Gravitational Redshifts: Main Sequence vs. Giants

The Pleiades, Anglo-Australian Observatory/Royal Observatory


One of the consequences of Einsteins theories of relativity is that everything will be affected by gravitational potentials, regardless of their mass. The effect of this is observed in experiments demonstrating the potential for gravity to bend light. But a more subtle realization is that light escaping such a gravitational well must lose energy, and since energy for light is related to wavelength, this will cause the light to increase in wavelength through a process known as gravitational redshifting.

Since the amount of redshift is dependent on just how deeply inside a gravitational well a photon is when it starts its journey, predictions have shown that photons being emitted from the photosphere of a main sequence star should be more redshifted than those coming from puffed out giants. With resolution having reached the threshold to detect this difference, a new paper has attempted to observationally detect this difference between the two.

Historically, gravitational redshifts have been detected on even more dense objects such as white dwarfs. By examining the average amount of redshifts for white dwarfs against main sequence stars in clusters such as the Hyades and Pleiades, teams have reported finding gravitational redshifts on the order of 30-40 km/s (NOTE: the redshift is expressed in units as if it were a recessional Doppler velocity, although it’s not. It’s just expressed this way for convenience). Even larger observations have been made for neutron stars.

For stars like the Sun, the expected amount of redshift (if the photon were to escape to infinity) is small, a mere 0.636 km/s. But because Earth also lies in the Sun’s gravitational well the amount of redshift if the photon were to escape from the distance of our orbit would only be 0.633 km/s leaving a distance of only ~0.003 km/s, a change swamped by other sources.

Thus, if astronomers wish to study the effects of gravitational redshift on stars of more normal density, other sources will be required. Thus, the team behind the new paper, led by Luca Pasquini from the European Southern Observatory, compared the shift among stars of the middling density of main sequence stars against that of giants. To eliminate effects of varying Doppler velocities, the team chose to study clusters, which have consistent velocities as a whole, but random internal velocities of individual stars. To negate the latter of these, they averaged the results of numerous stars of each type.

The team expected to find a discrepancy of ~0.6 km/s, yet when their results were processed, no such difference was detected. The two populations both showed the recessional velocity of the cluster, centered on 33.75 km/s. So where was the predicted shift?

To explain this, the team turned to models of stars and determined that main sequence stars had a mechanism which could potentially offset the redshift with a blueshift. Namely, convection in the atmosphere of the stars would blueshift material. The team states that low mass stars made up the bulk of the survey due to their number and such stars are thought to undergo greater amounts of convection than most other types of stars. Yet, it is still somewhat suspect that this offset could so precisely counter the gravitational redshift.

Ultimately, the team concludes that, regardless of the effect, the oddities observed here point to a limitation in the methodology. Trying to tease out such small effects with such a diverse population of stars may simply not work. As such, they recommend future investigations target only specific sub-classes for comparison in order to limit such effects.

Precession of the Equinoxes

Semi Major Axis
Solstice and Equinox - Credit: NASA

When he was first compiling his famous star catalogue in the year 129 BCE the Greek astronomer Hipparchus noticed that the positions of the stars did not match up with the Babylonian measurements that he was consulting. According to these Chaldean records, the stars had shifted in a rather systematic way, which indicated to Hipparchus that it was not the stars themselves that had moved but the frame of reference – i.e. the Earth itself.

Such a motion is called precession and consists of a cyclic wobbling in the orientation of Earth’s axis of rotation. Currently, this annual motion is about 50.3 seconds of arc per year or 1 degree every 71.6 years. The process is slow, but cumulative, and takes 25,772 years for a full precession to occur. This has historically been referred to as the Precession of the Equinoxes.

The name arises from the fact that during a precession, the equinoxes could be seen moving westward along the ecliptic relative to the stars that were believed to be “fixed” in place – that is, motionless from the perspective of astronomers – and opposite to the motion of the Sun along the ecliptic.

