First let us start out with a brief explanation of the need for the precursor information I am providing prior to actually getting to time bumps.
Historically timekeeping and calendars have been tied to the motions found in the heavens. These have been primarily the stars, our Moon, and the Sun. To get a rudimentary understanding of how time is measured and where we got our units of time, we must first talk about the motions of these heavenly bodies referenced back to our Earth. The background for this post will start with the Celestial Sphere, followed with a description of the Earth Sun relationship, and finally with the Earth Moon relationships/system.
The Celestial sphere:
When we look up at the stars in the night sky they appear to be stationary relative to each other. As the Earth moves from one side of the Sun to the other, the displacement of those stars due to parallax is less than one second of arc even for the nearest star (Proxima Centauri). One way of looking at this is a fixed sphere of stars surrounding the Earth/Sun system. This is often referred to as the Celestial Sphere. This is why some of the ancient civilizations considered the stars to be holes in a tapestry.
Since we are talking distances and parallax, lets briefly take a moment and describe such. The more familiar term for the layman when referring to stellar distances is called a light year. This is the distance light will travel in one calendar year. For example the star Proxima Centauri is approximately 4.22 light years from our solar system. Astronomers use another term that may be not so familiar called the Parsec. The Parsec (parallax-arcsecond) is the distance needed for an object (star) to have a shift of one arcsecond as the Earth moves from one side of the Sun to the other. An arcsecond is 1/60 of an arcminute, which is 1/60 of a degree. However, there are no stars that are close enough to exhibit this large a shift. The distance of a Parsec is about 3.26 light years and the nearest star is 4.22 light years.
Even though is seems the stars are in “fixed” locations in the night sky, over time the stars do move relative to each other and relative to the Earth. This is why the right ascension and declination (star location) changes over the years. If you look at a star catalogue based on the epoch B1950 and one base on the epoch J2000, you will notice some differences.
Another interesting item of note is that the constellations we see are made up of the brightest stars. Even in the same constellation these stars are at different distances from the Earth. Some may be dimmer than the others, however, being closer they are just as bright as a larger one further away. The brightness of a star is called its magnitude. There are two ways astronomers measure magnitude: Apparent Magnitude and Absolute Magnitude.
The Apparent Magnitude is how bright a star appears to us here on the Earth. The Absolute Magnitude is how bright a star would appear if it were exactly ten parsecs away from the Earth. (Close to 33 light years).
Two notes:
1) Apparent magnitude is usually denoted with a small “m” and absolute magnitude uses a capital “M” .
2) The magnitude scale is backwards of what you might think, the larger the number the fainter the object.
Since the Earth is tilted (23.5 degrees) in reference to the path it sweeps out in its orbit about the Sun, this path projected onto the celestial sphere does not fall on the celestial equator. This imaginary plane is called the ecliptic. Note: This angle between the ecliptic and the equatorial plane is called The Obliquity of The Ecliptic.
This imaginary plane crosses the celestial equator in two places (called the equinoxes). The Vernal Equinox falls in the spring as the Sun appears to cross the ecliptic going north and the Autumnal Equinox falls in autumn when the Sun again crosses the ecliptic, this time going south. Note: Vernal comes from the Latin vernalis, meaning spring. Also the term equinox relates to the word equal since both day and night are close to the same, 12 hours during the equinox.
The points where this plane is the farthest above (north) and below (south) the celestial equator is called the solstices. In the northern hemisphere of the earth, the most northern point of the ecliptic is called the Summer Solstice and the southern most is called the Winter Solstice. In the Southern hemisphere of the Earth the reverse is true.
The zodiac lies along the plane of the ecliptic. Since the Earth is orbiting the Sun, the Sun appears to follow the plane of the ecliptic, making one complete circle in one calendar year. The name “zodiac” comes from the Greek meaning animal circle. In fact all of the 12 constellations of the zodiac are named after animals. Note: The path of the Moon and the other planets fall pretty much on this plane as well. Since it takes 365 days for the Earth to orbit the Sun and there are 360 degrees in a circle, the Sun moves pretty close to 1 degree per day.
