What is the current size of the SN1006 supernova remnant?

You were so close! The answer was actually given lower down on the Wikipedia page: roughly 20 parsecs. Since one parsec is about 3.26 light-years, we can calculate that that comes out to about 64 light-years, as this seems to corroborate. The outer layers of the remnant are expanding outward at an outrageous rate - 11 million miles per hour, according to NASA. I invite you to do the calculations to figure out how large it will be at a certain point in time.

orbit - Exercise: 2D orbital mechanics simulation (python)

Just a little disclaimer beforehand: I have never studied astronomy or any exact sciences for that matter (not even IT), so I am trying to fill this gap by self-education. Astronomy is one of the areas that has captured my attention and my idea of self-education is head on applied approach. So, straight to the point - this is orbital simulation model that I am casually working on when I have time/mood. My major goal is to create complete solar system in motion and ability to plan spacecraft launches to other planets.

You are all free to pick this project up at any point and have fun experimenting!

update!!! (Nov10)

• velocity is now proper deltaV and giving additional motion now calculates
  sum vector of velocity


• you can place as many static objects as you
  like, on every time unit object in motion checks for gravity vectors
  from all sources (and checks for collision)


• greatly improved the performance of calculations


• a fix to account for interactive mod in matplotlib. Looks like that this is default option only for ipython. Regular python3 requires that statement explicitly.

Basically it is now possible to "launch" a spacecraft from the surface of the Earth and plot a mission to the Moon by making deltaV vector corrections via giveMotion(). Next in line is trying to implement global time variable to enable simultaneous motion e.g. Moon orbits Earth while spacecraft tries out a gravity assist maneuver.

Comments and suggestions for improvements are always welcome!

Done in Python3 with matplotlib library

import matplotlib.pyplot as plt
import math
plt.ion()

G = 6.673e-11  # gravity constant
gridArea = [0, 200, 0, 200]  # margins of the coordinate grid
gridScale = 1000000  # 1 unit of grid equals 1000000m or 1000km

plt.clf()  # clear plot area
plt.axis(gridArea)  # create new coordinate grid
plt.grid(b="on")  # place grid

class Object:
    _instances = []
    def __init__(self, name, position, radius, mass):
        self.name = name
        self.position = position
        self.radius = radius  # in grid values
        self.mass = mass
        self.placeObject()
        self.velocity = 0
        Object._instances.append(self)

    def placeObject(self):
        drawObject = plt.Circle(self.position, radius=self.radius, fill=False, color="black")
        plt.gca().add_patch(drawObject)
        plt.show()

    def giveMotion(self, deltaV, motionDirection, time):
        if self.velocity != 0:
            x_comp = math.sin(math.radians(self.motionDirection))*self.velocity
            y_comp = math.cos(math.radians(self.motionDirection))*self.velocity
            x_comp += math.sin(math.radians(motionDirection))*deltaV
            y_comp += math.cos(math.radians(motionDirection))*deltaV
            self.velocity = math.sqrt((x_comp**2)+(y_comp**2))

            if x_comp > 0 and y_comp > 0:  # calculate degrees depending on the coordinate quadrant
                self.motionDirection = math.degrees(math.asin(abs(x_comp)/self.velocity))  # update motion direction
            elif x_comp > 0 and y_comp < 0:
                self.motionDirection = math.degrees(math.asin(abs(y_comp)/self.velocity)) + 90
            elif x_comp < 0 and y_comp < 0:
                self.motionDirection = math.degrees(math.asin(abs(x_comp)/self.velocity)) + 180
            else:
                self.motionDirection = math.degrees(math.asin(abs(y_comp)/self.velocity)) + 270

        else:
            self.velocity = self.velocity + deltaV  # in m/s
            self.motionDirection = motionDirection  # degrees
        self.time = time  # in seconds
        self.vectorUpdate()

    def vectorUpdate(self):
        self.placeObject()
        data = []

        for t in range(self.time):
            motionForce = self.mass * self.velocity  # F = m * v
            x_net = 0
            y_net = 0
            for x in [y for y in Object._instances if y is not self]:
                distance = math.sqrt(((self.position[0]-x.position[0])**2) +
                             (self.position[1]-x.position[1])**2)
                gravityForce = G*(self.mass * x.mass)/((distance*gridScale)**2)