This precession is often referred to as a Platonic Year in astrological circles because of Plato’s recorded remark in the dialogue of Timaeus that a perfect year could be defined as the return of the celestial bodies (planets) and the fixed stars to their original positions in the night sky. However, it was Hipparchus who is first credited with observing this phenomenon, according to Greek astronomer Ptolemy whose own work was in part attributed to him.

The precession of the Earth’s axis has a number of noticeable effects. First of all , the positions of the south and north celestial poles appear to move in circles against the backdrop of stars, completing one cycle every 25, 772 years. Thus, while today the star Polaris lies approximately at the north celestial pole, this will change over time, and other stars will become the “north star”. Second, the position of the Earth in its orbit around the Sun during the solstices, equinoxes, or other seasonal times slowly changes.

The cause of this was first discussed by Sir Isaac Newton in his Philosophiae Naturalis Principia Mathematica where he described it as a consequence of gravitation. Though his equations were not exact, they have since been revised by scientists and his original theory proven correct.

It is now known that precessions are caused by the gravitational source of the Sun and Moon, in addition to the fact that the Earth is a spheroid and not a perfect sphere, meaning that when tilted, the Sun’s gravitational pull is stronger on the portion that is tilted towards it, thus creating a torque effect on the planet. If the Earth were a perfect sphere, there would be no precession.

Today, the term is still widely used, but generally in astrological circles and not within scientific contexts.

We have written many articles about the equinox for Universe Today. Here’s an article about the astronomical perspective of climate change, and here’s an article about the Vernal Equinox.

If you’d like more info on Earth, check out NASA’s Solar System Exploration Guide on Earth. And here’s a link to NASA’s Earth Observatory.

We’ve also recorded an episode of Astronomy Cast all about Gravity. Listen here, Episode 102: Gravity.


NASA: Precession

What is Gravitational Force?

Why Do Planets Orbit the Sun
The Solar System

Newton’s Law of Universal Gravitation is used to explain gravitational force. This law states that every massive particle in the universe attracts every other massive particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This general, physical law was derived from observations made by induction. Another way, more modern, way to state the law is: ‘every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses’.

Gravitational force surrounds us. It is what decides how much we weigh and how far a basketball will travel when thrown before it returns to the surface. The gravitational force on Earth is equal to the force the Earth exerts on you. At rest, on or near the surface of the Earth, the gravitational force equals your weight. On a different astronomical body like Venus or the Moon, the acceleration of gravity is different than on Earth, so if you were to stand on a scale, it would show you that you weigh a different amount than on Earth.

When two objects are gravitational locked, their gravitational force is centered in an area that is not at the center of either object, but at the barycenter of the system. The principle is similar to that of a see-saw. If two people of very different weights sit on opposite sides of the balance point, the heavier one must sit closer to the balance point so that they can equalize each others mass. For instance, if the heavier person weighs twice as much as the lighter one, they must sit at only half the distance from the fulcrum. The balance point is the center of mass of the see-saw, just as the barycenter is the balance point of the Earth-Moon system. This point that actually moves around the Sun in the orbit of the Earth, while the Earth and Moon each move around the barycenter, in their orbits.

Each system in the galaxy, and presumably, the universe, has a barycenter. The push and pull of the gravitational force of the objects is what keeps everything in space from crashing into one another.

We have written many articles about gravitational force for Universe Today. Here’s an article about gravity in space, and here’s an article about the discovery of gravity.

If you’d like more info on Gravity, check out The Constant Pull of Gravity: How Does It Work?, and here’s a link to Gravity on Earth Versus Gravity in Space: What’s the Difference?.

We’ve also recorded an entire episode of Astronomy Cast all about Gravity. Listen here, Episode 102: Gravity.

Ballistic Trajectory

The flight trajectory for the HEAT rocket. Credit: Copenhagen Suborbitals.


