05. Near-Earth Objects

Orbital Properties of Near-Earth Objects (NEOs)
Contact author(s): Sarah Greenstreet
Last verified to run: 2024-04-24
LSST Science Pipelines version: Weekly 2024_04
Container Size: medium
Targeted learning level: beginner

Description: An introduction to near-Earth objects (NEOs) in the DP0.3 dataset, including illustrations of the four NEOs sub-populations: Amors, Apollos, Atens, and Atiras.

Skills: Use TAP queries to retrieve Solar System objects. Plot properties and orbits of near-Earth objects.

LSST Data Products: TAP DP0.3 MPCORB (10-year) and SSObject tables.

Packages: lsst.rsp.get_tap_service

Credit: Originally developed by Sarah Greenstreet and the Rubin Community Science Team in the context of the Rubin DP0.

Get Support: Find DP0-related documentation and resources at dp0.lsst.io. Questions are welcome as new topics in the Support - Data Preview 0 Category of the Rubin Community Forum. Rubin staff will respond to all questions posted there.

1. Introduction¶

This tutorial demonstrates how to use TAP to query the MPCORB and SSObject tables. The tables are queried simultaneously and joined on their SSObjectId. The properties of near-Earth objects (NEOs; perihelia $q$<1.3 au) from the DP0.3 catalogs are examined. The NEOs in this sample are restricted to those with semimajor axes $a$<4.0 au, and eccentricities $e$<1.0. The orbital properties of NEOs are examined, including plotting the planet-crossing regions and boundaries that define the four NEO sub-populations: Amors, Apollos, Atens, and Atiras.

1.1 NEO Dynamical Classes¶

Near-Earth objects (NEOs) are traditionally divided into four dynamical sub-populations based on their orbital parameters compared to the orbit of the Earth. The Earth's orbit is nearly circular with a semimajor axis $a$ = 1.0 au and eccentricity $e$ = 0.017. This gives the Earth a perihelion distance $q$ = 0.983 au and an aphelion distance of $Q$ = 1.017 au. The four NEO sub-populations (Amors, Apollos, Atens, and Atiras) are divided using these orbital parameters of the Earth as follows:

Amors have orbits exterior to Earth's orbit with perihelia larger than Earth's aphelion and less than 1.3 au (1.017 au < $q$ < 1.3 au). Amors are always farther from the Sun than Earth.

Apollos are on Earth-crossing orbits with perihelia smaller than Earth's aphelion ($q$ < 1.017 au) and semimajor axes $a$ > 1.0 au. Apollos are Earth-crossers with semimajor axes larger than Earth's orbit.

Atens are on Earth-crossing orbits with aphelia larger than Earth's perihelion ($Q$ > 0.983 au) and semimajor axes $a$ < 1.0 au. Atens are Earth-crossers with semimajor axes smaller than Earth's orbit.

Atiras have orbits interior to Earth's orbit with aphelia smaller than Earth's perihelion ($Q$ < 0.983 au). Atiras are always closer to the Sun than Earth.

The schematic diagram shown below illustrates examples of Amor, Apollos, Aten, and Atira orbits compared to Earth's orbit.

This notebook provides examples of plotting the boundaries for each NEO sub-population in semimajor axis $a$ versus eccentricity $e$ and how to divide the NEOs queried from the DP0.3 tables into these four sub-populations.

1.2 Package Imports¶

The matplotlib and numpy libraries are widely used Python libraries for plotting and scientific computing. We will use these packages below, including the matplotlib.pyplot plotting sublibrary.

We also use the lsst.rsp package to access the TAP service and query the DP0 catalogs.

In [1]:

# general python packages
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

# LSST package for TAP queries
from lsst.rsp import get_tap_service

1.3 Define Functions and Parameters¶

1.3.1 Define functions for computing orbital parameters not included in the MPCORB table¶

Define a function to convert a given perihelion distance ($q$) and eccentricity ($e$) to an orbital semimajor axis ($a$). Their relationship is defined by $q=a*(1−e)$.

In [2]:

def calc_semimajor_axis(q, e):
    """
    Given a perihelion distance and orbital eccentricity,
    calculate the semi-major axis of the orbit.

    Parameters
    ----------
    q: ndarray
        Distance at perihelion, in au.
    e: ndarray
        Orbital eccentricity.

    Returns
    -------
    a: ndarray
        Semi-major axis of the orbit, in au.
        q = a(1-e), so a = q/(1-e)
    """

    return q / (1.0 - e)

Define a function to convert a given perihelion distance ($q$) and eccentricity ($e$) to an aphelion distance ($Q$). Their relationships are defined by: $Q=q*(1+e)/(1-e)$.

In [3]:

def calc_aphelion(q, e):
    """
    Given a perihelion distance and orbital eccentricity,
    calculate the semi-major axis of the orbit.

    Parameters
    ----------
    q: ndarray
        Distance at perihelion, in au.
    e: ndarray
        Orbital eccentricity.

    Returns
    -------
    Q: ndarray
        Distance at aphelion, in au.
        Q = q*(1+e)/(1-e)
    """

    return q * (1.0 + e) / (1.0 - e)

1.3.2 Define functions to return arrays needed for plotting planet-crossing curves on an ($a$, $e$) plot¶

Define a function to return an array with ($a$, $e$) values needed to plot a curve showing the aphelion at a planet's perihelion.

In [4]:

def get_Q_at_planets_q_curve_values(q_planet):

    a_Q = np.arange(0.001, q_planet, 0.001)
    e_Q = (q_planet / a_Q) - 1.0

    return (a_Q, e_Q)

Define a function to return an array with ($a$, $e$) values needed to plot a curve showing the perihelion at a planet's aphelion.