                x_pos = self.position[0] - x.position[0]
                y_pos = self.position[1] - x.position[1]

                if x_pos <= 0 and y_pos > 0:  # calculate degrees depending on the coordinate quadrant
                    gravityDirection = math.degrees(math.asin(abs(y_pos)/distance))+90

                elif x_pos > 0 and y_pos >= 0:
                    gravityDirection = math.degrees(math.asin(abs(x_pos)/distance))+180

                elif x_pos >= 0 and y_pos < 0:
                    gravityDirection = math.degrees(math.asin(abs(y_pos)/distance))+270

                else:
                    gravityDirection = math.degrees(math.asin(abs(x_pos)/distance))

                x_gF = gravityForce * math.sin(math.radians(gravityDirection))  # x component of vector
                y_gF = gravityForce * math.cos(math.radians(gravityDirection))  # y component of vector

                x_net += x_gF
                y_net += y_gF

            x_mF = motionForce * math.sin(math.radians(self.motionDirection))
            y_mF = motionForce * math.cos(math.radians(self.motionDirection))
            x_net += x_mF
            y_net += y_mF
            netForce = math.sqrt((x_net**2)+(y_net**2))

            if x_net > 0 and y_net > 0:  # calculate degrees depending on the coordinate quadrant
                self.motionDirection = math.degrees(math.asin(abs(x_net)/netForce))  # update motion direction
            elif x_net > 0 and y_net < 0:
                self.motionDirection = math.degrees(math.asin(abs(y_net)/netForce)) + 90
            elif x_net < 0 and y_net < 0:
                self.motionDirection = math.degrees(math.asin(abs(x_net)/netForce)) + 180
            else:
                self.motionDirection = math.degrees(math.asin(abs(y_net)/netForce)) + 270

            self.velocity = netForce/self.mass  # update velocity
            traveled = self.velocity/gridScale  # grid distance traveled per 1 sec
            self.position = (self.position[0] + math.sin(math.radians(self.motionDirection))*traveled,
                             self.position[1] + math.cos(math.radians(self.motionDirection))*traveled)  # update pos
            data.append([self.position[0], self.position[1]])

            collision = 0
            for x in [y for y in Object._instances if y is not self]:
                if (self.position[0] - x.position[0])**2 + (self.position[1] - x.position[1])**2 <= x.radius**2:
                    collision = 1
                    break
            if collision != 0:
                print("Collision!")
                break

        plt.plot([x[0] for x in data], [x[1] for x in data])

Earth = Object(name="Earth", position=(50.0, 50.0), radius=6.371, mass=5.972e24)
Moon = Object(name="Moon", position=(100.0, 100.0), radius=1.737, mass = 7.347e22)  # position not to real scale
Craft = Object(name="SpaceCraft", position=(49.0, 40.0), radius=1, mass=1.0e4)

Craft.giveMotion(deltaV=8500.0, motionDirection=100, time=130000)
Craft.giveMotion(deltaV=2000.0, motionDirection=90, time=60000)
plt.show(block=True)

How it works

It all boils down to two things:

1. Creating object like Earth = Object(name="Earth", position=(50.0, 50.0), radius=6.371, mass=5.972e24) with parameters of position on grid (1 unit of grid is 1000km by default but this can be changed too), radius in grid units and mass in kg.


2. Giving object some deltaV such as Craft.giveMotion(deltaV=8500.0, motionDirection=100, time=130000) obviously it requires Craft = Object(...) to be created in the first place as mentioned in previous point. Parameters here are deltaV in m/s (note that for now acceleration is instantaneous), motionDirection is direction of deltaV in degrees (from current position imagine 360 degree circle around object, so direction is a point on that circle) and finally parameter time is how many seconds after the deltaV push trajectory of the object will be monitored. Subsequent giveMotion() start off from last position of previous giveMotion().