Imagine you throw a ball as hard as you can. The ball flies through the air and lands some distance away. The harder you’re able to throw the ball, the further away it will land. When you throw the ball, it follows a ballistic trajectory, from where it takes off, to where it lands; its path is defined by the speed of its launch and the force of gravity pulling it down (and a little bit of atmospheric drag).

Now take this analogy further. Imagine you could throw the ball so hard that it flew all the way around the Earth and hit you on the back of the head. If you could throw the ball a little harder, it would go into orbit, continuously falling back to Earth, but with enough velocity to continue going around the planet. This speed is about 28,000 km/hour – it’s pretty hard to throw a ball that hard.

The first spacecraft were launched in a ballistic, or sub-orbital trajectory. They reached space, 100 km above the surface of the Earth, but they didn’t have enough energy to go into a true orbital trajectory. For example, the recently built SpaceShipOne doesn’t have any horizontal velocity. It travels straight up at a speed of about 1 km/s. Compare this to a low-Earth orbit escape velocity of 7.7 km/s. If a spacecraft is going to cover some horizontal distance, it needs have a maximum speed somewhere in between.

Spacecraft with a higher speed will travel along a ballistic trajectory. For example, the V2 rockets launched by Germany during World War II reached space and traveled about 330 km. Their maximum speed was 1.6 km/s. In intercontinental ballistic missile travels much faster, reaching a speed of 7 km/s and an altitude of 1200 km. Future intercontinental passenger flights might follow a similar trajectory.

We have written many articles about trajectory for Universe Today. Here’s an article about the Bolide, and here’s an article about the lunar orbit.

If you’d like more info on Trajectory, check out an article about Trajectories and Orbit, and here’s a link to Reduced Gravity Trajectory Page.

We’ve also recorded an entire episode of Astronomy Cast all about Gravity. Listen here, Episode 102: Gravity.

What Is Terminal Velocity?


The higher you are when you jump, the more it hurts when you hit the ground. That’s because the Earth’s gravity is constantly accelerating you towards its center. But there’s actually a maximum speed you reach, where the acceleration of the Earth’s gravity is balanced by the air resistance of the atmosphere. The maximum speed is called terminal velocity.

The terminal velocity speed changes depending on the weight of the object falling, its surface area and what it’s falling through. For example, a feather doesn’t weigh much and presents a very large surface area to the air as it falls. So its terminal velocity speed is much slower than a rock with the same weight. This is why an ant can fall off a tall building and land unharmed, while a similar fall would kill you. Keep in mind that this process happens in any gas or fluid. So terminal velocity defines the speed that a rock sinks when you drop it in the water.

So, let’s say you’re a skydiver jumping out of an airplane. What’s the fastest speed you’ll go? The terminal velocity of a skydiver in a free-fall position, where they’re falling with their belly towards the Earth is about 195 km/h (122 mph). But they can increase their speed tremendously by orienting their head towards the Earth – diving towards the ground. In this position, the skydiver’s velocity increases to more than 400 km/h.

The world skydiving speed record is held by Joseph Kittinger, who was able to fall at a speed of 988 km/h by orienting his body properly and jumping at high altitude, where there’s less wind resistance.

The gravity of the Earth pulls at you with a constant acceleration of 9.81 meters/second. Without any wind resistance, you’ll fall 9.81 meters/second faster every second. 9.81 meters/second the first second, 19.62 meters/ second in the next second, etc.

The opposing force of the atmosphere is called drag. And the amount of drag force increases approximately proportional to the square of the speed. So if you double your speed, you experience a squaring of the drag force. Since the drag force is going up much more quickly than the constant acceleration, you eventually reach a perfect balance between the force of gravity and the drag force of whatever you’re moving through.

Outside the Earth’s atmosphere, though, there’s no terminal velocity. You’ll just keep on accelerating until you smash into whatever’s pulling on you.

