Lecture 4 – Orbits of NEOs
Prof. Dr. E. Igenbergs (LRT)
Dr. D. Koschny (ESA)
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Image: © David A. Hardy/www.astroart.org'
Near-Earth objects – a threat for Earth?
Or: NEOs for engineers and physicists
Outline
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Recap – orbit types
Aten
Semimajor Axis < 1.0 AU
Aphelion > 0.983 AU
Earth Crossing
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 a: Semi-major axis
 b: Semi-minor axis
 p: semi-latus rectum
 e: eccentricity, e2 = 1 – b2/a2
 q: pericenter, periapsis, perihel
 q: true anomaly (orbit angle)
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Definition of an inertial system
 X-axis: to vernal equinox
 Z-axis: to North of Sun
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 W: right ascension of ascending node
 i: inclination
 w: argument of pericenter
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Position of asteroid as function of time
 From Newton’s laws, Kepler found the planets to orbit the
sun in an ellipse (p = semi-latus rectum; e = eccentricity,
q = orbit angle)(*):
r
p
1  e cos q
 Kepler came up with the concept of the mean anomaly M
(P = period):
M 
2  t
P
 Geometry see next page
(*): See U. Walter, Astronautics, p. 146ff
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 From the geometry:
M  E  e sin E
 Solve this analytically:
E  M  e sin(E previous )
 Again from geometry:
q
1 e
E
tan 
 tan
2
1 e
2
 This can be solved for q.
 So: M = f(t). Compute M, from that E. Then put in formulae
for q.
 See workshop for actually doing this
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The Vis-Viva Theorem
 From energy considerations:
2 1
v  GM   
r a
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Asteroid positions in the sky - 1
Project sky onto
detector
image
“Determining a plate solution”
Or: “solve the plate”
= find the distortion
coefficients of the optical system
Distortion coefficients
Determine photometric center of
asteroid
Email to MPC with measurements
COD J04
OBS P. Ruiz
MEA D. Koschny, M. Busch, A. Kn\"ofel
TEL 1.0-m f/4.4 reflector + CCD
ACK MPCReport file updated 2012.02.05
AC2 Detlef.Koschny@esa.int
NET PPMXL
DVK073 KC2012 01 24.15317 07 08 46.82
DVK073 KC2012 01 24.16225 07 08 46.33
DVK073 KC2012 01 24.16881 07 08 45.97
----- end -----
16:17:38
+24 30 18.9 20.2 R J04
+24 30 20.4 20.3 R J04
+24 30 21.2 20.7 R J04
Right Ascension/Declination
of asteroid
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Asteroid positions in the sky - 2
Commerial software is available for analysis – e.g. http://www.astrometrica.at
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COD J04
OBS P. Ruiz
MEA D. Koschny, M. Busch, A. Kn\"ofel
TEL 1.0-m f/4.4 reflector + CCD
ACK MPCReport file updated 2012.02.05
AC2 Detlef.Koschny@esa.int
NET PPMXL
DVK073 KC2012 01 24.15317 07 08 46.82
DVK073 KC2012 01 24.16225 07 08 46.33
DVK073 KC2012 01 24.16881 07 08 45.97
----- end -----
16:17:38
+24 30 18.9 20.2 R J04
+24 30 20.4 20.3 R J04
+24 30 21.2 20.7 R J04
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First orbit estimate
 ‘Tracklets’ observed and measured, sent to Minor Planet Center
 First orbit estimate done at MPC
(http://www.minorplanetcenter.org)
 Classify object as potential NEO
 Post on ‘NEO Confirmation Page’ for follow-up observations
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How is it done?
Convert RA/Dec
in vector
With position of Earth:
Convert to heliocentric
vector, guess distance
Guess velocity
from the distance
Propagate orbit –
compute distance
From this distance –
compute new guess
for velocity
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Method of Herget - 2
y
v
a2
a1
v
x
 a1 > a2
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Method of Herget - 2
y
e.g. assume a different
direction to begin with
=> the angles won’t fit
v
a2
a1
v
x
 a1 > a2
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Method of Herget
 Input: (RA, Dec)1; (RA, Dec)2 => l1, l2 in inertial
coordinates
 Guess distances r1, r2 => P1, P2
 There is only one orbit taking the object from P1 to P2 in
time Dt
 (1) Start by ‘guessing’ v = Dx/Dt
 (2) optionally: Include gravity in 1st order:
• Compute object’s acceleration at midpoint a = (r1 + r2)/2
• Assume that this a is close to the constant acceleration between the
two points (o.k. for timescales of weeks):
• v1 = Dr/Dt – aDt/2
 (3) Propagate orbit from t1 to t2 and compute r2
• This r2 will not be the initially guessed one
• From the difference: Compute new guess for velocity
• V1, new = v1 – Dr/Dt
 Repeat (1) – (3) until Dr is sufficiently small.