Questions:

1. Is this a valid algorithm to calculate orbits?


2. What are the obvious improvements to be made?


3. I have been considering "timeScale" variable that will optimize calculations, as it might not be necessary to recalculate vectors and positions for every second. Any thoughts on how it should be implemented or is it generally a good idea? (loss of accuracy vs improved performance)

Basically my aim is to start a discussion on the topic and see where it leads. And, if possible, learn (or even better - teach) something new and interesting.

Feel free to experiment!

Try using:

Earth = Object(name="Earth", position=(50.0, 100.0), radius=6.371, mass=5.972e24)
Moon = Object(name="Moon", position=(434.0, 100.0), radius=1.737, mass = 7.347e22)
Craft = Object(name="SpaceCraft", position=(43.0, 100.0), radius=1, mass=1.0e4)

Craft.giveMotion(deltaV=10575.0, motionDirection=180, time=322000)
Craft.giveMotion(deltaV=400.0, motionDirection=180, time=50000)

With two burns - one prograde at Earth orbit and one retrograde at Moon orbit I achieved stable Moon orbit. Are these close to theoreticaly expected values?

Suggested exercise: Try it in 3 burns - stable Earth orbit from Earth surface, prograde burn to reach Moon, retrograde burn to stabilize orbit around Moon. Then try to minimize deltaV.

Note: I plan to update the code with extensive comments for those not familiar with python3 syntax.

date time - September 26 twelve hours sunrise to sunset?

The day and night are not equal on the equinox, though they are about equal. Equinox is when the Earth's axis is not tilted towards the sun, or equivalently the path of the sun passes from the Northern to the Southern hemisphere.

Sunrise and sunset are measured from the "top" of the sun, not the centre, and the refraction of the atmosphere means that the sun is visible from the surface, even after it would have set were there no air to bend its light. Finally the Earth's elliptical orbit also varies the length of the day.

The day on which the day and night are equal varies with latitude. In Boston, at 42 degrees North (it's as far south as the French Riviera - always surprises me) the day is 12 hours long on Sept. 26.

Why does the Moon seem larger when it is close to the horizon?

Interesting fact. It Isn't bigger!

This is a well known optical illusion that dates back hundreds of years. It's all in your head, it seems bigger because your brain perceives size as something that is relative.

This is an effect called perspective distortion and is will understood and used in photography. Phil Plait's Blog (A.K.A - The Bad Astronomer) also writes on this matter and would be worth reading.

Consider the below picture:

As the focal length of a lens gets longer the difference in relative size of foreground and background objects becomes more pronounced. Your brain then interprets this and respectively the moon seems huge!

But why does it look yellowish?

It appears yellow sometimes because our atmosphere blocks and stretches certain wavelengths. So when the moon is low in the sky, it has to pass through the most atmosphere to get to you, stretching and distorting the light through the yellows and oranges.

observational astronomy - I'd like to become an astronomer. What experience do I need?

To get into scientific fields as a professional pretty much requires a doctorate these days. This can be in astronomy or your chosen specialty (see below), but many are also in physics; what the subject is called may depend on the institution you attend to get the Ph.D.

Realize that what you and the general public thinks of as "Astronomy" is likely not the same as what the professional scientific world calls "Astronomy". An "Astronomer" studies stars and groupings of stars. Someone who studies our Sun is usually called a "Solar Physicist" or something similar. Someone who studies other planets is a "Planetary Scientist"; this can be further subdivided into "Atmospheric Scientist" (which can be even further divided into "Atmospheric Chemist", "Atmospheric Dynamicist", etc.) and "Planetary Geologist", and so on. A scientist studying the origins of the universe is a "Cosmologist". If you have a particular area you are interested in, research this to find out what you need to learn to move into that field.

Regardless of the degree you pursue, you will need a lot of mathematics and science. During your undergraduate degree, I would recommend the following minimums: 3 semesters of Calculus, 1 semester of Ordinary Differential Equations, 1 semester of Linear Algebra (some schools combine ODE and LA), 3 semesters of Introductory Physics. I'd recommend a semester or two of introductory astronomy, although some schools don't consider that as absolutely necessary, surprisingly. Other useful undergraduate courses will depend on what you think you will want to specialize in, but would include more physics, more astronomy, chemistry, geology, meteorology, etc.