In [5]:

def get_q_at_planets_Q_curve_values(Q_planet):

    a_q = np.arange(Q_planet, 4.0, 0.001)
    e_q = (1.0 - (Q_planet / a_q))

    return (a_q, e_q)

Define the perihelia and aphelia for the terrestrial planets and the perihelion boundary for NEOs.

In [6]:

q_Earth = 0.983
Q_Earth = 1.017

q_Mercury = 0.307
Q_Mercury = 0.467

q_Venus = 0.718
Q_Venus = 0.728

q_Mars = 1.381
Q_Mars = 1.666

q_NEOs = 1.3

Set some plotting defaults to make plots look nice.

In [7]:

plt.style.use('tableau-colorblind10')
%matplotlib inline

params = {'axes.labelsize': 15,
          'font.size': 15,
          'legend.fontsize': 10,
          'xtick.major.width': 2,
          'xtick.minor.width': 1,
          'xtick.major.size': 6,
          'xtick.minor.size': 3,
          'xtick.direction': 'in',
          'xtick.top': True,
          'lines.linewidth': 2,
          'axes.linewidth': 2,
          'axes.labelweight': 1,
          'axes.titleweight': 1,
          'ytick.major.width': 1,
          'ytick.minor.width': 2,
          'ytick.major.size': 6,
          'ytick.minor.size': 3,
          'ytick.direction': 'in',
          'ytick.right': True,
          'figure.figsize': [6, 6],
          'figure.facecolor': 'White'
          }
plt.rcParams.update(params)

2. Orbital Parameters of Near-Earth Objects (NEOs)¶

2.1 Query the DP0.3 catalogs to create a table of NEOs and their orbital parameters¶

Get an instance of the Rubin TAP service client, and assert that it exists.

In [8]:

service = get_tap_service("ssotap")
assert service is not None

Query the DP0.3 catalogs, joining the MPCORB and SSObject tables on their ssObjectId. Limit the query to select sources in the MPCORB table with perihelia $q$ < 1.3 au, eccentricities $e$ < 1, and semimajor axes $a$ < 4 au to get only near-Earth objects, excluding interstellar objects with $e$ > 1 or $a$ < 0.

Define the table returned by this query as "uniqueNEOs" since it contains the IDs of unique solar system objects.

In [9]:

q_select = '1.3'
e_select = '1.0'
a_select = '4.0'

query = """
        SELECT
            mpc.ssObjectId, mpc.mpcDesignation,
            mpc.e, mpc.q, mpc.incl
        FROM
            dp03_catalogs_10yr.MPCORB as mpc
        INNER JOIN dp03_catalogs_10yr.SSObject as sso
        ON mpc.ssObjectId = sso.ssObjectId
        WHERE mpc.q < {} AND mpc.e < {}
        AND mpc.q/(1.0-mpc.e) < {} ORDER by mpc.mpcDesignation
        """.format(q_select, e_select, a_select)

uniqueNEOs = service.search(query).to_table().to_pandas()

Calculate the semimajor axis $a$ of each object's orbit, using the function defined above, and add it as a column to the uniqueNEOs table.

In [10]:

a = calc_semimajor_axis(uniqueNEOs['q'], uniqueNEOs['e'])
uniqueNEOs['a'] = a

Then use the function defined above to calculate the aphelion distance $Q$ of each object's orbit and add it as a column to the uniqueNEOs table.

In [11]:

Q = calc_aphelion(uniqueNEOs['q'], uniqueNEOs['e'])
uniqueNEOs['Q'] = Q

Print the uniqueNEOs table.

In [12]:

uniqueNEOs

Out[12]:

	ssObjectId	mpcDesignation	e	q	incl	a	Q
0	3351269693330531197	1929 SH	0.396081	1.123543	8.450610	1.860420	2.597297
1	-5234750409166262016	1932 EA1	0.435987	1.080947	11.883250	1.916528	2.752110
2	7991128850154218427	1936 CA	0.763999	0.441069	1.321700	1.868924	3.296779
3	-7789613295760699323	1937 UB	0.623179	0.622294	6.067580	1.651432	2.680570
4	8551587316774204226	1947 XC	0.712520	0.625595	2.521620	2.176135	3.726674
...	...	...	...	...	...	...	...
39465	-3152111153344676727	iso00020	0.912171	0.069097	155.764145	0.786725	1.504354
39466	-5397621441865963928	iso00026	0.921228	0.072821	109.157257	0.924457	1.776092
39467	4406479866270187902	iso00075	0.975119	0.087357	22.231391	3.510922	6.934487
39468	7151227162020825803	iso00094	0.967251	0.085642	48.439490	2.615082	5.144522
39469	2545331322875812297	iso00116	0.924412	0.067106	119.936046	0.887787	1.708469

39470 rows × 7 columns

2.2 Create semimajor axis $a$ versus eccentricity $e$ plots of the four dynamical sub-populations of NEOs¶

Plot the semimajor axis versus eccentricity of all NEOs in the uniqueNEOs table.

In [13]:

fig, ax = plt.subplots()

plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.scatter(uniqueNEOs.a, uniqueNEOs.e, s=0.1)
ax.set_xlabel('semimajor axis (au)')
ax.set_ylabel('eccentricity')
ax.minorticks_on()

plt.show()

In order to show the boundaries for the four NEO sub-populations (Amors, Apollos, Atens, and Atiras), arrays of ($a$, $e$) values along the Earth's perihelion and aphelion distances are needed along with those for the $q$ < 1.3 au NEO boundary.

