Trigonometric ParallaxResolution LimitsTimeKepler's LawsTwo-Body Problem

Reading for today: 3.1, 2.1-2.2

Reading for next lecture: 2.3 (p. 39-45)

Topics for TodayTopics for TodayAstonomy 62 Lecture #2

Declination = DEC = �

Right Ascension = RA = �

Equatorial Coordinate System

2=360°degrees1°=60' arcminutes 1'=60' ' arcseconds

2=24hhours 1h=60m minutes of RA1m=60s seconds of RA

Astonomy 62 Lecture #1

1. Where is the target?

2. How far is the target?2. How far is the target?

3. How bright is the target?

Basic Questons of Observatonal Basic Questons of Observatonal Astonomy:Astonomy:

Astonomy 62 Lecture #2

When p = 1'', d = 1 pc (parsec) = 3.086 x 1016m

a

a = 1 AU = 1.496 x 1011m

a

d= tan p≈ p

Trigonometic ParalaxTrigonometic Paralax

Astonomy 62 Lecture #2

Astonomy 62 Lecture #2

Problem: Calculate the distance for a star that shows a parallax angle of 0.3''.

Answer: d = 1''/0.3'' pc = 3.3 pc

Astonomy 62 Lecture #2

●Telescope diffraction limit

●Atmospheric Turbulence (Seeing)

Limits on Astometic Precision:Limits on Astometic Precision:

min≈

diff=1.22

D

min≈1 ”

Astonomy 62 Lecture #2

How do we define How do we define θθ??

=FWHM

Astonomy 62 Lecture #2

min≈1 ”=4.85×10−6 rad

Question: Find the diameter of the optical telescope for which the diffraction limit is equal to the atmospheric seeing limit,

Astonomy 62 Lecture #2

Left: Short exposure image of a star with a 10-cm ground-based telescope.Center: Short exposure with a 10-m telescope.Right: Long exposure with a 10-m telescope.

Image Credit: Vik Dhillon, Cambridge University

●Positions of 118,000 stars with precision 0.001'' (distances up to 1 kpc).

●106 stars with precision 0.02-0.03'' (distances up to 30-50 pc).

●Should measure parallaxes down to 4μas (distances up to 250 kpc).

Hipparcos – Hipparcos – High Precision Parallax High Precision Parallax Collecting Satellite Collecting Satellite (ESA, 1989-1993)(ESA, 1989-1993)

Astonomy 62 Lecture #2

SIM – SIM – Space Interferometry MissionSpace Interferometry Mission (NASA, projected launch >2020)(NASA, projected launch >2020)

●Positions of 109 stars with precision 20μas (distances and velocities up to 50 kpc).

Gaia – Gaia – Global Astrometric Interferometer Global Astrometric Interferometer for Astrophysics for Astrophysics (ESA, launch 2012)(ESA, launch 2012)

Measuring TimeMeasuring Time

Astonomy 62 Lecture #2

Universal Time (UT)Universal Time (UT) is the time on the Greenwich meridian.

Julian date (JD)Julian date (JD) is defined as the number of solar

days elapsed since noon on January 1, 4713 B.C. (according to the Julian calendar).

Noon Universal Time (UT) of January 21, 2011 corresponds to JD 2,455,583.0

Solar daySolar day is defined as one full revolution of the Earth with respect to the Sun. It is by definition 24 hours long.

Sidereal daySidereal day is defined as one full revolution of the

Earth with respect to the stars.

Heliocentric Julian date (HJD) Heliocentric Julian date (HJD) is JD measured with respect to the center of the Sun.

Measuring TimeMeasuring Time

Astonomy 62 Lecture #1

“...Accordingly, since noting prevents te eart fom moving, I suggest tat we should now consider also wheter several motons suit it, so tat it can be regarded as one of te planets. For, it is not te centr of al te revolutons.”

“Finaly, we should place te Sun himself at te centr of te Universe.”

Nicolaus Copernicus

Astonomy 62 Lecture #2

Copernican PrincipleCopernican Principle We are in no favored position in the

Universe.

Astonomy 62 Lecture #2

Kepler's Laws (1609, 1619)Kepler's Laws (1609, 1619)1st Law A planet orbits the Sun in an ellipse, with

the Sun at one focus.

2nd Law A line connecting a planet to the Sun sweeps out equal areas in equal time intervals.

3rd Law P2 = a3, where P is the orbital period of the planet (units = years) and a is it average distance from the Sun (units = AU).

Astonomy 62 Lecture #2

a = semi-major axis

b = semi-minor axis

e = eccentricity (0 < e < 1)

Equation for the ellipse:

r ' r=2 a , r=a 1−e2

1 ecos

Astonomy 62 Lecture #2

ra=a1e r

p=a1−e

Kepler's 1st LawKepler's 1st Law

Astonomy 62 Lecture #2

Astonomy 62 Lecture #2Solar SystemSolar System

Planet a (AU) eMercury 0.39 0.206Venus 0.72 0.0067Earth 1.00 0.017Mars 1.52 0.094Jupiter 5.20 0.049Saturn 9.58 0.057Uranus 19.2 0.046Neptune 30.0 0.011Pluto 39.5 0.24

Solar SystemSolar SystemAstonomy 62 Lecture #2

A line connecting a planet to the Sun sweeps out equal areas in equal time

intervals.

Kepler's 2nd LawKepler's 2nd Law

Astonomy 62 Lecture #2

P = orbital period of the planet

a = semi-major axis of the planet

P

1 yr 2

= a

1 AU 3

Kepler's 3rd LawKepler's 3rd Law

Astonomy 62 Lecture #2

1st Law An object at rest will remain at rest and an object in motion will remain in motion in a straight line at a constant speed unless acted upon by an unbalanced force.

2nd Law F=ma

3rd Law For every action there is an equal and opposite reaction.

Law of Gravitation Two masses M and m separated by a distance R experience the force:

F =GMm

R 2

Astonomy 62 Lecture #2

Newtn's Laws (1687)Newtn's Laws (1687)

m1

m2

Circular

Elliptical

m2

m1

Astonomy 62 Lecture #2

Binary OrbitsBinary Orbits

r=r2−r

1

m1r1m2

r 2=0

Astonomy 62 Lecture #2

Two-Body SystmTwo-Body Systm

m1r 1m2

r 2=0

r1=−

m2

m1m2

r

r2=m1

m1m2

r

r=r2−r

1

Astonomy 62 Lecture #2

Two-Body SystmTwo-Body Systm

In general, the two-body problem may be treated as an equivalent one-body problem with the reduced mass μ

moving about a fixed mass M at a distance r.

m1

m2

M=m1m

2

=m

1m

2

m1m

2

r2

r1

r=r1r

2

Astonomy 62 Lecture #2

ConclusionConclusion