If you aren't yet in college, take all the mathematics and physical science courses you can in high school... unless you're interested in becoming a astrobiologist (someone who studies living organisms on other planets - a speculative field at this point, but it does exist), in which case you should get some biology courses too!

Learning computer programming would be good also, as these fields often require programming for data processing and simulations. Amongst general purpose languages, the Python programming language may be the most popular in scientific environments. Others used include IDL and MatLab. Linux and Mac computers are used more in the science world than in the broader populace.

As noted in Jeremy's answer and its comments, job prospects aren't great for astronomers and related fields. That said, I don't want to discourage you, just prepare you for reality.

Of course, as another answer noted, if you just want to observe the skies, no degree is required, just time and patience.

cosmology - Which came first: black holes or galaxies?

_{Note: I am not an astrophysicist. If someone with the relevant expertise finds my analysis of the source paper to be invalid, I would appreciate your help with correcting my answer here.}

This is a somewhat complicated question to answer. Black holes do precede galaxies in the form of primordial black holes which arose in the early Universe from quantum fluctuations. Supermassive black holes (that you find at the center of galaxies) may not come from these, however. There are a few possible ways that supermassive black holes may have formed:

• Molecular cloud / dwarf galaxies collapsing to form supermassive stars (SMS) which later further collapsed into supermassive black holes (SMBH)


• Dwarf galaxies forming massive stars which collapse into massive black holes, which later coalesce into SMBHs


• Primordial black holes accreting sufficient matter to become massive black holes, which later coalesce into SMBHs

Depending on how strictly you define "galaxy" it does seem most likely that the supermassive black holes formed inside early galaxies, but a type of black hole (primordial black holes) did precede galaxies.

Source:

---

UPDATE 10/21/2013:

This just in, recent observations have ruled out the possibility that SMBHs gain mass only through merging with other black holes. For more, see this Astronomy.com article.

planetary formation - Can protoplanetary disks form main-sequence stars?

No, not really.

Stars can form in circumstellar disks, that are, in general, disks surrounding forming stars, but not in protoplanetary disks. Protoplanetary disks are, by definition, flat, rotationing disks composed of gas and dust, found around newly born low-mass stars (see the review by Williams & Cieza (2011)). There are two important point in this definition: protoplanetary disks orbit around low-mass stars, and, in particular, newly-born low-mass stars. These two points are important, because at this stage of the life of the disk and for this range of stellar mass, its mass it clearly too small to form any star.

That being said, it is not impossible to form stars in the protostellar disk of a forming star (that is one of the possibility to form multiple systems, that are still hard to explain), as discussed by various authors (see, for example Stamatellos et al. (2009), Vorobyov et al. (2013), Joos et al. (2013) and many others), but it is more likely to form brown dwarfs of very low-mass stars in these disks than main sequence stars (their mass is simply too small, once again).

That being said (bis repetita), you could, as pointed out by called2voyage, you could form a main sequence star in the circumstellar disk of a high-mass star (as a Wolf-Rayet type star).

the moon - Accidental or deliberate?

As the front image on the wikipedia-page already indicates, a total solar eclipse is not always total.
Earth's orbit is slightly elliptic, and so is the Moons orbit around the Earth.
Now take the Moon's slight orbital inclination into account and far from all total eclipses are really total.
Unlike stated usually.
In fact wiki states "On average, the Moon appears to be slightly smaller than the Sun as seen from the Earth, so the majority (about 60%) of central eclipses are annular."

On a sidenote: It's important to understand that if the inclination and eccentricity of the orbit would be zero, then we'd have a perfect total solar eclipse on earth every month!
But yeah you can look it up, that this is not the case. There are, after all, not every month news reports from ppl staring into the sun, are there ;)

On other note, we know that the Moon is increasing its distance to earth. This happens because Orbital angular momentum is converted into internal heat for both bodies through tidal forces.
This situation will evolve over time, so yes, it's a coincidence.

And no, I'm not aware of the Moon's distance having any strong implications for life.
Tell me, if this is not clarified enough from the above.

How do black holes evaporate?

