Okay, so I need to figure out this astronomy problem. Let's start by reading the question carefully.

The question is about a star with certain properties, and how the ratio of neutral titanium atoms in two energy levels changes when the star has spots versus when it doesn't. The options are A to D, with numerical factors. The key here is understanding how the effective temperature affects the ratio of these atoms in the photosphere, especially considering the presence of spots.

First, let me parse the given information. The star has a radius of 1.5 solar radii and a mass of 1.1 solar masses. Without spots, its effective temperature (Teff) is 6000 K. When 40% of the surface is covered by spots, the overall Teff drops to 5500 K. The problem states that in the photosphere, the ratio of Ti atoms in two energy levels (level 1 and level 2) decreases when the star has spots. We need to find the factor by which this ratio changes between the two cases (no spots vs. spots). The transition between the levels corresponds to a wavelength of about 1448 Å, which suggests it's a specific transition line, maybe in the near-ultraviolet or X-ray? Wait, 1448 Å is 14.48 nm, which is in the extreme ultraviolet. But wait, 1448 Å is 0.1448 meters, wait no, 1 Å is 1e-10 meters, so 1448 Å is 1.448e-7 meters. Wait, no, 1 meter is 1e10 Å, so 1448 Å is 1448e-10 meters. Wait, but maybe I should convert this to wavelengths in terms of the star's spectrum. But perhaps more importantly, the transition's wavelength relates to the energy difference between the two levels via the relation E = hc/λ. Since the star's temperature affects the excitation of the atoms, this affects the ratio of the number of atoms in the two levels.

The question says that the ratio decreases when the star has spots. So when the star has spots, the ratio (level 1/level 2) is lower than when there are no spots. So the ratio is lower with spots, meaning that with spots, there are fewer atoms in level 1 relative to level 2 compared to when there's no spots. Wait, no, the ratio is N1/N2. If the ratio decreases, that means N1/N2 is smaller when spots are present. So N1/N2 (with spots) is less than N1/N2 (without spots). So the ratio is lower.

But why does the presence of spots affect the temperature? Because spots are cooler regions on the star's surface. When the star has spots, part of its surface is cooler, so the overall effective temperature decreases. The effective temperature is the temperature of a blackbody that would emit the same total radiation as the star. So when 40% of the surface is cooler (spots), the average Teff drops from 6000 K to 5500 K. The Stefan-Boltzmann law says that the total luminosity is proportional to R² T⁴. But here, since spots are cooler, the average T is lower.

But the question is about the ratio of the number of atoms in two energy levels in the photosphere. This is a problem that relates to the Boltzmann equation, which describes the ratio of populations in different energy levels under thermal equilibrium. The Boltzmann factor is given by:

N1/N2 = (g1/g2) * exp(-(E2 - E1)/(k T))

Where g1 and g2 are the degeneracies of the two levels, E2 > E1, so E2 - E1 is positive. k is Boltzmann's constant, and T is the temperature.

But the problem states that the photosphere is in Local Thermal Equilibrium (LTE), so this equation should hold. The problem is that the star's overall Teff is 5500 K when spots are present, but wait, the question is about the ratio in the photosphere. Wait, the star's Teff is the effective temperature, which is an average over the entire surface. However, the photosphere's temperature might not be uniform. But when the star has spots, those spots are cooler (since they are dark spots, which are typically cooler than the surrounding photosphere), and the surrounding areas might be hotter. But the problem states that when 40% of the surface is covered by spots, the overall photospheric effective temperature decreases to 5500 K. Wait, but the photosphere's temperature is the same as the star's surface temperature in LTE. Wait, perhaps the problem is simplifying and considering that the entire photosphere's temperature is the same as the effective temperature. But that might not be the case. However, the problem states that in the stellar photosphere, the ratio of the number of atoms changes. So maybe the photospheric temperature is the same as the effective temperature of the star. Wait, but the effective temperature is an average over the entire surface. If the star has areas with different temperatures, how does that affect the photosphere's temperature?

Wait, perhaps the problem is assuming that the entire photosphere has a temperature equal to the effective temperature. Because when you have spots, the average Teff is lower, but the problem states that the photosphere's Teff is 5500 K when spots are present. Wait, but the effective temperature is not the same as the photospheric temperature. The effective temperature is the temperature of a blackbody that would emit the same total luminosity as the star. The photospheric temperature is the actual temperature of the photosphere, which is a bit deeper than the photosphere's surface. But in the problem, it's stated that when 40% of the surface is covered by spots, the overall photospheric effective temperature decreases to 5500 K. So perhaps the problem is using the term "photospheric effective temperature" to mean the average temperature of the photosphere, which is affected by the spots. Alternatively, it's possible that the problem is simplifying and considering that the entire photosphere has a temperature equal to the effective temperature. But that's not correct because the effective temperature is an average.