In [14]:

a_Q_Earth, e_Q_Earth = get_Q_at_planets_q_curve_values(q_Earth)

a_q_Earth, e_q_Earth = get_q_at_planets_Q_curve_values(Q_Earth)

a_q_NEO, e_q_NEO = get_q_at_planets_Q_curve_values(q_NEOs)

Plot the Earth-crossing curves as a function of $a$ and $e$ and label the NEO class regions.

In [15]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.axvline(x=1.0, color="blue")
ax.plot(a_Q_Earth, e_Q_Earth, "blue")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.plot(a_q_NEO, e_q_NEO, "black", linewidth=3)
ax.annotate('Amors', xy=(2.25, 0.5), xytext=(2.25, 0.5),
            rotation=30.0, weight='bold')
ax.annotate('Apollos', xy=(1.45, 0.75), xytext=(1.45, 0.75),
            weight='bold')
ax.annotate('Atens', xy=(0.7, 0.6), xytext=(0.7, 0.6),
            rotation=270.0, weight='bold')
ax.annotate('Atiras', xy=(0.3, 0.5), xytext=(0.3, 0.5),
            rotation=280.0, weight='bold')
ax.minorticks_on()
plt.xlabel("semimajor axis (au)")
plt.ylabel("eccentricity")
plt.title("NEO Class Boundaries")
plt.show()

To better understand the above ($a$, $e$) plot, remember that the Sun is at the origin ($a$, $e$) = (0, 0) and the Earth has $a$ = 1.0 au and $e$ = 0.017, so it sits on the x-axis where the three blue curves converge. The space between the two outermost blue curves is Earth-crossing space where the Earth-crossing NEOs (Apollos and Atens) sit. A semimajor axis $a$ = 1.0 au divides the Earth-crossing NEOs into Apollos ($a$ > 1.0 au) and Atens ($a$ < 1.0 au). The black curve marks the $q$ < 1.3 au boundary for NEOs by definition. Amors sit between the $q$ < 1.3 au curve and the outer edge of Earth-crossing space (i.e., perihelia > Earth's aphelion = $q$ > 1.017 au); Amors thus always remain farther from the Sun than the Earth. Atiras are opposite to the Amors in that they always remain closer to the Sun than the Earth and live interior to Earth-crossing space; Atiras thus have aphelia < Earth's perihelion = $Q$ < 0.983 au.

Next, in order to plot the four NEO sub-populations on an ($a$, $e$) plot like the one above, first define the parameters of each sub-population needed to extract them individually from the uniqueNEOs table.

To ensure that we don't miss any NEOs in our table by our defined parameters, total the number of objects in each extracted sub-population and assert that it equals the number of objects in the uniqueNEOs table.

In [16]:

Amor_params = (uniqueNEOs.q < q_NEOs) & (uniqueNEOs.q > Q_Earth)

Apollo_params = (uniqueNEOs.q < Q_Earth) & (uniqueNEOs.a > 1.0)

Aten_params = (uniqueNEOs.Q > q_Earth) & (uniqueNEOs.a < 1.0)

Atira_params = (uniqueNEOs.Q < q_Earth)

total_NEOs = len(uniqueNEOs[Amor_params]
                 + uniqueNEOs[Apollo_params]
                 + uniqueNEOs[Aten_params]
                 + uniqueNEOs[Atira_params])

assert total_NEOs == len(uniqueNEOs)

Plot the semimajor axis versus eccentricity of all NEOs in the uniqueNEOs table divided into the four dynamical sub-populations, overplot the boundaries for each sub-population, and label each NEO class region.

In [17]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.scatter(uniqueNEOs[Amor_params].a, uniqueNEOs[Amor_params].e,
           s=0.1, color='red')
ax.scatter(uniqueNEOs[Apollo_params].a, uniqueNEOs[Apollo_params].e,
           s=0.1, color='green')
ax.scatter(uniqueNEOs[Aten_params].a, uniqueNEOs[Aten_params].e,
           s=0.1, color='orange')
ax.scatter(uniqueNEOs[Atira_params].a, uniqueNEOs[Atira_params].e,
           s=0.25, color='purple')
ax.plot(a_Q_Earth, e_Q_Earth, "blue")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.axvline(x=1.0, color="blue")
ax.plot(a_q_NEO, e_q_NEO, "black", linewidth=5)
ax.annotate('Amors', xy=(2.25, 0.5), xytext=(2.25, 0.5),
            rotation=30.0, weight='bold')
ax.annotate('Apollos', xy=(1.45, 0.75), xytext=(1.45, 0.75),
            weight='bold')
ax.annotate('Atens', xy=(0.7, 0.6), xytext=(0.7, 0.6),
            rotation=270.0, weight='bold')
ax.annotate('Atiras', xy=(0.3, 0.5), xytext=(0.3, 0.5),
            rotation=280.0, weight='bold')
ax.minorticks_on()
plt.xlabel("semimajor axis (au)")
plt.ylabel("eccentricity")
plt.title("NEOs in DP0.3")
plt.show()

To better see the density of points in the above ($a$, $e$) scatter plot, next use a 2-D histogram plot where the bins are shown as hexagons and the color represents the number of data points within each bin, from purple/blue as the fewest to red regions containing the most objects. Again, overplot the NEO class boundaries with labels.