The evaporation occurs through Hawking radiation. This is a very slow and low energy process. So low that the cosmic microwave background radiation, which is just a few degrees above absolute zero, pours far more energy into the black hole than Hawking radiation takes away. So in principle a black hole cannot evaporate.

With the exception of conjectured atomic sized black holes, that is, as Hawking radiation would be more pronounced then.

Edit: See this answer on the physics SE: http://physics.stackexchange.com/a/26607/55483
The black hole would need to be less massive than the moon to radiate more energy than it absorbs. Bigger than atomic size, as I suggested.

What reason is there to doubt the existence of the hypothesised planet Tyche in the far distant solar system?

Please note, I do not necessarily believe in its existence, just after a scientific (hence, non-Wikipedia) reasoning to doubt the planet's existence, other than the "we would have seen it" argument. I am, nor are the articles below, referring to the alleged 'Nemesis' planet, as in the question What are the current observational constraints on the existence of Nemesis?

According to the article Astronomers Doubt Giant Planet 'Tyche' Exists in Our Solar System (Wolchover, 2011), the planet Tyche is a hypothesised massive planet

by John Matese and Daniel Whitmire of the University of Lousiana-Lafayette, is not new: They have been making a case for Tyche since 1999, suggesting that the giant planet's presence in a far-flung region of the solar system called the Oort cloud would explain the unusual orbital paths of some comets that originate there.

Further information is found in the scientists' research paper Persistent Evidence of a Jovian Mass Solar Companion in the Oort Cloud (Matese and Whitmire, 2010), describe the hypothesised planet to be

of mass 1 − 4 M(Jupiter) orbiting in the
innermost region of the outer Oort cloud. Our most restrictive prediction is that the orientation
angles of the orbit normal in galactic coordinates are centered on
, the galactic longitude of
the ascending node = 319 and i, the galactic inclination = 103 (or the opposite direction)
with an uncertainty in the normal direction subtending 2% of the sky.

They posit that the elongated orbit of Sedna is a result of the presence of this planet.

So, the question, what reason is there to doubt the existence of Tyche?

How do people measure the distance between the earth and the moon?

There are many ways and I'm not entirely sure who you mean with "how do people measure the distance" (does this exclude space observatories like e.g. Clementine, probes currently in lunar orbit like e.g. Lunar Atmosphere and Dust Environment Explorer a.k.a. LADEE, or any other currently available technology?), but one interesting and extremely precise way is by laser ranging, pointing them towards one or more of the retroreflectors that were left on the surface of the Moon by the Apollo landers, in what is known as the Lunar Laser Ranging Experiment. Since they are laid on the Moon's near side that is always pointed towards the Earth, these retroreflectors are available for measurements to any properly equipped researcher, no matter which country they're from.

By measuring the time it takes for the light to reflect back, we can infer distance by simply multiplying the c (speed of light in vacuum) with the time taken between the light signal being transmitted and received and then dividing all of it by two to get a single leg of the distance. I'll let you read the linked to Wikipedia page on LLRE for more information, but if you find that too boring, here's a YouTube video on Mythbusters: Moon Hoax Retroreflectors that explains it from the practical perspective.

As for results, again quoting that same Wikipedia page on LLRE:

Some of the findings of this long-term experiment are:

• The Moon is
  spiraling away from Earth at a rate of 3.8 cm (about 1.5 inches) per year. This rate
  has been described as anomalously high.


• The Moon probably has a
  liquid core of about 20% of the Moon's radius.


• The universal force
  of gravity is very stable. The experiments have put an upper limit on
  the change in Newton's gravitational constant G of less than 1 part in
  1011 since 1969. The likelihood of any "Nordtvedt effect" (a
  composition-dependent differential acceleration of the Moon and Earth
  towards the Sun) has been ruled out to high precision, strongly
  supporting the validity of the Strong Equivalence Principle.


• Einstein's theory of gravity (the general theory of relativity)
  predicts the Moon's orbit to within the accuracy of the laser ranging
  measurements.

So these distance measurements are so precise they're a lot more interesting and reliable than merely telling us how far away the Moon is. For one, it also blows any Moon landing hoax theories right out of the water. Links to many more official results can be found in references and external links sections of Wiki on LLRE.