Alternatively, perhaps the problem is considering that the presence of spots changes the average temperature of the photosphere, but the problem states that the transition corresponds to a wavelength of 1448 Å, which is a specific energy difference. So the key here is that the ratio of populations depends on the temperature. So the ratio N1/N2 is proportional to exp(-(E2-E1)/(k T)). If the temperature decreases, then the exponent becomes less negative, so the ratio increases. Wait, but the problem says that the ratio decreases when the star has spots. Wait, that seems contradictory. Let's think again.

Wait, let's clarify. The ratio N1/N2 is given by (g1/g2) exp(-(E2 - E1)/(k T)). Let's assume E2 > E1, so E2 - E1 is positive. Then, when T increases, the exponent becomes more negative, so exp(-(E2 - E1)/(k T)) becomes smaller. Therefore, N1/N2 decreases as T increases. Conversely, if T decreases, the exponent becomes less negative (since T is in the denominator), so exp(...) becomes larger, leading to a larger N1/N2.

But the problem states that when the star has spots (lower Teff), the ratio N1/N2 decreases. That matches with the formula. So when the star has spots, the effective temperature is lower (5500 K), so T is lower. Therefore, the exponent -(E2-E1)/(k T) becomes less negative (because T is smaller), so exp(...) is larger. Therefore, N1/N2 would be larger. Wait, but the problem says that the ratio decreases when the star has spots. That contradicts the formula. Wait, perhaps I made a mistake here.

Wait, let's double-check. The formula is N1/N2 = (g1/g2) exp(-(E2 - E1)/(k T)). If T decreases, then (E2 - E1)/(k T) increases, so the exponent becomes more positive. Wait, no. Let's consider T in the denominator. So if T decreases, (E2 - E1)/(k T) becomes larger. So the exponent -(E2 - E1)/(k T) becomes more negative. Therefore, exp(-(E2 - E1)/(k T)) becomes smaller. Therefore, N1/N2 decreases when T decreases. But according to the problem, when the star has spots (lower T), the ratio N1/N2 decreases. Wait, that matches. So when T decreases, N1/N2 decreases. So the problem states that when the star has spots (lower T), the ratio (N1/N2) decreases. Which is correct according to the formula.

Wait, but the problem says that "the ratio of the number of neutral atoms of Ti in two energetic levels (level 1 and level 2) decreases when the star has spots." So N1/N2 is lower when the star has spots. Which would happen if T decreases. So the formula holds. So the ratio is N1/N2 = (g1/g2) exp(-(E2 - E1)/(k T)). So when T decreases, the exponent becomes more negative, so N1/N2 decreases.

Therefore, the factor by which the ratio changes is (N1/N2 without spots) / (N1/N2 with spots) = [exp(-(E2 - E1)/(k T1))] / [exp(-(E2 - E1)/(k T2))] = exp( (E2 - E1)/(k) (1/T2 - 1/T1) )

Because T1 is 6000 K (no spots), T2 is 5500 K (spots). So the factor is exp( (E2 - E1)/(k) (1/5500 - 1/6000) )

But E2 - E1 is related to the wavelength of the transition. The energy difference between the two levels is given by E = hc/λ. So E2 - E1 = hc/λ. So the factor becomes exp( (hc/λ)/(k) (1/5500 - 1/6000) )

But we can compute this factor. Let's compute the exponent first.

Compute (1/5500 - 1/6000). Let's compute that in terms of 1/λ. Let's compute 1/5500 - 1/6000.

1/5500 = 1/5.5e3 = 0.000181818... per K

1/6000 = 0.000166666... per K

Difference: 0.0000151515... per K

Multiply by (hc/k). Let's compute hc/k.

h = Planck's constant = 6.626e-34 J·s

c = speed of light = 3e8 m/s

k = Boltzmann constant = 1.38e-23 J/K

So hc = 6.626e-34 * 3e8 = 1.9878e-25 J·m

hc/k = 1.9878e-25 / 1.38e-23 ≈ 0.01443 m·K ≈ 1.443e-2 m·K

So hc/k ≈ 1.443e-2 m·K

Multiply by the difference in inverse temperatures:

1.443e-2 m·K * 0.0000151515 per K = 1.443e-2 * 0.0000151515 m·K * per K

Wait, units here: m·K * per K is m/K. Wait, but actually, the exponent is (hc/λ)(1/T2 - 1/T1) / k. Wait, let me re-express the exponent correctly.

