Familiar Physical Laws

The Largest Scale of All - The Universe (Or Cosmos)

To the edge of the Big-bang horizon, the Universe is about 14 billion light years in extent[1]. The word “about” or “approximate” and its synonyms need some amplification.

Consider the elderly guard in the Smithsonian proudly telling a group of visiting school children that the dinosaur skeleton was 65 million, 37 years old. “How do you know that?” asked an alert teacher. “Well”, said the guard, “when I joined the staff here 37 years ago, they told me the dinosaurs were 65 million years old”!!!

Approximations are also relative to scale so we must bear this in mind if we are told some phenomenon is ± so many units.

One “light year” is both age and distance; one light year means that you would have to travel at the speed of light (186,000 miles/second) for one year. You can calculate that one light year is ~ 6 trillion miles (= 6 × 10¹² miles). That’s a six with 12 trailing zeros…

To visualize a single light year in miles, it’s about numerically equal the US GDP in $. If you assume a stack of a trillion one-dollar bills each 1/1000 of an inch thick, the stack would reach almost 100,000 miles! (The distance to the moon is 240,000 miles).

When I told you the Universe’s age was about 14 billion years, that’s equivalent to size of the Universe in units of light years. The age of the Universe is now claimed to be known to ± a few hundred million (light) years. This uncertainty is both a lot of time (and distance) on the scale of our puny existence, but not too bad relative to its scale.

The Universe turns out to about 10²³ miles in extent - a truly unimaginable distance. Of course the Universe is sparsely populated with galaxies of which ours is a smallish, middle aged, thin, flat, elliptical one. Our galaxy is about 100,000 light years (~ 6 x 10¹⁷ miles) in diameter. In case, you think space travelers have (or will) contact us, it’s not worth the frequent flyer miles! Unless someone can out-figure Einstein and travel with Jean-Luc and his crew in the Enterprise at warp speeds, we are likely to remain alone because it just takes too darn long to get here from there…

We also know the (approximate) mass of the Universe and the number of protons in it. The mass of our Universe is about 5 × 10⁵⁰ tonnes^[2]; it contains about 10⁸⁰ protons (called the “Eddington” number^[3] after a 20th century physicist). A recent estimate is that there are a staggering 7 × 10²² stars^[4] in the Universe. These are all staggering numbers, incomprehensibly too large for our imaginations to grasp).

The Solar System

We are one of approximately (again, that “approximately”!) 10¹¹ stars in our Galaxy located some 6 × 10¹⁶ miles from its center (and, as the late satirist and Sci Fi author Douglas Adams once wrote[5], we live in the “unfashionable western end of the Galaxy”!).

The distance from earth to our sun is some 93 million miles (less than 10 light minutes away, a mere hop and a skip). We are the third planet out from our sun. Our mass, M_e, is about 6 × 10²¹ tonnes (metric tonne = 1,000 kg), while the sun is about 2 × 10²⁷ tonnes, a factor 330,000 times as large as earth. Its gravity, even at 93 million miles is enough to keep us in orbit against the immense centrifugal acceleration the earth experiences as it whirls about the Sun on its yearly journey. And we are a small planet compared to Saturn (95 M_e) and Jupiter (3000 M_e). Be grateful for Jupiter! Its large gravitational mass saved our bacon a few years ago when the Shoemaker-Levy comet was attracted there and not here! Several impact plumes were larger than the whole diameter of earth…

The earth is just a mere 240,000 miles from the moon (which has only just over 1 % of our mass); gravity and Newtonian laws determines the orbit relationship. Our earth is about 8,000 miles in diameter.

On earth of course we see forces other than gravity acting. For example, tectonic plates float on a moving magma and shift in response to immense fluid drag forces from roiling actions in the molten core of the earth. It took the subcontinent of India just 60 million years to become unanchored from the Antarctic and to race across what is today the Indian Ocean. Its collision with mainland Asia is still evolving and the crush zone is a series of pressure waves that we call the Himalayas. Obviously more than gravity is evident at these scales! Such forces can be scaled by their governing laws and, while more complicated than the Reynolds number we referred to when speaking about simple fluid flows, the mathematics that describes these phenomena are simplified by proper scaling to the relevant parameters. We do not usually see these forces at play^[6] (but the “Big One” scheduled for California “soon” might be an obvious exception!).

Newton’s Law of Gravitation

It was Isaac Newton who related the force of gravity to distance and masses: for example the gravity force on you right now; if your body mass is m, the (usually expressed in units of newtons – we’ve already seen how big this unit is[7]). In this expression, g is the acceleration due to gravity (9.81 m/s²), G is the “Universal Gravitational Constant” (6.67 × 10^-11 N-m²/kg²), M_e is the mass of the earth (5.97 × 10²⁴ kg) and R is the distance to the center of the earth (6,370 km). Your mass conveniently cancels so you all experience the same acceleration due to gravity^[8], .

You can easily check out this formula!

g = 6.67 × 10-11 × 5.97 × 10²⁴/(6.370 × 10⁶)² = 9.81 m/s² (as it should)

It’s relatively easy to effectively cancel out the force: of gravity.  After all - aircraft fly, inverted roller coasters can be found within a few miles of any center of population, etc. But if you crash a car headlong into a tree, it’s not the force of gravity that is fatal – it’s the rate of deceleration of your mass that will kill you. In other words, gravitation is not the only important macroscopic force right here on earth.

Other forces and energy terms are also within common experience at this level. As you know, Newton’s 2nd Law of motion states that:

or, for an acceleration of “a” in m/s² for a mass m kg, you will need F newtons of force to produce it (or 1 newton º 1 kg × 1 m/s²).