In [18]:

fig, ax = plt.subplots(figsize=(8, 6))

plt.xlim([0., 4.])
plt.ylim([0., 1.])

ax.hexbin(uniqueNEOs.a, uniqueNEOs.e,
          gridsize=(int(4./0.02), 50),
          cmap='Spectral_r', bins='log',
          extent=(0, 4., 0, 1))
ax.plot(a_Q_Earth, e_Q_Earth, "blue")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.axvline(x=1.0, color="blue")
ax.plot(a_q_NEO, e_q_NEO, "black", linewidth=5)
ax.annotate('Amors', xy=(2.25, 0.5), xytext=(2.25, 0.5),
            rotation=30.0, weight='bold')
ax.annotate('Apollos', xy=(1.45, 0.75), xytext=(1.45, 0.75),
            weight='bold')
ax.annotate('Atens', xy=(0.7, 0.6), xytext=(0.7, 0.6),
            rotation=270.0, weight='bold')
ax.annotate('Atiras', xy=(0.3, 0.5), xytext=(0.3, 0.5),
            rotation=280.0, weight='bold')
ax.set_ylabel('eccentricity')
ax.set_xlabel('semimajor axis (au)')
ax.minorticks_on()

plt.show()

In the above plot, it is clear that in addition to other areas of overdensity, the Amors have an overdensity of points near 2.1 au < $a$ < 2.4 au and 0.4 < $e$ < 0.5 that is difficult to see in the scatter plot alone.

The above density plot prompts the question of what fraction of NEOs are in each of the sub-populations, which can easily be calculated.

In [19]:

perc_Amors = len(uniqueNEOs[Amor_params]) / total_NEOs * 100.
perc_Apollos = len(uniqueNEOs[Apollo_params]) / total_NEOs * 100.
perc_Atens = len(uniqueNEOs[Aten_params]) / total_NEOs * 100.
perc_Atiras = len(uniqueNEOs[Atira_params]) / total_NEOs * 100.

print("Percentage of NEOs that are Amors: ", "%4.1f" % (perc_Amors), "%")
print("Percentage of NEOs that are Apollos: ", "%4.1f" % (perc_Apollos), "%")
print("Percentage of NEOs that are Atens: ", "%4.1f" % (perc_Atens), "%")
print("Percentage of NEOs that are Atiras: ", "%4.1f" % (perc_Atiras), "%")

Percentage of NEOs that are Amors:  38.2 %
Percentage of NEOs that are Apollos:  55.3 %
Percentage of NEOs that are Atens:   6.5 %
Percentage of NEOs that are Atiras:   0.1 %

2.3 Create an ($a$, $e$) plot the terrestrial-planet-crossing regions and compare the the four NEO sub-populations¶

In addition to plotting the Earth-crossing curves on an ($a$, $e$) plot, the planet-crossing curves for all the terrestrial planets can also be plotted.

First, as with the Earth-crossing curves above, arrays of ($a$, $e$) values along the Mercury, Venus, and Mars' perihelion and aphelion distances are needed.

In [20]:

a_Q_Mercury, e_Q_Mercury = get_Q_at_planets_q_curve_values(q_Mercury)
a_q_Mercury, e_q_Mercury = get_q_at_planets_Q_curve_values(Q_Mercury)

a_Q_Venus, e_Q_Venus = get_Q_at_planets_q_curve_values(q_Venus)
a_q_Venus, e_q_Venus = get_q_at_planets_Q_curve_values(Q_Venus)

a_Q_Mars, e_Q_Mars = get_Q_at_planets_q_curve_values(q_Mars)
a_q_Mars, e_q_Mars = get_q_at_planets_Q_curve_values(Q_Mars)

Plot the terrestrial-planet-crossing curves as a function of $a$ and $e$.

In [21]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.plot(a_Q_Mercury, e_Q_Mercury, "green", label="Mercury-crossing")
ax.plot(a_q_Mercury, e_q_Mercury, "green")
ax.plot(a_Q_Venus, e_Q_Venus, "orange", label="Venus-crossing")
ax.plot(a_q_Venus, e_q_Venus, "orange")
ax.plot(a_Q_Earth, e_Q_Earth, "blue", label="Earth-crossing")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.plot(a_Q_Mars, e_Q_Mars, "red", label="Mars-crossing")
ax.plot(a_q_Mars, e_q_Mars, "red")
ax.minorticks_on()
ax.legend()
plt.xlabel("semimajor axis (au)")
plt.ylabel("eccentricity")
plt.title("Terrestrial-Planet-Crossing Regions for NEOs")
plt.show()

Each of the four NEO sub-populations can be plotted in addition to the terrestrial-planet-crossing curves in order to see which of the four inner planetary orbits each NEO class crosses.

In [22]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.scatter(uniqueNEOs[Amor_params].a, uniqueNEOs[Amor_params].e,
           s=0.1, color='red')
ax.scatter(uniqueNEOs[Apollo_params].a, uniqueNEOs[Apollo_params].e,
           s=0.1, color='green')
ax.scatter(uniqueNEOs[Aten_params].a, uniqueNEOs[Aten_params].e,
           s=0.1, color='orange')
ax.scatter(uniqueNEOs[Atira_params].a, uniqueNEOs[Atira_params].e,
           s=0.25, color='purple')
ax.plot(a_Q_Mercury, e_Q_Mercury, "green", label="Mercury-crossing")
ax.plot(a_q_Mercury, e_q_Mercury, "green")
ax.plot(a_Q_Venus, e_Q_Venus, "orange", label="Venus-crossing")
ax.plot(a_q_Venus, e_q_Venus, "orange")
ax.plot(a_Q_Earth, e_Q_Earth, "blue", label="Earth-crossing")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.plot(a_Q_Mars, e_Q_Mars, "red", label="Mars-crossing")
ax.plot(a_q_Mars, e_q_Mars, "red")
ax.plot(a_q_NEO, e_q_NEO, "black", linewidth=5)
ax.annotate('Amors', xy=(2.25, 0.5), xytext=(2.25, 0.5),
            rotation=30.0, weight='bold')
ax.annotate('Apollos', xy=(1.45, 0.75), xytext=(1.45, 0.75),
            weight='bold')
ax.annotate('Atens', xy=(0.7, 0.6), xytext=(0.7, 0.6),
            rotation=270.0, weight='bold')
ax.annotate('Atiras', xy=(0.3, 0.5), xytext=(0.3, 0.5),
            rotation=280.0, weight='bold')
ax.minorticks_on()
ax.legend()
plt.xlabel("semimajor axis (au)")
plt.ylabel("eccentricity")
plt.title("Terrestrial-Planet-Crossing NEOs")
plt.show()