The exponent is (E2 - E1)/(k) (1/T2 - 1/T1) = (hc/λ)/(k) (1/T2 - 1/T1)

So we can compute this as (hc/λk)(1/T2 - 1/T1)

But let's compute each part step by step.

First, compute hc/k: as above, 1.443e-2 m·K

Then, compute (1/T2 - 1/T1) = (1/5500 - 1/6000) K^{-1}

Calculate that:

1/5500 = 0.000181818...

1/6000 = 0.000166666...

Difference: 0.0000151515... per K

So (hc/k) * (1/T2 - 1/T1) = 1.443e-2 m·K * 0.0000151515 per K = 1.443e-2 * 0.0000151515 m·K/K

Wait, the units here: m·K * (1/K) = m. So the exponent is in terms of meters. But energy is in terms of Joules, so we need to convert this exponent into energy units. Wait, no, the exponent in the exponential function must be dimensionless. Because in the Boltzmann factor, the exponent is (E)/(kT), which is dimensionless. So perhaps I made a miscalculation in the units.

Wait, let's re-express the exponent correctly.

(E2 - E1) = hc/λ

E2 - E1 = hc/λ

So the exponent is (E2 - E1)/(k T) = (hc/λ)/(k T) = (hc)/(λ k T)

Therefore, the factor is exp( (hc)/(λ k) (1/T2 - 1/T1) )

Wait, no. Wait, the factor is N1/N2 (without spots) / (N1/N2 with spots) = [exp(-(E2-E1)/(k T1))] / [exp(-(E2-E1)/(k T2))] = exp( (E2-E1)/(k) (1/T2 - 1/T1) )

But E2-E1 = hc/λ. So the exponent is (hc/λ)/(k) (1/T2 - 1/T1)

Which is (hc)/(k λ) (1/T2 - 1/T1)

So the exponent is (hc)/(k λ) * (1/T2 - 1/T1)

So let's compute this exponent.

Given that λ = 1448 Å = 1448e-10 m

h = 6.626e-34 J·s

c = 3e8 m/s

k = 1.38e-23 J/K

T1 = 6000 K

T2 = 5500 K

First, compute (1/T2 - 1/T1) = (1/5500 - 1/6000) K^{-1} = (6000 - 5500)/(5500*6000) * K^{-1} = 500/(33,000,000) K^{-1} = 5e-2 / 3.3e7 K^{-1} = approximately 1.515e-9 K^{-1}

Wait, let's compute it exactly:

1/5500 ≈ 0.000181818

1/6000 ≈ 0.000166667

Difference: 0.0000151515 per K

So (1/T2 - 1/T1) = 0.0000151515 per K

Now, compute (hc)/(k λ) * (1/T2 - 1/T1)

First, hc = 6.626e-34 * 3e8 = 1.9878e-25 J·m

Divide by k λ: 1.9878e-25 J·m / (1.38e-23 J/K * 1448e-10 m) )

Compute denominator: 1.38e-23 * 1448e-10 = 1.38 * 1448 * 1e-33 = 2008.64 * 1e-33 = 2.00864e-30 J·m

So hc/(k λ) = 1.9878e-25 / 2.00864e-30 ≈ 9900 J/(m·K·m) Wait, units? Wait, J is kg·m²/s². Let's check units:

hc has units J·m

k has units J/K

λ has units m

So hc/(k λ) has units (J·m) / (J/K * m) ) = (J·m) / (J·m/K) ) = K

So hc/(k λ) is in Kelvin.

So 1.9878e-25 / 2.00864e-30 ≈ 9900 K

Multiply by (1/T2 - 1/T1) which is 0.0000151515 K^{-1}

So exponent = 9900 K * 0.0000151515 K^{-1} ≈ 0.15 K * K^{-1} = 0.15

So the exponent is approximately 0.15. Therefore, the factor is exp(0.15) ≈ 1.1618.

Wait, but that's the factor when T decreases. Wait, but according to the problem, when the star has spots, the ratio N1/N2 decreases. So (N1/N2 without spots) / (N1/N2 with spots) = exp(0.15) ≈ 1.16. So the ratio of the ratio (without spots to with spots) is approximately 1.16. But the problem asks for the factor by which the ratio changes when the star does not have spots compared to when it has spots. So that's (without spots)/(with spots) = 1.16. But the options are ~1.1, ~2.9, etc. Wait, but 1.16 is close to option A (~1.1). However, perhaps there's a miscalculation here.

Wait, let's double-check the calculation. Let's compute hc/(k λ) * (1/T2 - 1/T1) again.

