112.215 Modern NavigationThomas Herring ([email protected]),MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/~tah/12.2159/30/2009 12.215 Modern Naviation L05 2Review of Mondayʼs Class• Spherical Trigonometry– Review plane trigonometry– Concepts in Spherical Trigonometry• Distance measures• Azimuths and bearings– Basic formulas:• Cosine rule• Sine rule– Basic applications29/30/2009 12.215 Modern Naviation L05 3Todayʼs Class• Motion of the Earth and Sun– Geometry of Earth/Sun system– Astronomical coordinates– Motion of the Earth around the sun– Equation of Time• Astronomical positioning– Latitude and Longitude determination usingastronomical bodies• Error contributions to latitude and longitudemeasurements.9/30/2009 12.215 Modern Naviation L05 4Geometry of Earth Sun System• Figure below shows the basic geometrySUNωequatorωequatorωequatorωequatorNorthern HemisphereWinterNorthern HemisphereSpringNorthern HemisphereSummerNorthern HemisphereFallEarthʼs OrbitEcliptic39/30/2009 12.215 Modern Naviation L05 5Geometry of Earth Sun• The Earthʼs equator plane is inclined at ~23.5o to the orbit plane(called the ecliptic)• It takes ~365.25 solar days for one orbit (hence a leap-year every4 years, and the odd rule about leap years at century boundariesbecause the value is not exactly 365.25 solar days.• A solar-day is the length of time (on average) for the sun to movefrom noon to noon. Because the earth moves in its orbit by a littleduring the day, the length of time for stars to come to the sameplace in the sky is a little bit shorter.• A sidereal-day is the length of time for stars to come back to thesame point in the sky• There are ~366.25 sidereal days in a year (the extra day isbasically one rotation due to the orbit in one year).9/30/2009 12.215 Modern Naviation L05 6Time systems• There are a number of time systems encountered in astronomyand navigation.• Time keep by our watches is related to Universal TimeCoordinated (UTC). Used to be called Greenwich Mean Time(GMT)– UTC is based on atomic time standards (Cesium clocks) andis an average over Cesium clocks operated all around theworld. The US clocks are operated at the US NavalObservatory in Washington DC– The International Earth Rotation Service (IERS) (and formallythe Bureau International de Le Heure (BIH)) coordinates theseactivities and publishes corrections to the time systemsoperated in each country49/30/2009 12.215 Modern Naviation L05 7Time systems• The time defined by atomic clocks runs at a constant rate.Unfortunately, we tend to perceive time by the rotation of theEarth which is not uniform. There is a slowing of the rate ofrotation of Earth (about 1 second every 18 months) and there arefluctuations due mainly to changes in atmospheric winds andprocesses in the fluid core.• Time defined by the rotation of the Earth is called UT1 and issolar day system.• UTC has discontinuities, call leap-seconds, that are added tokeep it aligned with UT1. (When the atomic second was adoptedin the mid-1950s the rates were the same and so leap-secondswere not needed. They were introduced in the mid-1960s afterthe Earth rotation rate had slowed enough that the differencebetween UT1 and UTC had reached several seconds.9/30/2009 12.215 Modern Naviation L05 8Time systems• The difference between UT1 and UTC has bemeasured and the IERS coordinates thesemeasurements and published differences betweenUT1 and UTC. (They also decide when leap secondsneed to be added).• Sidereal time is derived from UT1 and measures timesin sidereal seconds. If we ran our watches on siderealtime, the stars would also be in the same place in thesky at the same time. (With solar time, the stars rise 4minutes earlier each night).59/30/2009 12.215 Modern Naviation L05 9Solar time• Solar time is based on the mean solar day, but the time that Sunreaches its highest point each day (around noon) varies throughout the year. The difference between noon at Greenwich andwhen the sun is at its highest point (or highest elevation angle) iscall the Equation of Time.• There are two components to the equation:– The Earthʼs orbit is eccentric (e=0.0167) and so moves atdifferent speeds through the orbit, causes an annual variation.– The equator is included to the orbit plane (obliquity of theecliptic) by ~23.5o and this causes a semi-annual variation.– Combination of the two effects cause changes in the time ofnoon at Greenwich by -14 to +16 minutes (see URL:http://www.nmm.ac.uk/9/30/2009 12.215 Modern Naviation L05 10Equation ofTimeGraphics fromURL given onprevious pageFor Longitudedeterminationusing the sun, thiseffect must beaccounted for (15minutes oftime~225 nauticalmiles)69/30/2009 12.215 Modern Naviation L05 11Astronomical position determination• To determine position using astronomical measurements (or asextant) requires relating positions of celestial objects to Earthcoordinates.• Celestial coordinates:– Declination: Measured from equator and it astronomicalcoordinate equivalent to latitude– Right Ascension: Angle measured along the equator (similarto longitude) but origin is the intersection of the equator andecliptic planes (called the first point of Aries).• Celestial coordinates are specified in a non-rotating or slowlyrotating frame. The diurnal rotation of the Earth is not in thecoordinates.9/30/2009 12.215 Modern Naviation L05 12Celestial coordinates• Since the celestial coordinates are given in a systemattached to the equator of the Earth, they wouldchange slowly with time due to precession (26,000year motion of the rotation axis, about an axisperpendicular to the orbit plane), and nutation(nodding of the rotation axis in space due togravitational torque on the equatorial bulge.• Because of these motions of the rotation axis (andhence in the equator) in space, celestial coordinatesare generated in a number of systems.79/30/2009 12.215 Modern Naviation L05 13Celestial coordinates• Fundamental coordinates of stars are given in a system whichcorresponds to the equator and ecliptic orientations at a specifictime, call coordinates of epoch. Current system is call J2000 andis the position at Jan 1.5, 2000.• The other common system is the coordinates of datecorresponding to the equator and ecliptic orientations at the dayof interest.• There is a mathematical relationship through the application of aseries of rotation matrices, that allow the systems to be related.• For