We have written many articles about the terminal velocity for Universe Today. Here’s an article featuring the definition of velocity, and here’s an article about the X-Prize Entrant completing the Drop Test

If you’d like more info on the Terminal Velocity, check out a Lecture on Terminal Velocity, and here’s a link to a NASA article entitled, The Way Things Fall.

We’ve also recorded an entire episode of Astronomy Cast all about Gravity. Listen here, Episode 102: Gravity.

GSU Hyperphysics


The eccentricity in Mars' orbit means that it is . Credit: NASA

When it comes to space, the word eccentricity nearly always refers to orbital eccentricity, or the eccentricity of the orbit of an astronomical body, like a planet, star, or moon. In turn, this relies on a mathematical description, or summary, of the body’s orbit, assuming Newtonian gravity (or something very close to it). Such orbits are approximately elliptical in shape, and a key parameter describing the ellipse is its eccentricity.

In simple terms, a circular orbit has an eccentricity of zero, and a parabolic or radial orbit an eccentricity of 1 (if the orbit is hyperbolic, its eccentricity is greater than 1); of course, if the eccentricity is 1 or greater, the ‘orbit’ is a bit of a misnomer!

In a planetary system with more than one planet (or for a planet with more than one moon, or a multiple star system other than a binary), orbits are only approximately elliptical, because each planet has a gravitational pull on every other one, and these accelerations produce non-elliptical orbits. And modeling orbits assuming the theory of general relativity describes gravity also leads to orbits which are only approximately elliptical (this is particular so for binary pulsars).

Nonetheless, orbits are nearly always summarized as ellipses, with eccentricity as one of the key orbital parameters. Why? Because this is very convenient, and because deviations from ellipses can be easily described by small perturbations.

The formula for eccentricity, in a two-body system under Newtonian gravity, is relatively easy to write, but, unfortunately, beyond the capabilities of the HTML coding of this webpage.

However, if you know the maximum distance of a body, from the center of mass – the apoapsis (apohelion, for solar system planets), ra – and the minimum such distance – the periapsis (perihelion), rp – then the eccentricity, e, of the orbit is just:

E = (ra – rp)/( ra+ rp)

Eccentricity of an Orbit (UCAR), Eccentricity of Earth’s Orbit (National Solar Observatory), and Equation of Time (University of Illinois) are websites with more on eccentricity.

Universe Today articles on eccentricity? Sure! For example: Measuring the Moon’s Eccentricity at Home, Buffy the Kuiper Belt Object, and Lake Asymmetry on Titan Explained.

Two Astronomy Cast episodes in which eccentricity is important are Neptune, and Earth; well worth listening to.

Gravity Formula

The gravity formula that most people remember, or think of, is the equation which captures Newton’s law of universal gravitation, which says that the gravitational force between two objects is proportional to the mass of each, and inversely proportional to the distance between them. It is usually written like this (G is the gravitational constant):

F = Gm1m2/r2

Another, common, gravity formula is the one you learned in school: the acceleration due to the gravity of the Earth, on a test mass. This is, by convention, written as g, and is easily derived from the gravity formula above (M is the mass of the Earth, and r its radius):

g = GM/r2

In 1915, Einstein published his general theory of relativity, which not only solved a many-decades-long mystery concerning the observed motion of the planet Mercury (the mystery of why Uranus’ orbit did not match that predicted from applying Newton’s law was solved by the discovery of Neptune, but no hypothetical planet could explain why Mercury’s orbit didn’t), but also made a prediction that was tested just a few years’ later (deflection of light near the Sun). Einstein’s theory contains many gravity formulae, most of which are difficult to write down using only simple HTML scripts (so I’m not going to try).

The Earth is not a perfect sphere – the distance from surface to center is smaller at the poles than the equator, for example – and it is rotating (which means that the force on an object includes the centripetal acceleration due to this rotation). For people who need accurate formulae for gravity, both on the Earth’s surface and above it, there is a set of international gravity formulae which define what is called theoretical gravity, or normal gravity, g0. This corrects for the variation in g due to latitude (and so both the force due to the Earth’s rotation, and its non-spherical shape).