First, compute hc:

h = 6.62607015e-34 J·s (exact value)

c = 299792458 m/s (exact value)

λ = 1448 Å = 1448e-10 m

So hc = 6.62607015e-34 * 2.99792458e8 ≈ 1.98644586e-25 J·m

k = 1.380649e-23 J/K

So hc/k = 1.98644586e-25 / 1.380649e-23 ≈ 0.01443 m·K

Then, hc/k * (1/λ) = 0.01443 m·K / 1448e-10 m = 0.01443 / (1448e-10) * K = 0.01443 / 1.448e-7 K ≈ 99600 K

Wait, because 1448e-10 m is 1.448e-7 m. So 0.01443 m·K / 1.448e-7 m = 0.01443 / 1.448e-7 K = approximately 99600 K

Then, multiply by (1/T2 - 1/T1) = 0.0000151515 K^{-1}

So exponent = 99600 K * 0.0000151515 K^{-1} ≈ 1.509

So exp(1.509) ≈ exp(1.5) ≈ 4.48. But 1.509 is a bit more. Let's compute exp(1.509):

We know that ln(4.5) ≈ 1.504, and ln(4.5) ≈ 1.504. Wait, but exp(1.509) is slightly higher than 4.5. Let me compute it more accurately.

Compute 1.509:

We know that e^1 = 2.71828

e^1.5 ≈ 4.4817 (since e^1.6 ≈ 4.953, e^1.5 ≈ 4.4817)

So e^1.509 ≈ e^(1.5 + 0.009) ≈ e^1.5 * e^0.009 ≈ 4.4817 * 1.00905 ≈ 4.4817 * 1.009 ≈ 4.529

So approximately 4.53. But the options are A: ~1.1, B: ~7.8, C: ~4.5, D: ~2.9. So 4.53 is close to option C (~4.5). But according to the problem, when the star has spots, the ratio N1/N2 decreases. So the factor is (without spots)/(with spots) = exp(1.509) ≈ 4.53. Therefore, the correct answer should be approximately 4.5, which is option C.

Wait, but earlier I thought the exponent was 0.15, leading to ~1.16. But that was due to a miscalculation. Let's recheck.

Wait, when I computed hc/(k λ), I think I made a mistake in the units. Let's recompute that step carefully.

hc = 6.626e-34 J·s * 3e8 m/s = 1.9878e-25 J·m

k = 1.38e-23 J/K

λ = 1448e-10 m

So hc/(k λ) = (1.9878e-25 J·m) / (1.38e-23 J/K * 1448e-10 m)

Denominator: 1.38e-23 * 1448e-10 = 1.38 * 1448 * 1e-33 = 2008.64e-33 = 2.00864e-30 J·m

So hc/(k λ) = 1.9878e-25 / 2.00864e-30 ≈ 9900 K

Multiply by (1/T2 - 1/T1) = 0.0000151515 K^{-1}

So exponent = 9900 * 0.0000151515 ≈ 0.15 K * K^{-1} = 0.15

Wait, but that gives exponent 0.15, leading to exp(0.15) ≈ 1.16. But according to the earlier step, when I computed hc/(k λ) as 99600 K, and multiplied by 0.0000151515 gives 1.509, leading to exp(1.509) ≈ 4.5. But there's a discrepancy here. Which one is correct?

Wait, let me recompute hc/(k λ):

hc = 6.626e-34 * 3e8 = 1.9878e-25 J·m

k = 1.38e-23 J/K

λ = 1448e-10 m

So hc/(k λ) = (1.9878e-25) / (1.38e-23 * 1448e-10)

Compute denominator:

1.38e-23 * 1448e-10 = 1.38 * 1448 * 1e-33 = (let's compute 1.38 * 1448)

1.38 * 1000 = 1380

1.38 * 448 = 1.38 * 400 = 552; 1.38 * 48 = 66.24 → total 552 + 66.24 = 618.24

So total 1.38 * 1448 = 1380 + 618.24 = 2000 + 18.24 = 2018.24

Wait, 1.38 * 1448:

Wait, 1.38 * 1000 = 1380

1.38 * 400 = 552

1.38 * 48 = 66.24

So 1380 + 552 + 66.24 = 1380 + 618.24 = 1998.24

Wait, 1.38 * 1448 = 1.38*(1000 + 400 + 48) = 1380 + 552 + 66.24 = 1380 + 618.24 = 1998.24

So denominator is 1998.24e-33 J·m

So hc/(k λ) = 1.9878e-25 / 1998.24e-33 = (1.9878 / 1998.24) * 1e8 ≈ (0.000994) * 1e8 = 99400 K

So approximately 99400 K.

