Astronomical Distances: AU, Light-Year, and Light-Travel Time

Two principal astronomical distance units discussed:
- Astronomical Unit (AU) – average Earth–Sun separation.
- Light-year (ly) – distance light covers in one Julian year.
Understanding these units provides scale for solar-system and interstellar distances.

Formal definition: mean distance from the centre of the Earth to the centre of the Sun.
Exact numerical value given in the lecture: $1\,\text{AU}=1.495978706\times10^{11}\;\text{m}.$
- Rough mental value often rounded to $1.50\times10^{11}\;\text{m}$ or $150\;\text{million km}.$
Serves as a convenient baseline for describing orbits within the Solar System.
Visual cue offered: imagine the Sun at one focus and Earth on a circular orbit with radius $1\,\text{AU}.$

Definition: Distance travelled by light in one year.
Formula foundation:
$\text{Distance}=v\,t\quad\Longrightarrow\quad 1\,\text{ly}=c\times(1\,\text{year}).$
Speed of light (exact, by definition of the metre):
$c=2.99792458\times10^{8}\;\text{m\,s}^{-1}\approx3.00\times10^{8}\;\text{m\,s}^{-1}.$
Time conversion for 1 Julian year used in the video:
- $365.25\;\text{days}\times24\;\text{h\,day}^{-1}\times3600\;\text{s\,h}^{-1}.$
Multiplying out gives the canonical light-year distance: $1\,\text{ly}=9.46\times10^{15}\;\text{m}.$
- Lecturer’s phrasing: “9.46 times 10 to the 15 meters.”

Kilometre conversion (divide by $10^{3}$ ):
$1\,\text{ly}=9.46\times10^{12}\;\text{km}.$
Mile conversion (using $1\,\text{mile}=1.6\,\text{km}$ ): $\frac{9.46}{1.6}\times10^{12}\;\text{miles}\approx5.9\times10^{12}\;\text{miles}.$
- Lecturer’s verbal mnemonic: “about 5,900,000,000,000 miles – nearly six trillion.”
Illustrative thought experiment: travelling at light speed, you would traverse ≈ $9.5\;\text{trillion km}$ in a single year.

Proxima Centauri – closest star after the Sun: $4.22\;\text{ly}.$
- Distance in km: $4.22\times9.46\times10^{12}\;\text{km}\approx3.8\times10^{13}\;\text{km}$ (≈38 trillion km).
Alpha Centauri A/B – next nearest system: slightly greater than Proxima, "a little brighter than our Sun," on the order of $\sim38$ trillion km as well.
Pedagogical point: even the nearest stars are tens of trillions of kilometres away, underscoring cosmic scale.

To find how many AU fit inside one light-year: $\frac{1\,\text{ly}}{1\,\text{AU}}=\frac{9.46\times10^{12}\;\text{km}}{1.495978706\times10^{8}\;\text{km}}=6.3236\times10^{4}\;\text{AU}.$
- Rounded figure quoted: 63,236 AU per light-year.
Concept image from lecture: if Earth–Sun distance is one unit, the nearest star would sit ≈63,000 times farther.

Using $d=1\,\text{AU}$ and $v=c$ :
$t=\frac{d}{c}=\frac{1.495978706\times10^{11}\;\text{m}}{2.99792458\times10^{8}\;\text{m\,s}^{-1}}\approx499\;\text{s}.$
Convert to minutes:
$\frac{499\;\text{s}}{60\;\text{s\,min}^{-1}}\approx8.3\;\text{min}.$
Interpretations & implications:
- We see the Sun as it existed 8 min 20 s ago.
- Instantaneous events (e.g.
  hypothetical “Sun disappears now”) would impact Earth only after this delay.
- General principle: we never witness any astronomical object in its current state; greater distances → longer look-back times.

Human perception of the cosmos is inherently time-delayed; astronomy is a study of history written in light.
Even at nature’s speed limit, interstellar travel faces daunting scales – trillions of kilometres or tens of thousands of AU.
The exercise strengthens appreciation for measurement systems, unit conversions, and scientific notation in communicating vast numbers.

$1\,\text{AU}=1.496\times10^{11}\;\text{m}\;(150\,\text{million km}).$
$c=2.998\times10^{8}\;\text{m\,s}^{-1}.$
$1\,\text{year}=365.25\times24\times3600\;\text{s}.$
$1\,\text{ly}=9.46\times10^{15}\;\text{m}=9.46\times10^{12}\;\text{km}=5.9\times10^{12}\;\text{miles}.$
$1\,\text{ly}=63,236\;\text{AU}.$
Light Sun→Earth travel time: $\approx8.3\;\text{min}.$