Here are some links that you can follow to learn more about gravity formulae (or gravity formulas): Newton’s theory of “Universal Gravitation” (NASA), International Gravity Formula(e) (University of Oklahoma), and Newton’s Law of Gravity (University of Oregon).

Many aspects of gravity, including a gravity formula or three, are covered in various Universe Today articles. For example, New Research Confirms Einstein, Milky Way Dwarf Galaxies Thwart Newtonian Gravity?, and Modifying Gravity to Account for Dark Matter. Here’s some information on 0 gravity.

Astronomy Cast’s episode Gravity gives you much more on not just one gravity formula, but several; and Gravitational Waves is good too. Be sure to check them out!

University of Nebraska-Lincoln

Gravity Equation

There is not one, not two, not even three gravity equations, but many!

The one most people know describes Newton’s universal law of gravitation:

F = Gm1m2/r2,
where F is the force due to gravity, between two masses (m1 and m2), which are a distance r apart; G is the gravitational constant.

From this is it straightforward to derive another, common, gravity equation, that which gives the acceleration due to gravity, g, here on the surface of the Earth:

g = GM/r2,
Where M is the mass of the Earth, r the radius of the Earth (or distance between the center of the Earth and you, standing on its surface), and G is the gravitational constant.

With its publication in the early years of the last century, Einstein’s theory of general relativity (GR) became a much more accurate theory of gravity (the theory has been tested extensively, and has passed all tests, with flying colors, to date). In GR, the gravity equation usually refers to Einstein’s field equations (EFE), which are not at all straight-forward to write, let alone explain (so I’m going to write them … but not explain them!):

G?? = 8?G/c4 T??

G (without the subscripts) is the gravitational constant, and c is the speed of light.

Finally, here’s a acceleration of gravity equation you’ve probably never heard of before:

a = ?(GMa0/r),

where a is the acceleration a star feels, due to gravity under MOND (MOdified Newtonian Dynamics), an alternative theory of gravity, M is the mass of a galaxy, r the distance between the star in the outskirts of that galaxy and its center, G the gravitational constant, and a0 a new constant.

Some websites which contain more on gravity equations, for your interest and enjoyment: Newton’s Theory of “Universal Gravitation” (NASA), Einstein’s equation of gravity (University of Wisconsin Madison – heavy), and Gravity Formula (University of Nebraska-Lincoln).

Universe Today, as you would expect, has several stories relevant to gravity equations; here are a few: See the Universe with Gravity Eyes, A Case of MOND Over Dark Matter, and Flyby Anomalies Explained?. Here’s an article about 0 gravity.

Gravity, an Astronomy Cast episode, has more on gravity equations, as do several Astronomy Cast Question Shows, such as September 26th, 2008, and March 31st, 2009.

University of Nebraska-Lincoln

Pioneer Anomaly

Artist impression of the Pioneer 10 probe (NASA)

Named after the Pioneer 10 and 11 space probes, the Pioneer anomaly refers to the fact that they seem to be moving a teensy bit different from how we think they should be moving (or, more technically, the spacecraft seem to be subject to an unmodeled acceleration whose direction is towards the Sun).

The anomaly was first noticed, by John Anderson, in 1980, when analysis of tracking data from the spacecraft showed a small, unexplained acceleration towards the Sun (this was first published in 1995, with the main paper appearing in 1998). Since then it has been studied continuously, by quite a few scientists.

The Pioneer anomaly is one of the (very few!) true mysteries in contemporary physics, and is a great example of how science is done.

The first step – which Anderson and colleagues took – was to work out where the spacecraft were, and how fast they were traveling (and in what direction), at as many times as they could. Then they estimated the effects of gravity, from all known solar system objects (from the Sun to tiny asteroids and comets). Then they estimated the effects of things like radiation pressure, and possible outgassing. Then … They also checked whether other spacecraft seemed to have experienced a similar anomalous acceleration (the net: not possible to get an unambiguous answer, because all others have known – but unmodelable – effects much bigger than the Pioneer anomaly). Several independent investigations have been conducted, using different approaches, etc.