Multiply by (1/T2 - 1/T1) = 0.0000151515 K^{-1}

Exponent = 99400 * 0.0000151515 ≈ 1.506

So exp(1.506) ≈ 4.51

So the factor is approximately 4.5. Therefore, the answer should be option C (~4.5).

But wait, the problem states that the ratio decreases when the star has spots. So the ratio without spots divided by the ratio with spots is exp(1.506) ≈ 4.5, which is option C.

But let's confirm once again. The formula for the ratio of populations is:

N1/N2 = (g1/g2) exp( -(E2-E1)/(k T) )

If T decreases, then (E2-E1)/(k T) increases, so exp( -(E2-E1)/(k T) ) decreases. Therefore, N1/N2 decreases. So when T decreases (spots present), the ratio N1/N2 decreases. Therefore, the ratio without spots is higher than with spots. So the factor is (without spots)/(with spots) = exp( (E2-E1)/(k) (1/T2 - 1/T1) )

Which we calculated as approximately 4.5, so option C.

But wait, the options are:

A. ~1.1

B. ~7.8

C. ~4.5

D. ~2.9

So the correct answer should be C.
</think>

To solve the problem, we analyze the ratio of neutral titanium atoms in two energy levels (N1/N2) in the stellar photosphere using the Boltzmann equation:

\[
\frac{N_1}{N_2} = \frac{g_1}{g_2} \exp\left(-\frac{E_2 - E_1}{kT}\right),
\]

where \(E_2 - E_1 = \frac{hc}{\lambda}\) is the energy difference between the two levels, \(k\) is Boltzmann's constant, \(T\) is the temperature, and \(g_1/g_2\) is the ratio of degeneracies. The factor by which the ratio changes when the star transitions from no spots (Teff = 6000 K) to spots (Teff = 5500 K) is given by:

\[
\text{Factor} = \exp\left(\frac{hc}{k\lambda} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)\right),
\]

where \(T_1 = 6000 \, \text{K}\) (no spots) and \(T_2 = 5500 \, \text{K}\) (spots). The wavelength of the transition is \(\lambda = 1448 \, \text{Å} = 1448 \times 10^{-10} \, \text{m}\).

### Step 1: Compute \(\frac{1}{T_2} - \frac{1}{T_1}\)
\[
\frac{1}{T_2} - \frac{1}{T_1} = \frac{1}{5500} - \frac{1}{6000} = \frac{6000 - 5500}{5500 \times 6000} = \frac{500}{33,000,000} \approx 1.515 \times 10^{-5} \, \text{K}^{-1}.
\]

### Step 2: Compute \(\frac{hc}{k\lambda}\)
Using constants:
- \(h = 6.626 \times 10^{-34} \, \text{J·s}\),
- \(c = 3 \times 10^8 \, \text{m/s}\),
- \(k = 1.38 \times 10^{-23} \, \text{J/K}\),
- \(\lambda = 1448 \times 10^{-10} \, \text{m}\),

\[
\frac{hc}{k\lambda} = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{(1.38 \times 10^{-23})(1448 \times 10^{-10})}.
\]

First, compute the numerator:
\[
hc = (6.626 \times 10^{-34})(3 \times 10^8) = 1.9878 \times 10^{-25} \, \text{J·m}.
\]

Next, compute the denominator:
\[
k\lambda = (1.38 \times 10^{-23})(1448 \times 10^{-10}) = 2.00864 \times 10^{-30} \, \text{J·m}.
\]

Now divide:
\[
\frac{hc}{k\lambda} = \frac{1.9878 \times 10^{-25}}{2.00864 \times 10^{-30}} \approx 9900 \, \text{K}.
\]

### Step 3: Compute the exponent
\[
\text{Exponent} = \frac{hc}{k\lambda} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) = 9900 \times 1.515 \times 10^{-5} \approx 0.15.
\]

### Step 4: Compute the factor
\[
\text{Factor} = \exp(0.15) \approx 1.16.
\]

However, this result contradicts the expected increase in the ratio (since the ratio decreases when the temperature decreases). Rechecking the calculation, the correct exponent is:

\[
\text{Exponent} = \frac{hc}{k\lambda} \left(\frac{1}{T_2} - \frac{1}{T_1}\right) = 99400 \times 1.515 \times 10^{-5} \approx 1.509.
\]

Thus:
\[
\text{Factor} = \exp(1.509) \approx 4.5.
\]

### Conclusion
The factor by which the ratio of the number of neutral titanium atoms in the two energy levels decreases when the star has spots compared to when it does not is approximately **4.5**, which corresponds to option **C**.

ANSWER: C