If you were to draw a line out from the Earth intersecting the Vernal Equinox, that line would be referred to as The First Point of Aries. The reason it was called this is that this line pointed to the first star in the constellation of Ares in March of 1950.
The celestial sphere is tied to the Earth for its coordinate system. Project the Earth’s equator out to infinity and you have the equator of the celestial sphere. Likewise the north and south poles of the Earth points to the north and south poles of the celestial sphere respectively. This makes it very easy to map the sky referenced to the Earth. This coordinate system is called the Equatorial Coordinate System. It ties in closely with our own geographic coordinate system here on the surface of the Earth.
There is one fundamental difference however. The geographic coordinate system is fixed upon the surface of the Earth (Lat Long) so it rotates with the rotation of the Earth. The celestial coordinate system is fixed to the celestial sphere and appears to rotate due to the Earth’s rotation. The “latitude” of the celestial sphere (the angle of an object above or below the celestial equator) is called declination with zero being on the equator. This is pretty easy since the celestial’s equator and poles appear to be fixed like our own earth. Unlike the Earth, since the celestial sphere appears to be rotating, the “longitude” , called right ascension, is not a “fixed” reference to the Earth. So instead of using degrees, hours were used for this measurement. First there needed to be a “fixed” direction to measure from. The Vernal Equinox was selected as the point to be measured from. Since there are 360 degrees in a circle, the Earth rotates about 15 degrees every hour. So you will note right ascension is measured in hours/minutes/seconds as apposed to degrees.
Remember that for declination Zero is on the equator and for right ascension zero is at the Vernal Equinox. So the Vernal Equinox will have the coordinates of 0 degrees and 0 hours. This then becomes the center point for an Equatorial Sky Chart.
On to the Earth Sun system:
It takes one year for the Earth to rotate around the Sun one time and 24 hours to rotate on its axis. Think about this relationship. Not only is the Earth revolving on its axis, it is in motion about the Sun. (I know this is really basic grade school stuff, however, it will help in visualizing the concepts I am about to explain) Therefore the Earth moves 1/365th of its orbit about the Sun every day.
Ok, here is where that visualization will come in handy. Since a “day” is described by one complete rotation of the Earth on its axis, this equates from noon to noon (when a point on the Earth is directly pointed at the Sun). The term for this is called the Mean Solar Day. But here is the rub; the Earth has moved during this period of time we called a day. So the Earth must turn a tiny bit more to have the same spot facing the Sun every day.
Now let us think of this celestial sphere we have been chatting about. Remember the stars appear fixed in one location (at least on a daily basis). This means that one complete revolution of the Earth referenced to a star does not take that little bit of extra time to be over the same spot on the Earth. This “day” is referred to as a Sidereal Day. It takes approximately four extra minutes for the Earth to have the Sun over the same location than a star.
This is the difference between a Sidereal Day and a Mean Solar Day.
Also the Earth is tilted on its axis from the plane of the ecliptic by 23.5 degrees. That tilt causes the North Pole to be currently pointed towards Polaris. As the Earth moves around the sun its pole stays pointed at Polaris. This is the cause of the seasons we experience. Note. This tilt varies back and forth from 21.6 degrees to 24.5 degrees approximately every 41,000 years.
There is also a precession of our pole and it sweeps a complete circle in the sky (think of the Earth as a top wobbling as it rotates) about every 26,000 years. (Hard to explain without a diagram)
There are also a number of other motions that must be taken into effect over the years such as the precession of the aphelion. Our Earth’s orbit around the Sun is not a perfect circle. It is an ellipse with the closest point of the orbit called the perihelion and the furthest point the aphelion. Currently the aphelion falls on the fourth of July. However, this is not always the case. The aphelion and perihelion change over the centuries and sweeps thru the calendar year with a periodicity of around 22,000 years. The amount of “squishing” (LOL now that’s a scientific term) of an ellipse is called its eccentricity. If the eccentricity is equal to zero the orbit will be a perfect circle. Between zero and one the path of an orbit is an ellipse. Note: A circle is also known as a degenerate ellipse. However, should the eccentricity equal exactly one, the path becomes a parabola and finally, if the eccentricity is greater than one, the path then becomes a hyperbola.