Next, define the parameters of each planet-crossing regions for each of the terrestrial planets in order to extract the planet-crossing NEO populations individually from the uniqueNEOs table.

In [23]:

Earth_crossing = (uniqueNEOs.q < Q_Earth) & (uniqueNEOs.Q > q_Earth)

Mercury_crossing = (uniqueNEOs.q < Q_Mercury) & (uniqueNEOs.Q > q_Mercury)

Venus_crossing = (uniqueNEOs.q < Q_Venus) & (uniqueNEOs.Q > q_Venus)

Mars_crossing = (uniqueNEOs.q < Q_Mars) & (uniqueNEOs.Q > q_Mars)

Make a 4-panel density plot of the planet-crossing NEO populations in the DP0.3 catalogs as a function of $a$ and $e$, with one panel for each of the four terrestrial planets; each density plot is individually normalized where the color represents the number of data points within each bin, from purple/blue as the fewest objects to red regions containing the most objects.

In [24]:

fig, axs = plt.subplots(2, 2)
fig.suptitle('Planet-Crossing NEOs in DP0.3')

axs[0, 0].set_xlim([0., 4.])
axs[0, 0].set_ylim([0., 1.])
axs[0, 0].hexbin(uniqueNEOs[Mercury_crossing].a,
                 uniqueNEOs[Mercury_crossing].e,
                 gridsize=(int(4./0.02), 50),
                 cmap='Spectral_r', bins='log',
                 extent=(0, 4., 0, 1))
axs[0, 0].plot(a_Q_Mercury, e_Q_Mercury, "green")
axs[0, 0].plot(a_q_Mercury, e_q_Mercury, "green")
axs[0, 0].plot(a_q_NEO, e_q_NEO, "black", linewidth=2)
axs[0, 0].set_title('Mercury-crossing')
axs[0, 0].minorticks_on()

axs[0, 1].set_xlim([0., 4.])
axs[0, 1].set_ylim([0., 1.])
axs[0, 1].hexbin(uniqueNEOs[Venus_crossing].a,
                 uniqueNEOs[Venus_crossing].e,
                 gridsize=(int(4./0.02), 50),
                 cmap='Spectral_r', bins='log',
                 extent=(0, 4., 0, 1))
axs[0, 1].plot(a_Q_Venus, e_Q_Venus, "orange")
axs[0, 1].plot(a_q_Venus, e_q_Venus, "orange")
axs[0, 1].plot(a_q_NEO, e_q_NEO, "black", linewidth=2)
axs[0, 1].set_title('Venus-crossing')
axs[0, 1].minorticks_on()

axs[1, 0].set_xlim([0., 4.])
axs[1, 0].set_ylim([0., 1.])
axs[1, 0].hexbin(uniqueNEOs[Earth_crossing].a,
                 uniqueNEOs[Earth_crossing].e,
                 gridsize=(int(4./0.02), 50),
                 cmap='Spectral_r', bins='log',
                 extent=(0, 4., 0, 1))
axs[1, 0].plot(a_Q_Earth, e_Q_Earth, "blue")
axs[1, 0].plot(a_q_Earth, e_q_Earth, "blue")
axs[1, 0].plot(a_q_NEO, e_q_NEO, "black", linewidth=2)
axs[1, 0].set_title('Earth-crossing')
axs[1, 0].minorticks_on()
axs[1, 0].xaxis.set_tick_params(labelbottom=False)

axs[1, 1].set_xlim([0., 4.])
axs[1, 1].set_ylim([0., 1.])
axs[1, 1].hexbin(uniqueNEOs[Mars_crossing].a,
                 uniqueNEOs[Mars_crossing].e,
                 gridsize=(int(4./0.02), 50),
                 cmap='Spectral_r', bins='log',
                 extent=(0, 4., 0, 1))
axs[1, 1].plot(a_Q_Mars, e_Q_Mars, "red")
axs[1, 1].plot(a_q_Mars, e_q_Mars, "red")
axs[1, 1].plot(a_q_NEO, e_q_NEO, "black", linewidth=2)
axs[1, 1].set_title('Mars-crossing')
axs[1, 1].minorticks_on()
axs[1, 1].xaxis.set_tick_params(labelbottom=False)

for ax in axs.flat:
    ax.set(xlabel='semimajor axis (au)', ylabel='eccentricity')
for ax in axs.flat:
    ax.label_outer()
plt.show()

The percentage of NEOs that are Earth-crossing or Mars-crossing or Venus-crossing or Mercury-crossing (as shown above) can also easily be calculated.

