function (blue line) included at the top of the plot. The fit predicts reaching about 21 exoplanets by 2014 and 38 by 2015. Predictions for longer periods are much more uncertain. Data source is available in Table 1.
Figure 2. Cumulative number of confirmed exoplanets detected in the last 22 years (red dots). The data was fitted with an exponential function (blue line) included at the top of the plot. The fit predicts reaching over 1,000 exoplanets by the end of 2013 and over 1,500 by 2015. Predictions for longer periods are much more uncertain. Data source is available in Table 1.

Table 1. Number per year and cumulative (total from all previous years) of discovered confirmed exoplanets and those potentially habitable. Data from the PHL's Exoplanet Catalog. The year 2011 was a good year for exoplanet discoveries. Figures 1 and 2 show the cumulative data plotted.
 Confirmed Potentially Habitable Year Per Year Cumulative Per year Comulative 1989 1 1 1990 0 1 1991 0 1 1992 3 4 1993 0 4 1994 0 4 1995 1 5 1996 6 11 1997 0 11 1998 7 18 1999 11 29 2000 19 48 2001 14 62 2002 30 92 2003 27 119 2004 31 150 2005 33 183 2006 29 212 2007 61 273 1 1 2008 61 334 0 1 2009 81 415 0 1 2010 114 529 1 2 2011 189 718 2 4 2012 147 865 3 7 2013 109 974 5 12

#### One-AU Exoplanets

posted Aug 12, 2013, 12:42 PM by Abel Mendez Torres   [ updated Aug 12, 2013, 1:01 PM ]

 Most known exoplanets are within one astronomical unit (AU) from their stars, a bias of our detection methods. Here are graphical representations of 417 stellar systems with planets within one AU from their star, out of 712 stars with 927 confirmed planets (some have multiple planets). Each system is represented in individual files for convenience but we plan to create a few posters out of these figures. Some interesting examples are shown below (click to enlarge) starting with our own Solar System. More specific explanations of the plots are at the end. You can download all 418 PNG figures as a single ZIP file, including our Sun, from this link. Images description: The star and planets are magnified for clarity, by x10 and x50 respectively, otherwise they would look like dots at the one AU scale of the plots. Since the scales are different for the stars and planets, the planets are always 5 times smaller than they appear compared to the star. Some stars are so big that they are out of the scale or overlapping the planets. The size of Earth, Neptune, and Jupiter is shown in each plot for reference. The stars are colored by their spectral color. Planets are colored red if hot, green warm (in the habitable zone), and blue cold. The size of the habitable zone, many times extending beyond one AU, is illustrated in light green.

#### Earth from Near and Deep Space

posted Jul 19, 2013, 2:33 AM by Abel Mendez Torres   [ updated Jul 23, 2013, 3:07 AM ]

 The Cassini spacecraft on Saturn and the Messenger spacecraft on Mercury are taking images of Earth in July 19, 2013. From their vantage point Earth will be just a few pixels, but these are rare events. Only Voyager, Cassini and Messenger have taken pictures of Earth from deep outer space before. However, not only these two spacecraft will be taking pictures of the full globe of Earth on that day. The GOES East and Meteosat meteorological satellites have similar view perspectives as Cassini and Messenger, respectively. To commemorate this event we will try to generate satellites images of Earth near the moments the pictures from Cassini and Messenger are taken. We have to accurately combine the satellite images, which are in black and white, with color information to generate true-color images. Our plan is to create a composite wallpaper with the Cassini and Messenger faraway images together with the Earth closeups. Figure 1. Simulated view of Earth at the moment of its picture from Cassini on Friday, July 19, 2013 between 5:27 to 5:42 PM EDT (21:27 and 21:47 UTC). Here shown exactly at 5:27 PM EDT. The GOES East satellite has a slightly west-displaced view of Earth than Cassini at the moment of the picture. Credit: NASA Solar System Simulator. Figure 2. Simulated view of Earth at the moment of its picture from Messenger on Friday, July 19, 2013 and Saturday, July 20, 2013 at 7:49 AM, 8:38 AM and 9:41 AM on both days  (11:49, 12:38 and 13:41 UTC). Here shown exactly at 7:49 AM EDT. The Meteosat satellite has a very similar view of Earth as Messenger at the moment of the picture. Credit: NASA Solar System Simulator. Figure 3. Actual image of Earth from space by the GEOS East meteorological satellite taken on July 18, 2013 at 5 PM EDT. Color of the image was synthetically incorporated from NASA Visible Earth imagery using the PHL's SER software. This image was created to test our software in preparation for the event. Click for larger version. Credit: PHL @ UPR Arecibo, NASA, NERC Satellite Receiving Station, Dundee University, Scotland. Here are the final results.Additional Information

#### Statistics of Nearby Earth-like Planets Around M-dwarfs Stars

posted Apr 8, 2013, 12:28 AM by Abel Mendez Torres   [ updated Apr 8, 2013, 1:20 AM ]

Here is an analysis on the frequency, number, distance, and probability of Earth-like planets (Earth-size within the habitable zone) around M-dwarf stars within 10 parsecs (~33 light years) from Earth. Table 1 shows these numbers for η(eta Earth) from four definitions of an Earth-like planet of Kopparapu (2013). These numbers assume that there are 248 M-dwarfs stars (RECONS) in 198 stellar systems (NSC) within 10 parsecs.

Table 1. Statistics of nearby Earth-like planets around M-dwarfs stars (see legend below for field definitions) for four definitions of Earth-like planet based on their size and orbit within a conservative or optimistic habitable zone.

 Planet Size Habitable Zone ηE NE δE (ly) pE 10 ly (%) dE 95% (ly) 0.5 — 1.4 RE conservative 0.48 +0.12/-0.24 119 +30/-60 6.38 ± 2.32 93.5 10.30 0.5 — 2.0 RE conservative 0.51 +0.10/-0.20 126 +25/-50 6.25 ± 2.27 94.6 10.10 0.5 — 1.4 RE optimistic 0.53 +0.08/-0.17 131 +20/-42 6.17 ± 2.24 95.1 9.97 0.5 — 2.0 RE optimistic 0.61 +0.07/-0.15 151 +17/-37 5.89 ± 2.14 96.9 9.51

Table Definitions:
ηE (eta Earth) = frequency of Earth-like planets per star (also called fraction or occurrence).
N= number of Earth-like planets.
δ(delta Earth) = mean distance between stellar systems with Earth-like planets.
pE 10 ly = probability of an Earth-like planet within 10 light years from Earth.
dE 95% = distance from Earth for the 95% confidence level of an Earth-like planet.

In summary, there are over one-hundred Earth-like planets around M-dwarf stars within 10 parsecs from Earth. Just one or two are probably transiting (transiting probability of ~0.6-1.6%). This also means that there are over 67 to 134 billion Earth-like planets around low mass stars in our galaxy. Assuming that there are 200 to 400 billion stars in the Milky Way where 70% of those are low mass stars (0.1 to 0.8 solar masses) (Bochanski et al., 2010).

References

Bochanski, J. J., Hawley, S. L., Covey, K. R., West, A. A., Reid, I. N., Golimowski, D. A., and Ivezić, v. (2010). The Luminosity and Mass Functions of Low-mass Stars in the Galactic Disk. II. The Field. The Astronomical Journal, 139, 6, 2679.

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