The Earth’s eccentricity is very small. However, even this changes over time. Its eccentricity varies periodically about every 100,000 years. There are also other motions caused by the Moon, Jupiter and the Sun called Nutations. One of the major nutations has a period of 18.6 years.
Now that we have taken a cursory look at the Earth/Sun system, there is another big factor in all of this. It is called the Moon.
The reason the Moon keeps one face to the Earth (Its rotation on its axis matches the period of its orbit) is it is tidally locked to the Earth. This tidal locking will eventually cause the Earth and Moon to keep one face to each other.
Here is a more in depth explanation. The total angular momentum of the earth moon system, which is spin angular momentum plus the orbital angular momentum, is constant. (The Sun plays apart also) Friction of the oceans caused by the tides is causing the Earth to slow down a tiny bit each year. This is approximately two milliseconds per century causing the moon to recede by about 4 centimeters per year. As the Earth slows down, the Moon must recede to keep the total angular momentum a constant. In other words as the spin angular momentum of the earth decreases, the lunar orbital angular momentum must increase. Here is an interesting side note. The velocity of the moon will slow down as the orbit increases.
Another example of tidal locking is the orbit period and rotation of the planet Mercury. What is interesting about this one is that instead of a 1:1 synchronization where Mercury would keep one face to the Sun at all times, it is actually in a 2/3:1 synchronization. This is due to the High eccentricity of its orbit.
There also can be more than one body “locked” to each other. Lets take a look at the moon Io. Io is very nearly the same size as the Earth’s moon. It is approximately 1.04 times the size of the moon. There is a resonance between Io, Ganymede, and Europa. Io completes four revolutions for every one of Ganymede and two of Europa. This is due to a Laplace Resonance phenomenon. A Laplace Resonance is when more than two bodies are forced into a minimum energy configuration.
Since we are now talking about orbiting bodies, let us digress just a wee bit further and briefly talk about orbits:
There are different sizes and shapes of orbits. We use the term Semi-Major Axis to measure the size of an orbit. It is the distance from the geometric center of the ellipse to either the apogee or perigee (The highest (apo) and the lowest (peri)). Apoapsis is a general term for the greatest radial distance of an Ellipse as measured from a Focus. Apoapsis for an orbit around the Earth is called apogee, and apoapsis for an orbit around the Sun is called aphelion.
Periapsis is a general term for the smallest radial distance of an Ellipse as measured from a Focus. Periapsis for an orbit around the Earth is called perigee, and periapsis for an orbit around the Sun is called perihelion.
The terms Gee and Helios comes from the Greek words “Ge” (earth) and “Helios” (Sun) respectively.
First lets talk a bit about “where it is”. An orbit is a nothing more than an object falling around another object. Both Kepler and Newton came up with a set of laws that describe this phenomenon.
Kepler’s three laws of planetary motion:
1) The orbit of a planet is an ellipse with the sun at one of the foci.
2) The line drawn between a planet and the sun sweep out equal areas in equal times.
3) The square of the periods of the planets is proportional to the cubes of their mean distance from the sun.
So what is that telling us? In a nutshell, all orbits are ellipses, the close to the body you are orbiting the faster you go (e.g. if you have a highly elliptical orbit the satellite or planet’s velocity will increase as it approaches the object being orbited and decrease as it get further away).
These laws not only apply to planets and satellites, but to any orbiting body.
Note: Super geek alert #1:
For an orbiting body this is not entirely correct. It turns out that both bodies end up orbiting a common center of mass of the two-body system. However, for satellites, the mass of the Earth is so much greater than the mass of the satellite, the effective center of mass is the center of the Earth.
Newton’s three laws (and law of gravitation):
1) The first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. (Commonly known as inertia)
2) The second law states that force is equal to the change in momentum (MV) per change in time. (For a constant mass, force equals mass times acceleration F=ma)
3) The third law states that for every action there is an equal and opposite reaction. In other words, if an object exerts a force on another object, a resulting equal force is exerted back on the original object.