In the last few years, much effort has gone into trying to find all the raw tracking data (this has been tough, many tapes have been misplaced, for example), and into extracting clean signals from this (also tough … the data were never intended to be analyzed this way, meta-data is sorely lacking, and so on).

And yet, the anomaly remains …

… there’s an unmodeled acceleration of approximately 9 x 10-10 m/s2, towards the Sun.

The Planetary Society has been funding research into the Pioneer anomaly, and has a great summary here! And you can be a fly on the wall at a meeting of a team of scientists investigating the Pioneer anomaly, by checking out this Pioneer Explorer Collaboration webpage.

Universe Today has several stories on the Pioneer anomaly, for example The Pioneer Anomaly: A Deviation from Einstein Gravity?, Is the Kuiper Belt Slowing the Pioneer Spacecraft?, and Ten Mysteries of the Solar System.

Astronomy Cast has two episodes covering the Pioneer anomaly, The End of Our Tour Through the Solar System, and the November 18th, 2008 Questions Show.

The Planetary Society

What is Loop Quantum Gravity?

In this illustration, one photon (purple) carries a million times the energy of another (yellow). Some theorists predict travel delays for higher-energy photons, which interact more strongly with the proposed frothy nature of space-time. Yet Fermi data on two photons from a gamma-ray burst fail to show this effect. The animation below shows the delay scientists had expected to observe. Credit: NASA/Sonoma State University/Aurore Simonnet

The two best theories we have, today, in physics – the Standard Model and General Relativity – are mutually incompatible; loop quantum gravity (LQG) is one of the best proposals for combining them in a consistent way.

General Relativity is a theory of spacetime, but it is not a quantum theory. Since the universe seems to be quantized in so many ways, one approach to extending GR is to quantize spacetime … somehow. In LQG, space is made up of a network of quantized loops of gravitational fields (see where the name comes from?), which are called spin networks (and which become spin foam when viewed over time). The quantization is at the Planck scale (as you would expect). LQG and string theory – perhaps the best known of theories which aim to both go deeper and encompass the Standard Model and General Relativity – differ in many ways; one of the most obvious is that LQG does not introduce extra dimensions. Another big difference: string theory aims to unify all forces, LQG does not (though it does include matter).

Starting with the Einstein field equations of GR, Abhay Ashtekar kicked of LQG in 1986, and in 1988 Carlo Rovelli and Lee Smolin built on Ashtekar’s work to introduce the loop representation of quantum general relativity. Since then lots of progress has been made, and so far no fatal flaws have been discovered. However, LQG suffers from a number of problems; perhaps the most frustrating is that we don’t know if LQG becomes GR as we move from the (quantized) Planck scale to the (continuum) scale at which our experiments and observations are done.

OK, so what about actual tests of LQG, you know, like in the lab or with telescopes?

Well, there are some, potential tests … such as whether the speed of light is indeed constant, and recently the Fermi telescope team reported the results of just such a test (result? No clear sign of LQG).

Interested in learning more? There is a lot of material freely available on the web, from easy reads like Quantum Foam and Loop Quantum Gravity and Lee Smolin’s Loop Quantum Gravity, to introductions for non-experts like Abhay Ashtekar’s Gravity and the Quantum, to reviews like Carlo Rovelli’s Loop Quantum Gravity, to this paper on an attempt to explain some observational results using loop quantum gravity (Loop Quantum Gravity and Ultra High Energy Cosmic Rays).

As you’d expect, Universe Today has several articles on, or which feature, loop quantum gravity; here is a selection What was Before the Big Bang? An Identical, Reversed Universe, Before the Big Bang?, and Before the Big Bang.

Source: Wikipedia