In [25]:

perc_Earth_crossing_NEOs = (
    len(uniqueNEOs[Earth_crossing]) / total_NEOs * 100.)
perc_Venus_crossing_NEOs = (
    len(uniqueNEOs[Venus_crossing]) / total_NEOs * 100.)
perc_Mercury_crossing_NEOs = (
    len(uniqueNEOs[Mercury_crossing]) / total_NEOs * 100.)
perc_Mars_crossing_NEOs = (
    len(uniqueNEOs[Mars_crossing]) / total_NEOs * 100.)

print("Percentage of NEOs that are Mercury-crossing: ", "%4.1f" %
      (perc_Mercury_crossing_NEOs), "%")
print("Percentage of NEOs that are Venus-crossing: ", "%4.1f" %
      (perc_Venus_crossing_NEOs), "%")
print("Percentage of NEOs that are Earth-crossing: ", "%4.1f" %
      (perc_Earth_crossing_NEOs), "%")
print("Percentage of NEOs that are Mars-crossing: ", "%4.1f" %
      (perc_Mars_crossing_NEOs), "%")

Percentage of NEOs that are Mercury-crossing:  13.3 %
Percentage of NEOs that are Venus-crossing:  31.5 %
Percentage of NEOs that are Earth-crossing:  61.8 %
Percentage of NEOs that are Mars-crossing:  92.3 %

Note that the Mars-crossing NEOs include all of the Amors, nearly all of the Apollos, and some of the Atens, covering the vast majority of the NEO population density at >90%. The Earth-crossing NEOs, Apollos and Atens, constitute 61.8% of the NEO population, consistent with their NEO percentages calculated above (Apollos = 55.3% of NEOs and Atens = 6.5% of NEOs). Venus- and Mercury-crossing NEOs include respectively fewer Apollos, Atens, and Atiras, where the Atens + Atiras are only 6.6% of the NEO population, resulting in respectively smaller percentages of the NEOs crossing the orbits of these planets.

It is also possible to extract the planet-crossing NEOs for multiple (or all) planets together as shown below.

In [26]:

V_AND_E_crossing = (
    (uniqueNEOs.q < Q_Venus) & (uniqueNEOs.Q > q_Earth))

Me_AND_V_AND_E_AND_Ma_crossing = (
    (uniqueNEOs.q < Q_Mercury) & (uniqueNEOs.Q > q_Mars))

The percentage of NEOs that are both Venus- and Earth-crossing can also easily be calculated, as can the percentage of NEOs that cross the orbits of Mercury, Venus, Earth, and Mars.

In [27]:

perc_V_AND_E_crossing_NEOs = (
    len(uniqueNEOs[V_AND_E_crossing]) / total_NEOs * 100.)
perc_Me_AND_V_AND_E_AND_Ma_crossing_NEOs = (
    len(uniqueNEOs[Me_AND_V_AND_E_AND_Ma_crossing]) / total_NEOs * 100.)

print("Percentage of NEOs that are both \
Venus- and Earth-crossing: ", "%4.1f" %
      (perc_V_AND_E_crossing_NEOs), "%")
print("Percentage of NEOs that are \
Mercury- and Venus- and Earth- and Mars-crossing: ", "%4.1f" %
      (perc_Me_AND_V_AND_E_AND_Ma_crossing_NEOs), "%")

Percentage of NEOs that are both Venus- and Earth-crossing:  31.4 %
Percentage of NEOs that are Mercury- and Venus- and Earth- and Mars-crossing:  11.5 %

NEOs on orbits that cross both Venus' and Earth's orbit can be additionally be plotted, as an example.

In [28]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 1.])
ax.scatter(uniqueNEOs[V_AND_E_crossing].a,
           uniqueNEOs[V_AND_E_crossing].e,
           s=0.1)
ax.plot(a_Q_Mercury, e_Q_Mercury, "green", label="Mercury-crossing")
ax.plot(a_q_Mercury, e_q_Mercury, "green")
ax.plot(a_Q_Venus, e_Q_Venus, "orange", label="Venus-crossing")
ax.plot(a_q_Venus, e_q_Venus, "orange")
ax.plot(a_Q_Earth, e_Q_Earth, "blue", label="Earth-crossing")
ax.plot(a_q_Earth, e_q_Earth, "blue")
ax.plot(a_Q_Mars, e_Q_Mars, "red", label="Mars-crossing")
ax.plot(a_q_Mars, e_q_Mars, "red")
ax.minorticks_on()
ax.legend()
plt.xlabel("semimajor axis (au)")
plt.ylabel("eccentricity")
plt.title("Venus- and Earth-Crossing NEOs")
plt.show()

2.4 Explore the inclination distribution of NEOs in DP0.3¶

First add a plot of the semimajor axis $a$ versus inclination $i$ of all NEOs in the uniqueNEOs table to the ($a$, $e$) plot.

In [29]:

fig, (ax1, ax2) = plt.subplots(2, 1)

ax1.set_xlim([0., 4.])
ax1.set_ylim([0., 90.])
ax1.scatter(uniqueNEOs.a, uniqueNEOs.incl, s=0.1)
ax1.set_ylabel('inclination (deg)')
ax1.set_title('NEOs in DP0.3')
ax1.minorticks_on()

ax2.set_xlim([0., 4.])
ax2.set_ylim([0., 1.])
ax2.scatter(uniqueNEOs.a, uniqueNEOs.e, s=0.1)
ax2.set_xlabel('semimajor axis (au)')
ax2.set_ylabel('eccentricity')
ax2.minorticks_on()

plt.show()

To better see the density of points in the above ($a$, $e$) and ($a$, $i$) scatter plot, use a 2-panel 2-D histogram plot where the bins are shown as hexagons and the color represents the number of data points within each bin, from purple/blue as the fewest to red regions containing the most objects. Again, overplot the NEO class boundaries with labels in the $a$ versus $e$ plot.