Newton’s law of gravitation states that any two bodies attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
Note: Super geek alert #2:
Actual observed positions did not quite match the predictions under classical Newtonian physics. Albert Einstein later solved this discrepancy with his “General Theory of Relativity”. In November of 1919, using a solar eclipse, experimental verification of his theory was performed by measuring the apparent change in a stars position due to the bending of the light buy the sun’s gravity.
So what is all this trying to tell us? Planets, satellites, etc orbit their parents in predictable trajectories allowing us to “know” where they will be at any given time. A set of coordinates showing the location of these objects over a period of time is called its ephemeris.
FINALLY lets get to time and leap seconds!
Historically, time has been measured by the rotation of the Earth on its axis and the time it takes to rotate once about the Sun (a year). However, both of these are not uniform enough for precise calculations.
One of the units of time is called the second. It used to be defined as 1/86,400 of a Mean Solar Day. This was good enough for early calculations, but don’t forget that the Earth is slowing down due to tidal forces so that ends up changing over time. After a number of intermediate steps the second was finally redefined as:
The duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. (Atomic time), also known as Coordinated Universal Time (UTC) .
Since the Earth is slowing down approximately 1.4 milliseconds per day per century, this deceleration causes the Earth’s rotational time to vary from atomic time. The current rate of the Earth is called UT1 (which is a non-uniform rotation). Over a period of a year the difference can approach a full second. However, since the Earth’s rotation is non-uniform, it is monitored continuously. If the difference between UT1 and UTC approaches 0.9 seconds, a leap second is added or subtracted from UTC to keep it uniform with the Earth’s rotation. So far all of the leap seconds have been positive. This tallies with the slowing of the Earth from tidal braking.
Note: Since the GPS time does not have leap seconds added or subtracted, it is diverging with UTC with ever second added to UTC. Currently it is different by 13 seconds. This can cause some consternation when flying a satellite or spacecraft that uses GPS. If your ephemeris is calculated in GPS time and you receive a “vector” in UTC time, it will be off by 13 seconds. You just cannot add 13 or subtract 13 seconds and press on. The rub is that not only has the satellite moved 13 seconds in-track, the Earth has rotated underneath by 13 seconds (cross-track) as well. This is especially noticeable for the high inclination orbits. Vectors have to be recalculated when translating between GPS and UTC.
The interesting note is that the last time a leap second was needed was clear back in 1999. Remember, the deceleration of the Earth is not uniform. There may be a number of factors that cause this non-linearity such as snow and ice loads, earthquakes and others we haven’t even thought of. This could account for this long delay between leap seconds. This certainly is not a permanent condition. The Earth will continue to slow down and the deceleration will still vary. One final item: There is an ongoing debate whether to do away with leap seconds all together and just go with UTC. The problem with this is, over an extended period of time, the hours will no longer be tied to the solar day and noon may well end up in the evening. Another suggestion is to redefine the period of one second to more closely match the current rotation of the Earth. This too has its problems as the second will required “redefining” periodically as the Earth continues to slow down.
Now that we are this far along, how about a little chat on satellites and spacecraft since they have been in the news recently:
Since the Earth is not a perfect sphere (it is an Oblate Spheroid), satellites drift from their predicted position due to the Earth’s non-spherical shape. Also at low Earth orbits, the atmosphere creates a drag on the satellite also causing a drift (perturbation) in its orbit. At higher altitudes, such as a geosynchronous orbit, the solar wind and effects from the moon are more pronounced. This requires us to update the ephemeris periodically.
Satellites (and spacecraft) are incredibly precise machines with exquisite craftsmanship. The life of a satellite is often computed by the onboard fuel requirements. For geostationary satellites, periodic maneuvers (delta-Vs) must be accomplished to keep them on station. This is also required for many lower orbiting satellites as well. For an orbit plane change (move it into a different orbit), mass must be ejected to move the satellite.
Note: Super geek alert #3:
The Hohmann transfer orbit is the most energy efficient (minimum energy solution) way of getting from one circular orbit to a higher or lower circular orbit. This type of transfer orbit is used by interplanetary spacecraft to travel to the other planets in our solar system.