There is also a plethora of other ways to measure the distance to the Moon, with different precision and equipment requirements, but I'll let others add some other methods in their answers. If however you had some specific "people measuring the distance" equipped with specific measuring devices in mind, don't forget to mention that in your question. ;)

What is the cause of the variation from high and low mean obliquity periods of Mars?

It is reasonably well known that Mars has a greater obliquity range than Earth, due to Mars lacking a stabilising influence of a large moon. However, the Martian obliquity seems to have gone through periods of high and low mean obliquity, as shown in the top chart below:

Image source: Mars Climate Modeling Group

What is the current accepted theory as to why there are periodic high and low mean obliquities on Mars?

(Please note, I am not asking about orbital eccentricity)

gas giants - Can small gas planets exist?

This may help you

KOI-314c is small even when compared to Uranus or Neptune.

Also, look for "Mini-Neptune" on Wikipedia.

So, it seems those "dwarf-giant" planets are possible.

Although I do not have the required knowledge, I wonder if it is possible to exist a planet smaller than the earth that still could be considered a gas planet.

For example, just think of something mars-sized or between mars and the earth in size and mass (by size I mean not only the "solid" part, but including the atmosphere outer layers, just as one would see at the eyepiece - Venus, Titan). Now, think of it as a heftier version of Titan with a much extended and denser atmosphere... could that be called a Gas planet? Besides, nature don't have to fit exactly to our classifications/expectations.

I know Hydrogen and Helium are very volatile and have a small atomic mass, but I guess in a very cold environment, for example few AUs around a red/brown dwarf, a small planet with a small mass (that means low escape velocity) could retain all of its original atmosphere for aeons.

I dare say that even the most creative sci-fi authors coudn't predict the variety of planets that are being discovered nowadays!

What is the difference between a dwarf spheroidal galaxy and a globular cluster?

The main difference is if it is gravitationally bound to another galaxy or not.

A globullar cluster is a group of old stars inside a galaxy. It is not independent.

A galaxy is a group of stars, and a dwarf sheroidal galaxy is a group of (mainly) old stars, gravitationally bound in itself, but not bound inside a larger body (even when it can be orbiting a local group center of mass).

If Segue 1 is orbiting around Milky Way's center, it is a globullar cluster. If it is orbiting with Milky way about Local Group's center near Andromeda, then it is a dwarf spheroidal galaxy.

Why can't light escape from a black hole?

I like to think of this in terms of escape velocity.

Escape velocity is the speed needed to escape the gravitational pull of a given object. For the Earth, that speed is 11.2 km/second (Mach 34!). When rockets blast off from Earth, they aren't trying to achieve a certain height or altitude, they're trying to reach a certain speed, the escape velocity.

Once a rocket reaches 11.2 kips*, it has attained the speed needed to leave the Earth completely. If a rocket fails to attain that speed, regardless of its height, it will fall back to the Earth. (You can imagine a magical balloon that slowly lifts you up into space, up past the ISS and most satellites, and then you let go: since you aren't going fast enough, you will fall back down, past all the satellites, and crash into the earth.)

Smaller gravitational bodies, like the moon, have smaller escape velocities. That's why the lunar landers were able to leave the moon with such a small ascent stage, compared to the massive Saturn V it took to leave Earth: they only had to go 2.4 km/second.

To escape the Sun, you'd have to go 617.5 km/second!

Fortunately for us, light goes faster than 617.5 kips, so we're able to see the light created on the Sun. However, as you increase the mass of an object, eventually the escape velocity would meet or exceed 299,792km/s, the speed of light. At that point not even light itself can go fast enough to escape the gravity well, and will always be pulled back down into the black hole.

*Short for "ki​lometers p​er s​econd"

solar system - What was the length of year 1 million years back?

(Disclaimer: As I already pointed out in a comment to the question above, I never did a calculation with $H_0$ before and I might be utterly, horrible wrong with my interpretation.)