In [30]:

fig, axs = plt.subplots(2, 1, figsize=(6, 6))

axs[0].hexbin(uniqueNEOs.a,
              uniqueNEOs.incl,
              gridsize=(int(4.2/0.02), 50),
              cmap='Spectral_r', bins='log',
              extent=(0, 4.2, 0, 90))
axs[0].set_ylabel('inclination (deg)')
axs[0].minorticks_on()
axs[0].set_xlim(0., 4.)
axs[0].set_ylim(0., 90.)

axs[1].hexbin(uniqueNEOs.a,
              uniqueNEOs.e,
              gridsize=(int(4.2/0.02), 50),
              cmap='Spectral_r', bins='log',
              extent=(0, 4.2, 0, 1))
axs[1].plot(a_Q_Earth, e_Q_Earth, "blue")
axs[1].plot(a_q_Earth, e_q_Earth, "blue")
axs[1].axvline(x=1.0, color="blue")
axs[1].plot(a_q_NEO, e_q_NEO, "black", linewidth=5)
axs[1].set_ylabel('eccentricity')
axs[1].set_xlabel('semi-major axis (au)')

axs[1].set_xlim(0., 4.)
axs[1].set_ylim(0., 1.)
axs[1].minorticks_on()
plt.show()

A 2-panel scatter plot of ($a$, $i$; top) and ($a$, $e$; bottom) with all NEOs in the uniqueNEOs table split into the 4 sub-populations can also be plotted.

In [31]:

fig, (ax1, ax2) = plt.subplots(2, 1)

ax1.set_xlim([0., 4.])
ax1.set_ylim([0., 90.])
ax1.scatter(uniqueNEOs[Amor_params].a,
            uniqueNEOs[Amor_params].incl,
            s=0.1, color='red',
            label="Amors")
ax1.scatter(uniqueNEOs[Apollo_params].a,
            uniqueNEOs[Apollo_params].incl,
            s=0.1, color='green',
            label="Apollos")
ax1.scatter(uniqueNEOs[Aten_params].a,
            uniqueNEOs[Aten_params].incl,
            s=0.1, color='orange',
            label="Atens")
ax1.scatter(uniqueNEOs[Atira_params].a,
            uniqueNEOs[Atira_params].incl,
            s=0.1, color='purple',
            label="Atiras")
ax1.minorticks_on()
ax1.set_ylabel("inclination (deg)")
ax1.set_title("NEOs in DP0.3")

ax2.set_xlim([0., 4.])
ax2.set_ylim([0., 1.])
ax2.scatter(uniqueNEOs[Amor_params].a,
            uniqueNEOs[Amor_params].e,
            s=0.1, color='red',
            label="Amors")
ax2.scatter(uniqueNEOs[Apollo_params].a,
            uniqueNEOs[Apollo_params].e,
            s=0.1, color='green',
            label="Apollos")
ax2.scatter(uniqueNEOs[Aten_params].a,
            uniqueNEOs[Aten_params].e,
            s=0.1, color='orange',
            label="Atens")
ax2.scatter(uniqueNEOs[Atira_params].a,
            uniqueNEOs[Atira_params].e,
            s=0.1, color='purple',
            label="Atiras")
ax2.plot(a_Q_Earth, e_Q_Earth, "blue")
ax2.plot(a_q_Earth, e_q_Earth, "blue")
ax2.axvline(x=1.0, color="blue")
ax2.plot(a_q_NEO, e_q_NEO, "black", linewidth=5)
ax2.annotate('Amors',
             xy=(2.4, 0.5), xytext=(2.4, 0.5),
             rotation=20.0, weight='bold', fontsize=10)
ax2.annotate('Apollos',
             xy=(1.45, 0.75), xytext=(1.45, 0.75),
             weight='bold', fontsize=10)
ax2.annotate('Atens',
             xy=(0.7, 0.6), xytext=(0.7, 0.6),
             rotation=270.0, weight='bold', fontsize=10)
ax2.annotate('Atiras',
             xy=(0.4, 0.5), xytext=(0.4, 0.5),
             rotation=280.0, weight='bold', fontsize=10)
ax2.minorticks_on()
ax2.set_xlabel("semimajor axis (au)")
ax2.set_ylabel("eccentricity")

plt.show()

It can also be helpful to make a 4-panel scatter plot showing the semimajor axes $a$ versus inclinations $i$ of each NEO sub-population individually.

In [32]:

fig, axs = plt.subplots(2, 2)
fig.suptitle('NEO Inclinations in DP0.3')

axs[0, 0].set_xlim([0., 4.])
axs[0, 0].set_ylim([0., 90.])
axs[0, 0].scatter(uniqueNEOs[Atira_params].a,
                  uniqueNEOs[Atira_params].incl,
                  s=0.25, color='purple')
axs[0, 0].set_title('Atiras')
axs[0, 0].minorticks_on()

axs[0, 1].set_xlim([0., 4.])
axs[0, 1].set_ylim([0., 90.])
axs[0, 1].scatter(uniqueNEOs[Aten_params].a,
                  uniqueNEOs[Aten_params].incl,
                  s=0.1, color='orange')
axs[0, 1].set_title('Atens')
axs[0, 1].minorticks_on()

axs[1, 0].set_xlim([0., 4.])
axs[1, 0].set_ylim([0., 90.])
axs[1, 0].scatter(uniqueNEOs[Apollo_params].a,
                  uniqueNEOs[Apollo_params].incl,
                  s=0.1, color='green')
axs[1, 0].set_title('Apollos')
axs[1, 0].minorticks_on()
axs[1, 0].xaxis.set_tick_params(labelbottom=False)

axs[1, 1].set_xlim([0., 4.])
axs[1, 1].set_ylim([0., 90.])
axs[1, 1].scatter(uniqueNEOs[Amor_params].a,
                  uniqueNEOs[Amor_params].incl,
                  s=0.1, color='red')
axs[1, 1].set_title('Amors')
axs[1, 1].minorticks_on()
axs[1, 1].xaxis.set_tick_params(labelbottom=False)

for ax in axs.flat:
    ax.set(xlabel='semimajor axis (au)', ylabel='inclination (deg)')
for ax in axs.flat:
    ax.label_outer()
plt.show()