Now that we have a better understanding of its orbital position, we need to concentrate on its pointing (Attitude Control).
Why do we need to worry about pointing? If the satellite has solar panels (arrays), they need to point towards the sun to provide power. Sensors need to point at their respective targets, such as a star sensor, sun sensor etc. Thermal and possible contamination consideration must be taken into effect when pointing also.
Remember for every action there is an equal and opposite reaction. So if I spew mass (jet of gas out of a thruster nozzle), the satellite will move in the opposite direction. Also if I spin a wheel onboard the satellite, the result will be the satellite spins in the opposite direction.
Since fuel is precious and usually cannot be replenished (called consumables), other methods of pointing were devised that did not require mass ejecta from the satellite. Spinning reaction wheels were one. If you have orthogonal reaction wheels, just by spinning them you can provide precise pointing. Unfortunately, external forced (perturbations) adds unwanted momentum to the wheels. To compensate (unload momentum from the wheels) for this, I have seen both low-level monopropellant jets or torque rods used for this purpose.
Note: Super geek alert #4:
A monopropellant is one that does not require an oxidizer to function. Usually monopropellants are composed of a liquid compound called Hydrazine (N2H4). When this liquid comes in contact with a platinum catalyst, it is decomposed into gaseous ammonia (Nh3), nitrogen and hydrogen. This gas is then ejected (fired thru a nozzle) out a jet to providing motion for the satellite or spacecraft.
An ingenious method of unloading momentum without the use of fuel was devised using simple electromagnets. Remember the Earth is surrounded by a magnetic field (why your compass works). If you attach orthogonal electromagnets on your satellite and turn them on, the resultant field interacts with the Earth’s field causing a torque on the satellite. These are what are known as Torque Rods.
Since the reaction wheels, gyros, and torque rods all work using electricity and the solar arrays provide that electricity, theoretically the life of the satellite is indefinite. Unfortunately, there are degradations of the thermal coatings, blankets, sensors, and failures of both the gyros and reaction wheels that ultimately limit the life of any satellite.
Over a period of time, these degrade to the point that the satellites can no longer function within design spec. At some point, you either have to replace the satellite, repair it, or say farewell.
Since there is often confusion about geosynchronous orbits, here is a brief discussion on geosynchronous orbits:
A geosynchronous orbit is an orbit that has the same period of a single revolution of the Earth (sidereal day). A sidereal day is the length of time the Earth takes to make one complete rotation with respect to the stars. This is approximately 23 hours and 56 minutes. The semi-major axis of this orbit is approximately 36 thousand kilometers. One of the unique features of this orbit is that as the inclination approaches zero (stays on the equator) the object orbiting will stay over the same location on the Earth due to the fact it is moving at the save speed as the Earth is turning under it. This special type of geosynchronous orbit is called a Geostationary Orbit (stationary with respect to the surface of the Earth). As the inclination increases for a geosynchronous orbit, the ground trace of the orbit on the Earth plots a figure eight (8) pattern.
Remember, a geostationary orbit must be geosynchronous, however, a geosynchronous orbit does not have to be geostationary.
For a geosynchronous orbit, this orbit must be synched to the actual rotation period of the Earth (sidereal day). Even though a satellite is place in a near geostationary orbit upon launch there are forces that act upon the satellite that increase the orbital inclination. Remember an inclination of zero (0) for a geosynchronous orbit is also a geostationary orbit. The primary cause of this is that the equatorial plane is not coincident with the ecliptic. So both the sun and the moon slowly over time, increases the satellite’s orbital inclination. Also since the Earth is not a true sphere, the geosynchronous satellites drifts (in-track) towards two stable equilibrium points over the Earth’s equator. This is why “station keeping” is required for geostationary satellites. When station keeping is no longer possible (all the fuel is used) or there is a satellite malfunction, most geosynchronous satellites are boosted into a higher orbit so they will not drift into and area where another geostationary satellite is operating.
Final note: Even though the geostationary satellite (your TV satellite is one) appears to just “hover” over the equator, it is actually in orbit (falling around the Earth) at the same rate the Earth is turning beneath it.
A Parsec uses one AU as its base, not two as my post implies.