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Follow-up observations allow better
orbit determination
 To confirm new objects
 Done by many teams, e.g. ESA/OGS telescope
 Amateurs play a significant role
‘The Koschny Observatory’ – 40 cm
Cassegrain (B12)
‘Optical Ground Station’ of ESA – 1-m RitcheyCretien (J04)
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Detailed orbit estimate
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Detailed orbit estimate - 2
 Take into account perturbing planets, large asteroids
• Including close flybys at Earth
 Non-gravitational effects
• Outgassing (comets)
• Yarkowsky effect (see following page)
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Yarkowsky effect
 Slows down or speeds
up the asteroid
 => change in semi-major
axis
 Transport mechanism of
main-belt asteroids into
Kirkwood gaps?
 Acts on time scales of
million of years
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Effect of Yarkowsky
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Tools for orbit computation - 1
 SPICE (http://naif.jpl.nasa.gov/naif/toolkit.html)
• Data format definition for orbit data, planetary constants, time
• Software library for geometrical computations – available in
FORTRAN, C, IDL, Matlab – wrapper for Python available
• Provides well-tested routines for many mathematical routines – e.g.
conversion from Kepler elements to cartesian coordiantes and vice
versa
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Tools for orbit computations - 2
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Tools for orbit computations - 2
JPL HORIZONS
 A web-based tool for getting information on solar system
objects
 Can produce e.g. state vectors, ephemerides
 Can be called from your own software or used
interactively
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http://ssd.jpl.nasa.gov/horizons.cgi#top
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Summary
 We were reminded of the NEO orbit types
 We heard some basics about orbits – ellipses, Kepler
 We can compute the position of an object as a function of
time in the orbit: Kepler equations
 We know how to estimate a first orbit: Method of Herget
 We learned who is doing these things in the international
context: MPC, Uni Pisa, JPL
 We heard what is needed for a more detailed orbit
estimate: Pertubations, Yarkowsky
 We saw some existing tools for performing computations
in the solar system: SPICE; HORIZONS
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Workshop
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Task 1
 What is the typical angular velocity in arcsec/sec of a
main-belt asteroid as seen from the Earth when the
object is in opposition?
• Assume a distance of 2.5 AU, circular orbit
 ESA’s 1-m telescope is an f/4.4 system. The CCD camera
has a pixel scale of 13.5 um. If you want to limit the
image smear to less than 1.5 pixel, how long can your
maximum exposure time be?
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Task 2
 How long will it take for a main-belt object of 100 m and
average thermal conductivity and a = 2.2 AU to drift to
the next resonance?
 What is needed to predict the Yarkowsky effect?
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Task 3
 Write a ‘program outline’ to compute the position of an
asteroid
 Perform the computation for an object with a = 1.5 AU, e
= 0.5
• How much time does the object spend between 0.983 and 1.02 AU?
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Task 4
 Here we develop a simplified method of guessing an
initial velocity/direction for an asteroid (“modified
Herget”)
• Given: 3 observations, direction 0 deg,
y
• Main-belt asteroids: at 2 AU
r2
a2
vt = ?
a1
g
r1
x
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Task 4 – step by step
 Assume r1, r3 – must be on lines l1, l3
 (1) Compute v = (r3 – r1)/(2t)
 (2) Make sure v hits l2 (change g accordingly)
 (3) Compute new intercept with l3, given the new direction of v
v in km/s
angle in deg
 Repeat (1) – (3)
iterations
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Task 5
 Is the object a potential NEO?
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Task 4
 Is the object a potential NEO?
• Hint: Criterion: Semi-major axis; use Vis-Viva Theorem
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