So far the above plots showing the inclination distribution of NEOs in the DP0.3 catalogs has been limited to inclinations $i<90$ deg. Now extend the inclination range in a ($a$, $i$) scatter plot beyond 90 deg to the retrograde ($90<i<180$ deg) region and label the prograde ($0<i<90$ deg) and retrograde regions.

In [33]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([0., 180.])
ax.scatter(uniqueNEOs.a, uniqueNEOs.incl, s=0.1)
ax.axhline(y=90.0, color="black", linestyle="dotted")
ax.annotate('prograde', xy=(0.05, 80.0), xytext=(0.05, 80.0))
ax.annotate('retrograde', xy=(0.05, 95.0), xytext=(0.05, 95.0))
plt.xlabel("semimajor axis (au)")
plt.ylabel("inclination (deg)")
plt.title("NEOs in DP0.3")

Out[33]:

Text(0.5, 1.0, 'NEOs in DP0.3')

The retrograde (i>90 deg) NEOs in the DP0.3 catalogs can be extracted and plotted with their object designations labeled.

In [34]:

high_i = (uniqueNEOs.incl > 90.0) & (uniqueNEOs.q < 1.3)
print(len(uniqueNEOs[high_i]))
print(uniqueNEOs[high_i][['mpcDesignation', 'q', 'a', 'e', 'incl']])

6
      mpcDesignation         q         a         e        incl
3643        2009 HC8  0.488301  2.526936  0.806762  154.360980
10351       LPCC1165  0.088383  3.776360  0.976596  116.977934
39464       iso00008  0.071941  0.919391  0.921751  158.874391
39465       iso00020  0.069097  0.786725  0.912171  155.764145
39466       iso00026  0.072821  0.924457  0.921228  109.157257
39469       iso00116  0.067106  0.887787  0.924412  119.936046

In [35]:

fig, ax = plt.subplots()
plt.xlim([0., 4.])
plt.ylim([90., 180.])
ax.scatter(uniqueNEOs[high_i].a, uniqueNEOs[high_i].incl)
ax.annotate(uniqueNEOs[high_i].iloc[0].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[0].a,
                uniqueNEOs[high_i].iloc[0].incl),
            xytext=(uniqueNEOs[high_i].iloc[0].a+0.1,
                    uniqueNEOs[high_i].iloc[0].incl-2.),
            fontsize=10)
ax.annotate(uniqueNEOs[high_i].iloc[1].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[1].a,
                uniqueNEOs[high_i].iloc[1].incl),
            xytext=(uniqueNEOs[high_i].iloc[1].a-0.65,
                    uniqueNEOs[high_i].iloc[1].incl-2.),
            fontsize=10)
ax.annotate(uniqueNEOs[high_i].iloc[2].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[2].a,
                uniqueNEOs[high_i].iloc[2].incl),
            xytext=(uniqueNEOs[high_i].iloc[2].a+0.1,
                    uniqueNEOs[high_i].iloc[2].incl-2.),
            fontsize=10)
ax.annotate(uniqueNEOs[high_i].iloc[3].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[3].a,
                uniqueNEOs[high_i].iloc[3].incl),
            xytext=(uniqueNEOs[high_i].iloc[3].a+0.1,
                    uniqueNEOs[high_i].iloc[3].incl-2.),
            fontsize=10)
ax.annotate(uniqueNEOs[high_i].iloc[4].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[4].a,
                uniqueNEOs[high_i].iloc[4].incl),
            xytext=(uniqueNEOs[high_i].iloc[4].a+0.1,
                    uniqueNEOs[high_i].iloc[4].incl-2.),
            fontsize=10)
ax.annotate(uniqueNEOs[high_i].iloc[5].mpcDesignation,
            xy=(uniqueNEOs[high_i].iloc[5].a,
                uniqueNEOs[high_i].iloc[5].incl),
            xytext=(uniqueNEOs[high_i].iloc[5].a+0.1,
                    uniqueNEOs[high_i].iloc[5].incl-2.),
            fontsize=10)
plt.xlabel("semimajor axis (au)")
plt.ylabel("inclination (deg)")
plt.title("Retrograde NEOs in DP0.3")

Out[35]:

Text(0.5, 1.0, 'Retrograde NEOs in DP0.3')

Exercises for the learner:¶

1. Plot the subset of Apollos that are only Venus-crossing (i.e., exclude the Earth-crossing Apollos).

2. Compute the percentage of NEOs on retrograde (incl > 90 deg) orbits.

3. Make a 4-panel plot showing the 1-D inclination distributions for each of the four NEO sub-populations individually.

4. How many objects in the uniqueNEOs table are interstellar objects (with prefix "iso" in their mpcDesignation)? Query the DP0.3 tables to include the interstellar objects and plot their semimajor axes, eccentricities, and inclinations.

In [ ]: