Notes on Constellations, Andromeda, and Light Travel Time

Four bright stars form the Great Square of Pegasus.
If you follow the top two stars to the left, you reach a few bright stars in the constellation Andromeda.
The right side of the W-shaped Cassiopeia points to the Andromeda stars.
These stars help locate the faint fuzzy light of the Andromeda Galaxy (M31).
This sequence provides a practical star-hopping method to find M31 in the night sky.

The Andromeda Galaxy is located at a distance of about $d_{M31} \,\approx\, 2.5 \times 10^{6} \ \text{ly}$ (2.5 million light years).
Light entering our eyes from M31 left its origin about $t_{M31} \,\approx\, 2.5 \times 10^{6} \ \text{yr}$ ago. In other words, we see Andromeda as it was 2.5 million years in the past.
Through a telescope, the Andromeda Galaxy is revealed as a vast spiral-shaped collection of more than $N_{\star} \,\approx\, 1.0 \times 10^{11}$ stars (i.e., about 100,000,000,000 stars).
The term for the “look-back” effect is that we are seeing an object as it appeared in the past due to the finite speed of light.

The disk of the Milky Way galaxy has a diameter of about $D_{MW} \,\approx\, 1.0 \times 10^{5} \ \text{ly}$ (100,000 light years).
The far side of the Milky Way is located roughly $\Delta D \,=\, 1.0 \times 10^{5} \ \text{ly}$ farther away from us than the near side.
Question: How does the light travel time from the far side compare to the near side?
- Answer: The travel time from the far side is greater.
- Reason: Since the far side is $\Delta D = 1.0 \times 10^{5} \ \text{ly}$ farther away, the light from the far side that reaches us today must have left about \Delta t \,=\, 1.0 \× 10^{5} \ \text{yr} before the light that currently reaches us from the near side.
Consequence: The photo of our galaxy not only contains light that has traveled for about $t_{MW} \,=\, 2.5 \times 10^{6} \ \text{yr}$ (in the sense of the Milky Way’s distance scale to us for the Andromeda image) but also includes light that left at different times over a period of roughly $\Delta t \,=\, 1.0 \times 10^{5} \ \text{yr}$ due to depth along the line of sight within the disk.

This example demonstrates that when we study the universe, space and time become intertwined.
Key idea: look-back time connects spatial distances with temporal history because light takes time to travel across space.
For distant objects, we are effectively looking back in time: we see the universe as it was when the light began its journey.

Look-back time is directly related to distance in light-years: for any object at distance $D$ , the light-travel time is approximately $t \approx D$ years (in years and light-years units).
Observing nearby extended structures (like the Milky Way’s disk) reveals a spread of arrival times corresponding to internal depth, not just the time offset to Earth.
For distant galaxies, the look-back time becomes a tool to study the history of the universe, star formation rates, galaxy evolution, and cosmology.
Practical implication: when interpreting images, astronomers must account for the fact that different parts of an object can reflect light emitted at different times, effectively showing a time-lrupted snapshot.

Andromeda Galaxy distance: $d_{M31} \approx 2.5 \times 10^{6} \ \text{ly}$
Andromeda look-back time: $t_{M31} \approx 2.5 \times 10^{6} \ \text{yr}$
Andromeda star count: $N_{\star} \approx 1.0 \times 10^{11}$
Milky Way disk diameter: $D_{MW} \approx 1.0 \times 10^{5} \ \text{ly}$
Far side distance offset: $\Delta D = 1.0 \times 10^{5} \ \text{ly}$
Far side light-travel time difference: $\Delta t = 1.0 \times 10^{5} \ \text{yr}$
Milky Way look-back region span in a single image: $\Delta t \,\approx\, 1.0 \times 10^{5} \ \text{yr}$
Conceptual statement: space and time become intertwined when studying the universe.