If you completely ignore the slowly changing orbit of earth and only take expansion of space into account and assume the Hubble-parameter to be pretty constant in the timeframe of 1 My, we can calculate the difference of the orbital period of earth using Keppler's third law [3]:

$T = 2pisqrt(a^3/GM)$

for

$a = 1.4959789*10^{11} m$ (semi-major axis of earth today) [1]
$G = 6.67*10^{-11} Nm^2/kg^2$ (gravitational constant)
$M = 1.988435*10^{30} kg$ (mass sun) [1]

We also assume: $H_0 = 2.3*10^{-18} s^-1$ [2] (Hubble parameter then and today in SI-units) which basically means "in every second a meter get $2.3*10^{-18} m$ longer".

Instead of taking the length of an (siderial) orbital period of earth from some source, let's calculate it manually first and take it as a reference.

$T_{today} = 2 pi sqrt((1.4959789*10^{11}m)^3/(6.67*10^{-11} Nm^2/kg^2 * 1.988435*10^{30} kg))$ = 365 days 8 hours 56 minutes 13.45 seconds

Pretty close and a good reference for more calculations.

Now, what was earth's semi-major axis 1 million years ago, only taking into account a constant $H_0$?

$x - (2.3*10^{-18} s^-1 * 1 My * x) = 1.4959789*10^{11} m$
Solving for $x$ leads to $x = 1.49598*10^{11} m$.

(Sorry for the lousy precision; I only have Wolfram Alpha at my hands right now.)

The old semi-major axis is a little smaller. Using Keppler's law again we can calculate the orbital period again:

$T_{old} = 2 pi sqrt((1.496*10^{11} m)^3/(6.67*10^{-11} Nm^2/kg^2 * 1.988435*10^{30} kg))$ = 365 days 8 hours 56 minutes 48.26 seconds

So, subtracting both times from another we can say that 1 My ago the year was indeed 34.81 seconds shorter.

However. This probably doesn't mean much; the orbit changes slightly over time anyway; the Hubble-parameter is not considered a constant any more, it changes slightly over time; and while this was an interesting question I don't trust my interpretation much and hope that someone else who's more qualified than me could enlighten the question better than I ever could.

(I hope I didn't botch anything somewhere. I need more coffee.)

[1] Source: Wolfram Alpha
[2] Source for Hubble-parameter in SI-units taken from the German Wikipedia: http://de.wikipedia.org/wiki/Hubble-Konstante#Definition
[3] http://en.wikipedia.org/wiki/Orbital_period#Small_body_orbiting_a_central_body

light - Assumption of our universe being the surface of a 4-D sphere in order to describe relative uniformity in the cosmic microwave background?

To a high degree of accuracy, the cosmic background radiation we observe is homogeneous and isotropic... except for the CMBR dipole anisotropy, which is there because we are moving at $369pm 0.9;mathrm{km/s}$ relative to the CMB rest frame, in which this anisotropy would vanish.

Relevantly, the Friedmann-Robertson-Walker family of solutions of general relativity used to model the Big Bang cosmology assume that the universe is, on the large scale, homogeneous and isotropic.

It seems as if this is necessary to assume in order for the CMB to make an sense. ... Our position in space would dictate what we would observe in terms of background radiation.

That's not the case. There are four distinct homogeneous and isotropic spatial geometries: the Euclidean space $mathbf{E}^3$, the hyperbolic space $mathbf{H}^3$, the real projective space $mathbf{RP}^3$, and the sphere $S^3$. In any of those cases, the universe would look the same all around us and be independent of our position. Thus, the sphere is possible but not necessary. The first two of those cases correspond to the "flat" and "open" Big Bang cosmologies, respectively, and the last two are variations of the "closed" cosmology.

And if we don't require the universe to be completely homogeneous and isotropic, but merely look like it in the part of it we observe, then much more exotic geometries are also possible. With cosmological inflation, virtually any large-scale geometry can be "blown up" to look flat, homogenous, and isotropic within our observable portion--even though it might not be s beyond the our cosmological horizon.

Otherwise, an instantaneous infinite Euclidean space model would work.

It does work, in the sense of being consistent with observations. In fact, in the standard ΛCDM cosmological model, space is infinite and Euclidean.